diff --git "a/0tE0T4oBgHgl3EQfuAGv/content/tmp_files/load_file.txt" "b/0tE0T4oBgHgl3EQfuAGv/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/0tE0T4oBgHgl3EQfuAGv/content/tmp_files/load_file.txt" @@ -0,0 +1,697 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf,len=696 +page_content='1 The Real Number n-Degree Pythagorean Theorem Jeffrey S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Lee1,2,4 Gerald B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Cleaver1,3 1Early Universe Cosmology and Strings Group 2Space Physics Numerical Modeling Group Center for Astrophysics, Space Physics, and Engineering Research 3Department of Physics 4Department of Geosciences Baylor University One Bear Place Waco, TX 76706 Jeff_Lee@Baylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='edu Gerald_Cleaver@Baylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='edu Key Words: ∞-degree Pythagorean Theorem, Law of Cosines, maximum triangle area, minimum triangle area Word Count: 4203 Abstract This paper extends the Pythagorean Theorem to positive and negative real exponents to take the form n n n a b c + = and makes use of the definition 1 b a \uf067 = \uf0b3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For the case of n + \uf0ce , 1 n \uf0b3 is necessary for the vertex angle to be real, and there are no restrictions on γ beyond its definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, for n − \uf0ce , two significant restrictions that are necessary for n n n a b c + = to yield real vertex angles have been discovered by this work: 1 2 \uf067 \uf0a3 \uf03c , and n cannot exceed a critical value which is γ-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Additionally, the areas of the associated triangles have been determined as well as the conditions for those areas to be maxima or minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Introduction For centuries, the Pythagorean Theorem has been significant in the foundation of mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Although the finally proven Fermat’s Last Theorem [1] definitively establishes the non-existence of Pythagorean Triples for n n n a b c + = with | 2 n n \uf0ce \uf03e , the Pythagorean Theorem can be extended to higher degrees which are not required to be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, only positive integers possess the physical representation of dimension for n n n a b c + = (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', 1 n = defines a scalar sum of straight line segments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2 n = defines a scalar sum of areas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 3 n = defines a scalar sum of volumes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' and 4 n \uf0b3 defines a scalar sum of hypervolumes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 0 n = is mathematically meaningless because it results in 0 0 0 2 1 a b c + = → = ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2 The extension of the Pythagorean Theorem to higher dimensions using a variety of methods has been extensively addressed in the literature [2-15], most frequently by considering a formulation such as 2 2 Total 1 n k k a a = =\uf0e5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Rather, the n-degree Pythagorean Theorem contains a relationship between the ratio of the adjacent side lengths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', γ) and the vertex angle (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' n\uf0ce indicates that the triangles to which n n n a b c + = is being applied are not necessarily right angled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 1 \uf067 \uf0b3 is defined as a precondition with no loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' If 0 1 \uf067 \uf03c \uf0a3 , the 1 \uf067 \uf0b3 triangle has merely been rotated within its plane, and 0 \uf067 \uf03c implies the geometrically uninteresting imposition upon the triangle of non-physical side lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The vertex angle θ is a function of γ and n and is therefore denoted ( ) ,n \uf071 \uf067 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For 0 1 n \uf03c \uf03c , the vertex angle is complex (not addressed in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For 1 2 n \uf03c \uf03c , the triangle is obtuse with ( ) 90 , 180 n \uf071 \uf067 \uf03c \uf03c , and for 2 n \uf03e , the triangle is acute with ( ) , 90 n \uf071 \uf067 \uf03c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In each instance of | 1 n n + \uf0ce \uf0b3 , ( ) 0 , 180 n \uf071 \uf067 \uf0a3 \uf0a3 , and there are no restrictions on the ratio of the adjacent side lengths (other than 1 \uf067 \uf0b3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' When n − \uf0ce , the situation changes significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In order that ( ) ,n \uf071 \uf067 \uf0ce , the restriction 1 2 \uf067 \uf0a3 \uf03c must be imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, even with 1 2 \uf067 \uf0a3 \uf03c , there is one additional compulsory restriction for ( ) ,n \uf071 \uf067 \uf0ce : if 1 \uf067 \uf0b9 , the degree of the negative real exponent Pythagorean Theorem must not exceed a critical value which is dependent on γ ( ) ( ) ( ) crit crit i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', 1 , and thus, 1 n n n n \uf067 \uf067 \uf0a3 \uf0b9 \uf0b3 \uf0b9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' If ( ) crit 1 n n \uf067 \uf03e \uf0b9 , the vertex angle is complex (also not addressed in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For | 1 n n + \uf0ce \uf0b3 , the areas of the associated triangles, with fixed a and γ values, reach a maximum which occurs when the triangle is right isosceles ( ) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', 1 and 90 \uf067 \uf071 = = ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' the areas increase as n → \uf0a5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Conversely, if the perimeter of a triangle is fixed, the triangle area approaches a maximum value for increasing n and approaches 0 for decreasing γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For ( ) crit | if 1 n n n \uf067 \uf067 − \uf0ce \uf0a3 \uf0b9 , the areas of the associated triangles with a fixed perimeter are maximized for 2 \uf067 = and as n → −\uf0a5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For finite n, the maximum area occurs when ( ) ,n \uf071 \uf067 is a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The n-Degree Pythagorean Theorem with Positive Real Exponents For the general scalene triangle in Figure 1, the n-degree Pythagorean Theorem can be written as n n n a b c + = (1) which must also conform to the standard Law of Cosines, 2 2 2 2 cos c a b ab \uf071 = + − , (2) 3 with n + \uf0ce .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In this work, the extension of the Pythagorean exponents to numbers other than n + \uf0ce does not extend with it the Law of Cosines because unlike in [10, 12] because an n-dimensional simplex does not arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Here, the objective is the solution for the vertex angle θ such that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (1) conforms to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 1: A scalene triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Using 1 b a \uf067 = \uf0b3 , rewriting eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (1) as ( ) 2 2 n n n c a b = + , (3) and equating eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (2) and (3) yields ( ) 2 2 1 1 1 cos 2 n n \uf067 \uf067 \uf071 \uf067 + − \uf0e9 \uf0f9 + − + \uf0ea \uf0fa = \uf0ea \uf0fa \uf0ea \uf0fa \uf0eb \uf0fb , (4) where \uf071 + denotes that the vertex angle arises from a version of the n-degree Pythagorean Theorem in which n + \uf0ce .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' If 0 n = , as stated above, no triangle exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' there is no 0-degree (or 0-dimensional) Pythagorean Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' If 0 1 n \uf03c \uf03c , \uf071 is a complex angle for all side ratios γ and is not applicable here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, if 1 n \uf0b3 , \uf071 is always a real angle, and therefore, a corresponding real triangle does exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' As expected, for the 1 n = case, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) becomes ( ) ( ) 1 1 cos 1 180o n \uf071 + − = = − = which is sensible because the triangle has collapsed into a straight line which is independent of the side ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For the 2 n = case, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) θb c a b θa θ 4 becomes ( ) ( ) 1 2 cos 0 90o n \uf071 + − = = = , and the traditional Pythagorean Theorem with a 90o vertex angle, also independent of the side ratio, is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' It is important to note that the vertex angle exists only for combinations of n and γ such that the argument of the inverse cosine function in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) is not greater than 1 or less than -1, ( ) 2 2 1 1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', 1 2 n n \uf067 \uf067 \uf067 \uf0e6 \uf0f6 + − + \uf0e7 \uf0f7 \uf0a3 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, the stipulation that 1 \uf067 \uf0b3 ensures that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) will always result in real angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Therefore, there are no forbidden side ratios, and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) applies without restriction for 1 n \uf0b3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 The 1 ≤ n ≤ 2 Case If 1 2 n \uf0a3 \uf0a3 , the range of vertex angles is between 90o and 180o (as shown above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Plots of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) are shown in Figure 2 and Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 2: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio and the Pythagorean exponent for 1 ≤ n ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Vertex Angle (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') 2 180 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='9 170 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 160 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='7 150 Pythagorean Exponent 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 140 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 130 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 120 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='3 110 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 100 06 2 3 4 5 6 8 6 10 Side Ratio5 Figure 3: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio and the Pythagorean exponent for 1 ≤ n ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For any given value of n, there is a corresponding value of γ that results in a maximum vertex angle which is seen in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' That vertex angle and the side ratio that gives rise to it are found by equating the first derivative of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' ( )( ) 2 2 0 1 1 1 0 n n n n d d \uf071 \uf067 \uf067 \uf067 \uf067 + − = \uf0de − + − + = (5) The solution to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (5) for ( ) n \uf067 is not analytic, and the total number of complex solutions grows rapidly with increasing n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, it is clear that for all values of n, 1 \uf067 = is a solution – it is the only positive real solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Thus, the largest vertex angle occurs in an isosceles triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Therefore, by substituting 1 \uf067 = into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4), the maximum vertex angle can be found (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2 1 max cos 1 2 n n \uf071 + − − \uf0e6 \uf0f6 = − \uf0e7 \uf0f7 \uf0e8 \uf0f8 , (6) where max \uf071 + denotes the maximum value of \uf071 + for any degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The second derivative of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) is extremely unruly, and therefore, the confirmation that max \uf071 + is a maximum angle was performed numerically with Maple®.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' A plot of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (6) is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 6 Figure 4: Maximum vertex angle versus Pythagorean exponent for 1 ≤ n ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 The n ≥ 2 Case For 2 n \uf0b3 , plots of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) are shown in Figure 5 and Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In this case, the vertex angles do not exceed 90o, and there is a minimum vertex angle for which eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' That vertex angle and the side length that gives rise to it are also found by equating the first derivative of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) to zero (as was done above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Given that for all values of n, 1 \uf067 = is once again the only positive real solution to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (5), and by substituting 1 \uf067 = into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4), the minimum vertex angle can be found (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Thus, as was the case in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 for the largest vertex angle for 1 2 n \uf0a3 \uf0a3 , the smallest vertex angle for 2 n \uf03e occurs when the triangle is isosceles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2 1 min cos 1 2 n n \uf071 + − − \uf0e6 \uf0f6 = − \uf0e7 \uf0f7 \uf0e8 \uf0f8 , (7) where min \uf071 + denotes the minimum value of \uf071 + for any degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 180 170 (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') 160 Maximum Vertex Angle 150 140 130 120 110 100 90 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='9 2 Pythagorean Exponent7 Figure 5: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio and the Pythagorean exponent for n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 6: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio and the Pythagorean exponent for n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The side ratio is extended to the excluded regime of γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The confirmation that min \uf071 + is a minimum angle was also performed numerically with Maple®.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' A plot of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (7) is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Vertex Angle (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') 10 06 6 85 Pythagorean Exponent 80 6 75 70 4 65 3 2 09 1 2 3 4 5 7 8 6 10 Side Ratio8 Figure 7: Minimum vertex angle versus Pythagorean exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In the limit that the Pythagorean exponent becomes infinite, ( ) 2 2 1 1 1 lim lim cos 2 n n n n \uf067 \uf067 \uf071 \uf071 \uf067 − \uf0a5 →\uf0a5 →\uf0a5 \uf0ec \uf0fc \uf0e9 \uf0f9 + − + \uf0ef \uf0ef \uf0ea \uf0fa = = \uf0ed \uf0fd \uf0ea \uf0fa \uf0ef \uf0ef \uf0ea \uf0fa \uf0eb \uf0fb \uf0ee \uf0fe , (8) which must be evaluated as a piecewise function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='Case 1 ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='γ < 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='0 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='lim cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf071 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0a5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='→\uf0a5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0e9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0f9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='+ − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0e6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf03c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0e7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0e8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='Case 2 ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='( ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 1 1 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='lim cos ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='o ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf071 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0a5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='→\uf0a5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0e9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0f9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='+ − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='Case 3 ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf03e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf071 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf067 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0e9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0f9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='+ − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0e6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0f6 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ef ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ea ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0ee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='\uf0fe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='i This case is included for completeness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' even though it was specified that γ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 06 85 (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') Minimum Vertex Angle 80 75 70 65 60 2 3 4 5 6 7 6 10 11 12 13 14 15 Pythagorean Exponent9 Case 2, illustrated graphically by 100 n = in Figure 6, indicates that the infinite degree Pythagorean Theorem generates an equilateral triangle, and expectedly, it is the only case that does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For case 3 with an infinite side ratio, ( ) lim lim 1 90o n \uf067 \uf071 \uf067 \uf0a5 →\uf0a5 →\uf0a5 \uf0e9 \uf0f9 \uf03e = \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Thus, the infinite degree Pythagorean Theorem for a triangle with an infinite side ratio (a straight line) requires the same vertex angle as the standard ( ) 2 n = Pythagorean Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' A plot of the behavior of the three cases of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (8) is shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 8: Plot of the infinite degree Pythagorean Theorem which was produced with n = 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The n-Degree Pythagorean Theorem with Negative Real Exponents Section 2 can be adapted to examine real triangles that conform to the negative exponent n-degree Pythagorean Theorem, and a very different picture emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Once again, eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (2) and (3) can be equated to form ( ) 2 2 1 1 1 cos 2 n n \uf067 \uf067 \uf071 \uf067 − − \uf0e9 \uf0f9 + − + \uf0ea \uf0fa = \uf0ea \uf0fa \uf0ea \uf0fa \uf0eb \uf0fb , (9) where \uf071 − denotes that the vertex angle arises from a version of the n-degree Pythagorean Theorem in which n − \uf0ce .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Unlike the positive exponents case, there is an infinite set of ( ) ,n \uf067 for which the always positive argument of the inverse cosine function exceeds 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' If 1 \uf067 = , then eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9) yields ( ) 2 2 2 1 1 1 1 1 1 1 cos cos 1 2 2 n n \uf071 − \uf0e6 \uf0f6 − \uf0e7 \uf0f7 − − \uf0e8 \uf0f8 \uf0e9 \uf0f9 \uf0e6 \uf0f6 + − + \uf0ea \uf0fa = = − \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0ea \uf0fa \uf0e8 \uf0f8 \uf0ea \uf0fa \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (10) 10 2 1 1 2 1 n \uf0e6 \uf0f6 − \uf0e7 \uf0f7 \uf0e8 \uf0f8 − \uf03c for all n, and isosceles triangles are permitted for all degrees of the negative real exponent Pythagorean Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, as γ increases, ( ) 2 2 1 1 2 n n \uf067 \uf067 \uf067 + − + (the argument of the inverse cosine function in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9)) also increases and can exceed 1, and the vertex angle becomes complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Therefore, there exists a γ- dependent critical value of the Pythagorean degree, ( ) crit n \uf067 , which is the largest exponent for a given side ratio that will produce a real vertex angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' This occurs when ( ) crit crit 2 2 1 1 1 2 n n \uf067 \uf067 \uf067 + − + = , which can be simplified as ( ) crit crit 1 1 1 n n \uf067 \uf067 + = − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (11) Equation (11) must be solved numerically, and its solution, ( ) crit n \uf067 , is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 9: The Pythagorean critical degree as a function of the side ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Thus, if ( ) crit n n \uf067 \uf03e , then \uf071 is a complex angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' If, on the other hand, ( ) crit n n \uf067 \uf0a3 , \uf071 will be real, and a corresponding real triangle exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The upper limit of γ is not immediately apparent, but consider the argument of the inverse cosine function in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For \uf071 − to be a real angle, ( ) 2 2 1 1 1 2 n n \uf067 \uf067 \uf067 + − + \uf0a3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' As such, ( ) ( ) 2 2 1 1 n n \uf067 \uf067 − \uf0a3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, the term ( ) 2 1 1 n n \uf067 + \uf0b3 for all γ and all n including n → −\uf0a5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Side Ratio 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='9 2 0 1 2 Critical Degree 3 4 5 6 7 811 This requires that ( ) 2 min 1 1 \uf067 \uf0e9 \uf0f9 − \uf03c \uf0eb \uf0fb (for n \uf0b9 −\uf0a5), and therefore, 2 \uf067 \uf03c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Consequently, the n-degree Pythagorean Theorem with negative real exponents cannot be applied to real triangles with side ratios greater than or equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' As indicated by Figure 10 and Figure 11, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 2 \uf067 \uf03c \uf03c is required to produce a real vertex angle for ( ) crit n n \uf067 \uf0a3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, by definition 1 \uf067 \uf0b3 , and therefore, 1 2 \uf067 \uf0a3 \uf03c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Also different from the 1 n = and 2 n = cases is that the 1 n = − and 2 n = − cases produce non- constant but equal γ-dependent angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9): when 1 n = − or 2 n = − , ( ) 2 1 2 1 cos 2 2 1 \uf067 \uf067 \uf071 \uf067 \uf067 − − \uf0e9 \uf0f9 + = − \uf0ea \uf0fa + \uf0ea \uf0fa \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 The n ≤ ncrit(γ) Case For ( ) crit n n \uf067 \uf0a3 , plots of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9) are shown in Figure 10 and Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Additionally, a maximum value of the vertex angle exists for a given side ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' As in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1, that vertex angle and the side ratio that gives rise to it are found by equating the first derivative of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9) (instead of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4)) to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 10: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio and the Pythagorean exponent for n ≤ ncrit(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The missing section at the top and on the right side of the figure are due to n > ncrit(γ) for a given value of γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' the border of this section has the equation ncrit(γ) as shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The side ratio is extended to the excluded regime of γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Vertex Angle (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') 0 60 1 50 Pythagorean Exponent 40 30 20 8 10 9 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 2 Side Ratio12 Figure 11: The vertex angle of the n-degree Pythagorean Theorem as a function of the side ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The gaps between the plots and the horizontal axis are due to n > ncrit(γ) at those values of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The side ratio is extended to the excluded regime of γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' As was the case with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (5), the solution to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (12) for ( ) n \uf067 is not analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, it is clear that for all values of n, 1 \uf067 = is once again a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' By substituting 1 \uf067 = into eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9), the vertex angle can be found (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (13)) which expectedly has the same mathematical form as eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' ( )( ) 2 2 0 1 1 1 0 n n n n d d \uf071 \uf067 \uf067 \uf067 \uf067 − − = \uf0de − + − + = (12) 2 1 max cos 1 2 n n \uf071 − − − \uf0e6 \uf0f6 = − \uf0e7 \uf0f7 \uf0e8 \uf0f8 , (13) where max \uf071 − denotes the maximum value of \uf071 − for a given degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The second derivative of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9) is the same as the second derivative of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4), and therefore, the mathematical confirmation that max \uf071 − is a maximum angle was also performed numerically with Maple®.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' A plot of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (13) is shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 60 n=-1 n=-2 50 n=-3 n = -100 40 30 20 10 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 Side Ratio13 Figure 12: Maximum vertex angle versus Pythagorean exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In the limit that the Pythagorean exponent becomes infinitely negative, ( ) 2 2 1 1 1 lim lim cos 2 n n n n \uf067 \uf067 \uf071 \uf071 \uf067 − −\uf0a5 →−\uf0a5 →−\uf0a5 \uf0ec \uf0fc \uf0e9 \uf0f9 + − + \uf0ef \uf0ef \uf0ea \uf0fa = = \uf0ed \uf0fd \uf0ea \uf0fa \uf0ef \uf0ef \uf0ea \uf0fa \uf0eb \uf0fb \uf0ee \uf0fe , (14) which is like the case with positive exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Even with the imposed condition that 1 2 \uf067 \uf0a3 \uf03c , eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (14) must be evaluated as a piecewise function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Case 1 ( ) = γ 1 ( ) 1 60o \uf071 \uf067 −\uf0a5 = = Case 2 ( ) 1 < γ < 2 ( ) 1 1 2 cos 2 \uf067 \uf071 \uf067 − −\uf0a5 \uf0e6 \uf0f6 \uf03c \uf03c = \uf0e7 \uf0f7 \uf0e8 \uf0f8 Case 3 ( ) − → γ 2 ( ) 2 0 \uf071 \uf067 − −\uf0a5 → = 60 50 (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=') Maximum Vertex Angle 40 30 20 10 0 15 12 6- 6 3 0 Pythagorean Exponent14 Case 1 produces the same result as did case 2 for positive exponents ( ) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', 1 \uf067 = – a maximum vertex angle of 60o and thus, an equilateral triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Also, like the positive exponents case, the equilateral triangle is never actually realized because although γ can equal 1, n cannot be infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' As illustrated in Figure 13, the negative infinite degree Pythagorean Theorem can produce real triangles, and the requirement that 1 2 \uf067 \uf0a3 \uf03c is retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Furthermore, a straight line will result when ( ) ( ) ( ) crit crit 2 2 1 1 1 lim lim cos 0 2 n n n n n n \uf067 \uf067 \uf067 \uf067 \uf071 \uf067 − → → \uf0ec \uf0fc \uf0e9 \uf0f9 + − + \uf0ef \uf0ef \uf0ea \uf0fa = = \uf0ed \uf0fd \uf0ea \uf0fa \uf0ef \uf0ef \uf0ea \uf0fa \uf0eb \uf0fb \uf0ee \uf0fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 13: Plot of the negative infinite degree Pythagorean Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The Areas of the Associated Triangles The area of the scalene triangle in Figure 1, with a real vertex angle, is 1 sin 2 A ab \uf071 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Its side lengths are a, b a \uf067 = , and from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (1), ( ) 1 1 n n c a \uf067 = + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Therefore, its area is 2 1 sin 2 A a \uf067 \uf071 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (15) Applying 2 2 cos 1 sin \uf071 \uf071 = − and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4) or eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (9) depending on whether n + \uf0ce or n − \uf0ce , respectively, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (15) can be written in terms of n and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 15 ( ) 2 2 2 2 2 4 1 1 2 n n a A \uf067 \uf067 \uf067 \uf0e9 \uf0f9 \uf0e6 \uf0f6 = − + − + \uf0e7 \uf0f7 \uf0ea \uf0fa \uf0e8 \uf0f8 \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (16) The restrictions on γ remain: 1 \uf067 \uf0b3 for n + \uf0ce , and 1 2 \uf067 \uf0a3 \uf03c for n − \uf0ce .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, there are no restrictions on a (other than 0 a \uf03e ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 The Maximum Areas of Triangles for the n-Degree Pythagorean Theorem with Positive Real Exponents A plot of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (16) is shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' It is clear as if a is constant and γ increases, the area of the associated triangle increases without bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Also, if a is constant, increasing n increases the area, but this is a negligible effect compared to increasing γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, the effect that n has on the area increases as γ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 14: Triangle area (in dimensionless units) with unit side length (a = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In the left-side figure, the effect of n on the area is difficult to distinguish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, in the right-side figure, the effect of n on the area is clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Determining the conditions that maximize the area of a triangle can be done with an “angular” approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (15), the maximum area of a triangle with given values of a and γ clearly occurs when 90 \uf071 = (corresponding to 2 n = ) and is 2 1 2 A a \uf067 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In summary, If γ and n are fixed, increasing a will increase the triangle’s area without bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' If a and n are fixed, increasing γ will increase the triangle’s area without bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Therefore, no absolute maximum area triangle exists because the triangle will experience infinite dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The only minimum area is the trivial solution ( ) 0 A = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Most significant from this analysis is that a right triangle, with fixed values of a and γ, has the maximum area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Triangle Area Triangle Area 10 5 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='9 8 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 Pythagorean Exponent 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 Pythag orean Exponent 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='7 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 3 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 0 2 3 4 5 6 7 8 9 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='9 Side Ratio Side Ratio16 The above result can, of course, be determined from the dependence of \uf071 + on n and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (4), ( ) 2 2 1 1 1 cos 90 2 n n \uf067 \uf067 \uf071 \uf067 + − \uf0e9 \uf0f9 + − + \uf0ea \uf0fa = = \uf0ea \uf0fa \uf0ea \uf0fa \uf0eb \uf0fb which gives ( ) 2 2 1 1 n n \uf067 \uf067 + = + (17) because ( ) 2 2 1 1 0 2 n n \uf067 \uf067 \uf067 + − + = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The only real solution to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (17) for n is 2 n = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' An alternative and more rigorous approach to finding the degree which gives rise to the maximum area case for a given value of γ is from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (16);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 0 A n \uf0b6 = \uf0b6 gives ( ) ( ) ( ) 2 2 2 1 1 1 1 ln ln 1 0 1 n n n n n n n n \uf067 \uf067 \uf067 \uf067 \uf067 \uf067 \uf067 \uf0e9 \uf0f9 \uf0e9 \uf0f9 \uf0e6 \uf0f6 + + − + − + = \uf0ea \uf0fa \uf0e7 \uf0f7 \uf0ea \uf0fa + \uf0e8 \uf0f8 \uf0eb \uf0fb \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (18) But ( ) 2 1 0 n n \uf067 + \uf0b9 for any value of γ or n, and therefore, ( ) ( ) 2 2 1 1 1 ln ln 1 0 1 n n n n n n \uf067 \uf067 \uf067 \uf067 \uf067 \uf067 \uf0e9 \uf0f9 \uf0e9 \uf0f9 \uf0e6 \uf0f6 + − + − + = \uf0ea \uf0fa \uf0e7 \uf0f7 \uf0ea \uf0fa + \uf0e8 \uf0f8 \uf0eb \uf0fb \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (19) ( ) 2 2 1 1 0 n n \uf067 \uf067 \uf0e9 \uf0f9 + − + = \uf0ea \uf0fa \uf0eb \uf0fb iff 2 n = (as shown with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' ( ) 1 ln ln 1 0 1 n n n n \uf067 \uf067 \uf067 \uf067 \uf0e9 \uf0f9 \uf0e6 \uf0f6 − + \uf0b9 \uf0ea \uf0fa \uf0e7 \uf0f7 + \uf0e8 \uf0f8 \uf0eb \uf0fb , regardless of the value of γ, although it asymptotically approaches zero for increasing n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' As expected, this somewhat more complex approach gives the same result as the “angular approach” used above – the maximum area triangle for positive real exponents occurs when 2 n = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Thus, the n-degree Pythagorean Theorem that produces triangles with the largest area for a given a and γ is the standard Pythagorean Theorem, and the triangles it produces are right angled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Even though the triangles that result from 1 2 n \uf0a3 \uf03c have vertex angles which are larger than 90o, they have smaller areas than 2 1 2 A a \uf067 = because sin 1 \uf071 \uf03c for obtuse \uf071 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 17 For a given a and a given n (not necessarily 2 n = ), the value of γ that produces the triangle with the maximum area would be calculated from the derivative of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (16) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', 0 A \uf067 \uf0e6 \uf0f6 \uf0b6 = \uf0e7 \uf0f7 \uf0b6 \uf0e8 \uf0f8 which yields ( ) ( ) 2 2 1 2 2 1 1 1 1 4 n n n n n \uf067 \uf067 \uf067 \uf067 \uf0e6 \uf0f6 − \uf0e7 \uf0f7 − \uf0e8 \uf0f8 \uf0e9 \uf0f9 \uf0e9 \uf0f9 + − + − + = \uf0ea \uf0fa \uf0ea \uf0fa \uf0eb \uf0fb \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (20) However, equation (20) has no solution for 0 n \uf03e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' This is physically reasonable because increasing γ for a given value of a continuously dilates the triangle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' therefore, there is no finite maximum area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The smallest of all of the areas of these continuously enlarging triangles (with a fixed side length a) occurs when 1 \uf067 = and is 3 2 1 2 2 min 2 1 4 n n n n A a − − \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e9 \uf0f9 = − \uf0ea \uf0fa \uf0ea \uf0fa \uf0eb \uf0fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (21) Equation (21) (a plot of which is shown in Figure 15) represents the areas of the members of an infinite set of isosceles triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Each of these triangles has the minimum possible area among all of the various triangles of the same degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The triangle in this infinite set with the largest area occurs from the infinite degree case, is denoted by max min A \uf0a5 , and is given by the limit of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' max 2 min min lim 6 n A A a \uf0a5 →\uf0a5 = = , (22) Figure 15: The minimum area (in dimensionless units) of an isosceles triangle with unit side length (a = 1) for the n-degree Pythagorean Theorem with positive real exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The red line indicates max min A \uf0a5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 The Area of a Triangle with a Fixed Perimeter (n > 0) To further examine these ideas, it is sensible to consider the area of a triangle with a fixed perimeter P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Then, determine the values of γ and n that produce the maximum area for the n + \uf0ce case (and then in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2, for the n − \uf0ce case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The triangle perimeter is trivially ( ) 1 1 1 n n P a b c a \uf067 \uf067 \uf0e6 \uf0f6 = + + = + + + \uf0e7 \uf0f7 \uf0e8 \uf0f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (23) The area, in terms of the fixed perimeter, is denoted P A , and it results from combining eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (16) and (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' ( ) ( ) 2 2 2 2 2 1 4 1 1 2 1 1 P n n n n P A \uf067 \uf067 \uf067 \uf067 \uf067 \uf0ec \uf0fc \uf0ef \uf0ef \uf0e9 \uf0f9 \uf0ef \uf0ef = − + − + \uf0ed \uf0fd \uf0ea \uf0fa \uf0e9 \uf0f9 \uf0eb \uf0fb \uf0ef \uf0ef + + + \uf0ea \uf0fa \uf0ef \uf0ef \uf0eb \uf0fb \uf0ee \uf0fe (24) From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (24): for a fixed P, and as a function of n, the area of the triangle is a maximum when 1 \uf067 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' This area is denoted by max P A and is given by 1 1 2 max 2 1 4 16 16 1 2 n n n P n n P A + \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 − \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e7 \uf0f7 − = \uf0e7 \uf0f7 \uf0e7 \uf0e6 \uf0f6 \uf0f7 + \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 , (25) Equation (25) represents the areas of an infinite set of maximum area isosceles triangles with a fixed perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The triangle in that set with the largest area is the infinite degree member which is denoted by max P A \uf0a5 , and its area is given by 1 1 2 2 max 2 1 4 16 3 lim 16 36 1 2 n n n P n n n P A P \uf0a5 + \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 →\uf0a5 − \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e6 \uf0f6 − = = \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0e6 \uf0f6 \uf0f7 \uf0e8 \uf0f8 + \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (26) 19 From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (24), as \uf067 → \uf0a5 , the areas of the triangles approach zero (as shown in Figure 17), as they take the form of a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Also, from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (24) (and as shown in Figure 16), triangles with a fixed P will see their areas (as a function of γ) asymptotically increase as n → \uf0a5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Those areas are denoted P A\uf0a5 and are given by ( ) 2 2 2 4 1 1 4 2 1 P A P \uf067 \uf067 \uf0a5 \uf0e9 \uf0f9 − = \uf0ea \uf0fa + \uf0ea \uf0fa \uf0eb \uf0fb , (27) Figure 16: Area (in dimensionless units) of Pythagorean triangles with unit perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 17: Area (in dimensionless units) of Pythagorean triangles with unit perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 The Maximum Areas of Triangles for the n-Degree Pythagorean Theorem with Negative Real Exponents For n − \uf0ce , a plot of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (16) is shown in Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The area relationship for positive real exponents also applies to the case of negative real exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, for all ( ) crit n n \uf067 \uf0a3 , the vertex angles are real and acute, and if a and γ are fixed, the maximum area triangle has the largest vertex angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 18: Triangle area (in dimensionless units) with unit side length (a = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The missing section at the top right and on the right side of figure are due to n > ncrit(γ) for a given value of γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' the border of this section has the equation ncrit(γ) as shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' To determine the triangle with the maximum area, consider eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (15) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' \uf05b \uf05d ( ) \uf05b \uf05d ( ) ( ) ( ) 1 2 2 2 1 2 2 2 2 2 2 max 1 1 1 1 1 max sin 1 min cos 1 min 2 2 2 2 n n A a a a \uf067 \uf067 \uf067 \uf071 \uf067 \uf071 \uf067 \uf067 \uf0ec \uf0fc \uf0e9 \uf0f9 \uf0e6 \uf0f6 \uf0ef \uf0ef + − + \uf0ea \uf0fa \uf0e7 \uf0f7 \uf0ef \uf0ef = = − = − \uf0ed \uf0fd \uf0ea \uf0fa \uf0e7 \uf0f7 \uf0ef \uf0ef \uf0e7 \uf0f7 \uf0ea \uf0fa \uf0e8 \uf0f8 \uf0eb \uf0fb \uf0ef \uf0ef \uf0ee \uf0fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (28) In the special case of an isosceles triangle ( ) 1 \uf067 = , ( ) 2 2 2 1 1 min 1 2 2 n n n n \uf067 \uf067 \uf067 − \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e6 \uf0f6 + − + \uf0e7 \uf0f7 = − \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 for 1 n \uf0a3 − because ( ) crit 1 n \uf067 = does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Triangle Area 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='45 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 Pythagorean Exponent 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='35 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='15 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='05 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='8 2 Side Ratio21 The maximum triangle area is then 1 1 1 2 1 2 2 max 1 sin 2 1 4 2 n n n n A a a \uf067 \uf067 \uf071 − − \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e7 \uf0f7 = \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e6 \uf0f6\uf0e6 \uf0f6 = = − \uf0e7 \uf0f7\uf0e7 \uf0f7 \uf0e7 \uf0f7\uf0e7 \uf0f7 \uf0e8 \uf0f8\uf0e8 \uf0f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (29) If the negative infinite degree case of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (28) is considered, the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' ( ) 2 2 1 1 lim min 2 2 n n n \uf067 \uf067 \uf067 \uf067 →−\uf0a5 \uf0ec \uf0fc \uf0e6 \uf0f6 + − + \uf0ef \uf0ef \uf0e7 \uf0f7 = \uf0ed \uf0fd \uf0e7 \uf0f7 \uf0ef \uf0ef \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0ee \uf0fe for any side ratio that conforms to 1 2 \uf067 \uf0a3 \uf03c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For this negative infinite degree case, the maximum triangle area is 2 2 2 max 1 1 sin 4 2 4 A a a \uf067 \uf071 \uf067 \uf067 −\uf0a5 = = − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (30) Equation (30) necessarily complies with the requirement that 1 2 \uf067 \uf0a3 \uf03c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Additionally, three interesting results emerge about this infinite set of negative infinite degree triangles: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In this set, there are two values of γ that produce triangles with equal areas: 1 \uf067 = and 3 \uf067 = , and that maximum area is 1, 3 2 2 max 1 3 sin 2 4 A a a \uf067 \uf067 \uf067 \uf071 −\uf0a5 = = = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2 \uf067 = necessarily gives zero area because the triangle has collapsed into a straight line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In this infinite set, the triangle with the largest area is the member for which 2 \uf067 = , and its area is max 2 2 max 1 1 sin 2 2 A a a \uf067 \uf071 −\uf0a5 = = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='1 The Area of a Triangle with a Fixed Perimeter (n < 0) If, on the other hand, the perimeter is fixed, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (24) applies for 1 2 \uf067 \uf0a3 \uf03c , and the largest area triangle will be isosceles, and it will have an area of 2 3 36 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' This area is expected because the conditions that 1 \uf067 = and n → −\uf0a5 result in equilateral triangle in which 3 P a = , and therefore, 2 2 3 3 36 4 P a = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' From eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (24), if lim P n A →−\uf0a5 is taken, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (31) results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' As seen in Figure 19 (a plot of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (31)), the area will asymptotically increase as n → −\uf0a5 and will asymptotically decrease toward zero as 2 \uf067 → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (31) determines the area for a fixed perimeter, arbitrary side ratio, and infinite degree triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 22 ( ) 2 2 2 1 4 4 2 P A P \uf067 \uf067 \uf067 −\uf0a5 \uf0e9 \uf0f9 = − \uf0ea \uf0fa + \uf0ea \uf0fa \uf0eb \uf0fb , (31) where P A−\uf0a5 is the asymptotic area of the triangles with a fixed perimeter and for which n → −\uf0a5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Among the infinite set of triangles whose areas are given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' (31), the triangle with the largest area is isosceles (as stated above), and its area is 2 3 36 P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 19: Area of Pythagorean triangles with unit perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Figure 20: Area of Pythagorean triangles with unit perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Summary The Pythagorean Theorem has been extended to positive and negative real exponents unfettered by the physical requirement of dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The relationship between the ratio of the adjacent sides and the vertex angle was determined for a given degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' It was found that for positive exponents, the stipulation that 1 \uf067 \uf0b3 can be applied to all degrees, and no complex vertex angles arose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For 1 2 n \uf03c \uf03c , an obtuse triangle results, and if 2 n \uf03e , the triangle is acute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' However, for negative exponents to produce real vertex angles, the restriction 1 2 \uf067 \uf0a3 \uf03c is necessary but not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The additional requirement that if 1 2 \uf067 \uf03c \uf03c , then ( ) crit n n \uf067 \uf0a3 needs to be imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' The areas of the associated triangles for positive and negative real exponents were explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' With fixed a and γ values, the areas for | 1 n n + \uf0ce \uf03e are maximized when the triangle is right isosceles requiring 1 \uf067 = and 2 n = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Additionally, triangle areas increase as n → \uf0a5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Alternatively, if the perimeter of a triangle is kept constant, the triangle area approaches a maximum value with increasing n and approaches 0 for decreasing γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' For the n − \uf0ce case, as n → −\uf0a5 , the triangle with a fixed perimeter and the maximum area has a side ratio of 2 \uf067 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' In contrast, if the degree is finite and ( ) crit n n \uf067 \uf0a3 (if 1 \uf067 \uf0b9 ), the maximum area occurs when the vertex angle ( ) ,n \uf071 \uf067 is a maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 24 Works Cited 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Faltings, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', The proof of Fermat’s last theorem by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Taylor and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Wiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Notices of the AMS, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 42(7): p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 743-746.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Agarwal, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', Pythagorean theorem before and after Pythagoras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Math, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 30: p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 357-389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=' Amir-Moéz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE0T4oBgHgl3EQfuAGv/content/2301.02600v1.pdf'} +page_content=', R.' 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