"가우스와 정17각형의 작도"의 두 판 사이의 차이

수학노트
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(사용자 2명의 중간 판 49개는 보이지 않습니다)
1번째 줄: 1번째 줄:
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<h5>간단한 소개</h5>
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==개요==
  
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* 가우스는 정17각형이 자와 컴파스로 작도가능함을 증명함.
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* 가우스는 정17각형이 자와 컴파스로 작도가능함을 증명함
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* 대수적으로 보자면, <math>x^{16}+x^{15}+\cdots+x+1=0</math>의 풀이를 반복적인 2차방정식의 풀이로 환원할 수 있는가의 문제.
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* 대수적으로, <math>z^{16}+z^{15}+\cdots+z+1=0</math>의 풀이를 반복적인 2차방정식의 풀이로 환원할 수 있는가의 문제
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* 쉬운 예를 들자면, <math>x^4+x^3+x^2+x+1=0</math> 은 다음과 같은 순서로 풀수 있음.
+
* 16차 방정식을 2차방정식 네번 푸는 문제로 바꾸는 것
 +
** 이 아이디어를 좀더 간단한 예를 통해 이해하기 위해서는 [[정오각형]] 항목 중 꼭지점의 평면좌표를 참조
  
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양변을 <math>x^2</math>으로 나누면, <math>x^2+x+1+\frac{1}{x}+\frac{1}{x^2}=0</math> 을 얻게됨.
 
  
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'''<math>t=x+\frac{1}{x}</math>''' 로 치환하면, 원래의 방정식에서 다음 식을 얻을 수 있음.
 
  
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<math>x^2+\frac{1}{x^2}+x+\frac{1}{x}+1=(x+\frac{1}{x})^2+(x+\frac{1}{x})-1=t^2+t-1=0</math>
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==증명==
  
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방정식을 풀어가면,
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* <math>\zeta=e^{2\pi i \over 17}</math>  로 두자. 이 값을 대수적으로 구하는 것이 목표.
 +
* <math>(3^1, 3^2,3^3, 3^4, 3^5, 3^7, 3^8, 3^9, 3^{10}, 3^{11}, 3^{12}, 3^{13}, 3^{14}, 3^{15}, 3^{16}) \equiv (3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4,12, 2, 6, 1) \pmod {17}</math>
 +
*  이 순서대로 2로 나눈 나머지에 따라서 분류
 +
** <math>A_0 = \zeta^{9} + \zeta^{13} + \zeta^{15} + \zeta^{16}+\zeta^{8} + \zeta^{4} + \zeta^{2} +\zeta^{1}</math>
 +
** <math>A_1 = \zeta^3 + \zeta^{10} + \zeta^{5} + \zeta^{11}+\zeta^{14} + \zeta^{7} + \zeta^{12} +\zeta^{6}</math>
 +
** <math>A_0+A_1= -1</math>, <math>A_{0}A_{1} = -4</math>, <math>A_0>A_1</math>
 +
** <math>A_0 = \frac{-1 + \sqrt{17}}{2}</math> , <math>A_1= \frac{-1 - \sqrt{17}}{2}</math>
 +
*  이번에는 4로 나눈 나머지에 따라서 분류
 +
** <math>B_0 = \zeta^{13}+  \zeta^{16}+ \zeta^4 +  \zeta^1 </math>
 +
** <math>B_1= \zeta^3 + \zeta^5 + \zeta^{14} + \zeta^{12}</math>
 +
** <math>B_2= \zeta^9 + \zeta^{15} + \zeta^8 +\zeta^2</math>
 +
** <math>B_3 =\zeta^{10} + \zeta^{11} + \zeta^{7} +\zeta^{6}</math>
 +
** <math>B_0+B_2=A_0</math>, <math>B_0B_2= -1</math>, <math>B_0>0</math>
 +
** <math>B_0 = \frac{-1 + \sqrt{17} + \sqrt{34 - 2\sqrt{17}}}{4}</math>, <math>B_2 = \frac{-1 + \sqrt{17} - \sqrt{34 - 2\sqrt{17}}}{4}</math>
 +
** <math>B_1+B_3=A_1</math>, <math>B_1B_3= -1</math>, <math>B_{1}> 0</math>
 +
** <math>B_1 = \frac{-1 - \sqrt{17} + \sqrt{34 + 2\sqrt{17}}}{4}</math>, <math>B_3 = \frac{-1 - \sqrt{17} - \sqrt{34 + 2\sqrt{17}}}{4}</math>
 +
*  이번에는 8로 나눈 나머지에 따라서 분류
 +
** <math>C_0= \zeta^{16}+  \zeta^1</math>, <math>C_4= \zeta^{13} +\zeta^4</math>, <math>C_0 > C_1</math>
 +
** <math>C_0+C_4=B_0</math>, <math>C_0C_4=B_1</math>
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** <math>C_0= \frac{B_0+\sqrt{B_0^2-4B_1}}{2}= \frac{-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+  \sqrt{68+12\sqrt{17}-4{\sqrt{170+38\sqrt{17}}}} }{8}</math>
 +
** <math>C_4= \frac{B_0 - \sqrt{B_0^2-4B_1}}{2}</math>
 +
*  이제 마무리
 +
** <math>\zeta =\frac{{C_0} + \sqrt{{C_0}^2 - 4}}{2}</math>
 +
** <math>\cos \frac{2\pi}{17}= \frac{-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+  \sqrt{68+12\sqrt{17}-4{\sqrt{170+38\sqrt{17}}}} }{16}</math>
  
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+
  
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<math>t^2+t-1=0</math>
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<math>t=\frac{-1\pm\sqrt{5}}{2}</math>
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==가우스합과의 관계==
  
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<math>t=x+\frac{1}{x}</math>
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*  참고로 위에서 <math>A_0-A_1</math> 은 [[가우스 합|가우스합]] 임을 알 수 있음.
 +
** <math>\{3, 10, 5, 11, 14, 7, 12,  6\}</math> 는 <math>\pmod {17}</math> 에 대하여 이차비잉여
 +
** <math>\{9, 13, 15, 16, 8, 4, 2, 1\}</math>는  <math>\pmod {17}</math> 에 대하여 이차잉여
 +
** 따라서 <math>A_{0}A_{1}</math>를 계산하는 대신에 <math>A_0-A_1=\sqrt{17}</math> 를 활용할 수도 있음.
  
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<math>x^2-tx+1=0</math>
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<math>x=\frac{t+\sqrt{t^2-4}}{2}</math>
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==재미있는 사실==
  
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 을 얻게 됨. 
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* 17은 [[페르마 소수|페르마소수]]이다
  
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따라서 x는 유리수에서 시작하여, 사칙연산에 제곱근을 사용하여 표현가능하고, 따라서 자와 컴파스를 활용하여 작도가능.
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* <math>x^{16}+x^{15}+\cdots+x+1=0</math> 의 경우에도 본질적으로는 위의 경우와 다르지 않으나, 2차방정식을 네번 풀어야 하고, 좀더 복잡해짐. 자세한 사항은 아래의 참고할 만한 자료를 볼 것.
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==역사==
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* 1796 가우스
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* [[수학사 연표]]
  
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<h5>관련된 학부 과목과 미리 알고 있으면 좋은 것들</h5>
 
  
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* [[추상대수학]]<br>
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** 순환군 (cyclic groups)
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==메모==
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* http://pballew.blogspot.com/2011/04/constructructable-polygons-and-x17-1.html
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==관련된 학부 과목과 미리 알고 있으면 좋은 것들==
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 +
* [[추상대수학]]
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** [[순환군]]
 
** 가해군 (solvable groups)
 
** 가해군 (solvable groups)
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* [[초등정수론]]<br>
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* [[초등정수론]]
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** 합동식
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** [[합동식과 군론]]
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** 원시근 (primitive roots)
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** [[원시근(primitive root)]]
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** [[오일러의 totient 함수]]
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<h5>관련된 대학원 과목</h5>
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==관련된 항목들==
  
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* [[그리스 3대 작도 불가능문제]]
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* [[정다각형의 작도]]
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* [[원시근(primitive root)]]
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* [[가우스(1777 - 1855)]]
  
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<h5>관련된 다른 주제들</h5>
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*  그리스 3대 작도 불가능문제<br>
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==매스매티카 파일 및 계산 리소스==
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** The duplication of the cube
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* https://docs.google.com/leaf?id=0B8XXo8Tve1cxZGVmNzY1NmEtMThiNi00OTdkLTgxYTQtYTZhMzdlZTM4Mzkw&sort=name&layout=list&num=50
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** The trisection of an angle
 
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** The quadrature of the circle
 
  
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<h5>표준적인 도서 및 추천도서</h5>
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* [http://www.amazon.com/Problems-Elementary-Geometry-Phoenix-Editions/dp/0486495515 Famous Problems of Elementary Geometry] (Dover Phoenix Editions)<br>
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==사전형태의 자료==
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** Felix Klein
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* http://en.wikipedia.org/wiki/Heptadecagon
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 +
 
 +
==관련논문==
 +
* Garcia, Stephan Ramon, Trevor Hyde, and Bob Lutz. ‘Gauss’ Hidden Menagerie: From Cyclotomy to Supercharacters’. arXiv:1501.07507 [math], 29 January 2015. http://arxiv.org/abs/1501.07507.
 +
 
 +
==관련도서==
 +
 
 +
* [http://www.amazon.com/Problems-Elementary-Geometry-Phoenix-Editions/dp/0486495515 Famous Problems of Elementary Geometry] (Dover Phoenix Editions)
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** 펠릭스 클라인 Felix Klein
 
** 얇은 책으로, 대수방정식과 함께 고대 그리스 3대 작도 불가능문제를 소개함.
 
** 얇은 책으로, 대수방정식과 함께 고대 그리스 3대 작도 불가능문제를 소개함.
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* [http://www.amazon.com/Functions-Integrals-Translations-Mathematical-Monographs/dp/0821805878 Elliptic functions and elliptic integrals]
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** Viktor Prasolov, Yuri Solovyev
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*  Lectures on Elementary Number Theory
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** Hans Rademacher
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==동영상==
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* [http://www.youtube.com/watch?v=E2QMzRcjKkM 정17각형의 작도 과정을 보여주는 동영상], Youtube
  
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<h5>위키링크</h5>
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[[분류:작도]]
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[[분류:추상대수학]]
  
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* http://en.wikipedia.org/wiki/Heptadecagon
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== 노트 ==
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*  
 
  
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===말뭉치===
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# Then, on 30 March 1796, the 19 year old Gauss discovered that it was possible to construct the regular heptadecagon (17-gon).<ref name="ref_47ccf09f">[https://mathpages.com/home/kmath487.htm Constructing the Heptadecagon]</ref>
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# One of the nicest actual constructions of the 17-gon is Richmond's (1893), as reproduced in Stewart's "Galois Theory".<ref name="ref_47ccf09f" />
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# Gauss was clearly fond of this discovery, and there's a story that he asked to have a heptadecagon carved on his tombstone, like the sphere incribed in a cylinder on Archimedes' tombstone.<ref name="ref_47ccf09f" />
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# On the other hand, if proximity to the actual remains is not important, then the heptadecagon on the monument to Gauss in his native town of Brunswick, or even the figure above, may suffice.<ref name="ref_47ccf09f" />
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# Gauss's "heptadecagon', a 17-sided polygon that showed the relationship between geometry and algebra.<ref name="ref_275a9462">[https://www.inverse.com/article/44309-johann-carl-friedrich-gauss-math-statistics-accomplishments Johann Carl Friedrich Gauß Changed History With His 17-Sided Shape]</ref>
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# But what’s widely considered his first important discovery is his construction of a 17-sided polygon called a heptadecagon, using only a ruler and a compass.<ref name="ref_275a9462" />
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# A regular heptadecagon is represented by the Schläfli symbol {17}.<ref name="ref_3d3204f1">[https://en.wikipedia.org/wiki/Heptadecagon Heptadecagon]</ref>
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# Constructing a regular heptadecagon thus involves finding the cosine of 2 π / 17 {\displaystyle 2\pi /17} in terms of square roots, which involves an equation of degree 17—a Fermat prime.<ref name="ref_3d3204f1" />
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# The explicit construction of a heptadecagon was given by Herbert William Richmond in 1893.<ref name="ref_3d3204f1" />
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# These 4 symmetries can be seen in 4 distinct symmetries on the heptadecagon.<ref name="ref_3d3204f1" />
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# Gauss proved in 1796 (when he was 19 years old) that the heptadecagon is constructible with a compass and straightedge.<ref name="ref_05209e8b">[https://mathworld.wolfram.com/Heptadecagon.html Heptadecagon -- from Wolfram MathWorld]</ref>
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# The first explicit construction of a heptadecagon was given by Erchinger in about 1800.<ref name="ref_05209e8b" />
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# You now have points and of a heptadecagon.<ref name="ref_05209e8b" />
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# Connect the adjacent points for to 17, forming the heptadecagon.<ref name="ref_05209e8b" />
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# A regular heptadecagon, or 17 sided polygon, was known to have existed by mathematicians for many years, but creating one proved to be a greater challenge.<ref name="ref_c4d14414">[https://interestingengineering.com/create-regular-heptadecagon-using-math How to Create a Regular Heptadecagon using Math!]</ref>
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# A regular heptadecagon was first created by 19-year-old Carl Friedrich Gauss in 1796 in a groundbreaking proof.<ref name="ref_c4d14414" />
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# The proof in regards to this 17-gon's construction marked the first major breakthrough in polygon construction in over 2,000 years.<ref name="ref_c4d14414" />
 +
# Apart from a heptadecagon, there are also heptadecagrams, which are 17 sided star polygons.<ref name="ref_c4d14414" />
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# A regular heptadecagon has internal angles each measuring 158.823529411765 degrees.<ref name="ref_fc48496f">[http://academickids.com/encyclopedia/index.php/Heptadecagon Academic Kids]</ref>
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# The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796.<ref name="ref_fc48496f" />
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# Gauß proved in 1796 (when he was 19 years old) that the heptadecagon is Constructible with a Compass and Straightedge.<ref name="ref_b1207714">[https://archive.lib.msu.edu/crcmath/math/math/h/h188.htm Heptadecagon]</ref>
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# The following elegant construction for the heptadecagon (Yates 1949, Coxeter 1969, Stewart 1977, Wells 1992) was first given by Richmond (1893).<ref name="ref_b1207714" />
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# The following animation of a heptadecagon editing.<ref name="ref_8efd6d53">[https://kids.kiddle.co/Heptadecagon Heptadecagon facts for kids]</ref>
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# Go to google.com and search about Gauss' 17-gon construction.<ref name="ref_5c6bd9ed">[http://mathgardenblog.blogspot.com/2014/06/construct-15gon.html Math Garden: How to construct a regular polygon with 15 sides]</ref>
 +
# He described in his Disquitiones Arithmeticae, a major work on number theory, how to construct a regular 17-gon with Euclidean tools.<ref name="ref_853b9c26">[https://mathcs.clarku.edu/~djoyce/elements/bookIV/propIV16.html Euclid's Elements, Book IV, Proposition 16]</ref>
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# The eighteen year old Gauss began his scientific diary with his construction of the regular 17-gon.<ref name="ref_b543aaf5">[https://www.maa.org/news/on-this-day/1796-3-30 Mathematical Association of America]</ref>
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===소스===
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<references />
  
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<h5>참고할만한 자료</h5>
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== 메타데이터 ==
  
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* [[1993332/attachments/912170|The constuction of the Regular Polygon of 17 sides]][[1993332/attachments/912172|]] (페이지내 첨부파일)<br>
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===위키데이터===
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** 클라인의 [http://www.amazon.com/Problems-Elementary-Geometry-Phoenix-Editions/dp/0486495515 Famous Problems of Elementary Geometry] 에서 발췌.
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* ID :  [https://www.wikidata.org/wiki/Q542476 Q542476]
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* 정17각형의 작도 과정을 보여주는 동영상
+
===Spacy 패턴 목록===
 +
* [{'LEMMA': 'heptadecagon'}]
 +
* [{'LOWER': '17'}, {'OP': '*'}, {'LEMMA': 'gon'}]
 +
* [{'LEMMA': 'heptakaidecagon'}]

2021년 2월 10일 (수) 00:55 판

개요

  • 가우스는 정17각형이 자와 컴파스로 작도가능함을 증명함
  • 대수적으로, \(z^{16}+z^{15}+\cdots+z+1=0\)의 풀이를 반복적인 2차방정식의 풀이로 환원할 수 있는가의 문제
  • 16차 방정식을 2차방정식 네번 푸는 문제로 바꾸는 것
    • 이 아이디어를 좀더 간단한 예를 통해 이해하기 위해서는 정오각형 항목 중 꼭지점의 평면좌표를 참조


증명

  • \(\zeta=e^{2\pi i \over 17}\) 로 두자. 이 값을 대수적으로 구하는 것이 목표.
  • \((3^1, 3^2,3^3, 3^4, 3^5, 3^7, 3^8, 3^9, 3^{10}, 3^{11}, 3^{12}, 3^{13}, 3^{14}, 3^{15}, 3^{16}) \equiv (3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4,12, 2, 6, 1) \pmod {17}\)
  • 이 순서대로 2로 나눈 나머지에 따라서 분류
    • \(A_0 = \zeta^{9} + \zeta^{13} + \zeta^{15} + \zeta^{16}+\zeta^{8} + \zeta^{4} + \zeta^{2} +\zeta^{1}\)
    • \(A_1 = \zeta^3 + \zeta^{10} + \zeta^{5} + \zeta^{11}+\zeta^{14} + \zeta^{7} + \zeta^{12} +\zeta^{6}\)
    • \(A_0+A_1= -1\), \(A_{0}A_{1} = -4\), \(A_0>A_1\)
    • \(A_0 = \frac{-1 + \sqrt{17}}{2}\) , \(A_1= \frac{-1 - \sqrt{17}}{2}\)
  • 이번에는 4로 나눈 나머지에 따라서 분류
    • \(B_0 = \zeta^{13}+ \zeta^{16}+ \zeta^4 + \zeta^1 \)
    • \(B_1= \zeta^3 + \zeta^5 + \zeta^{14} + \zeta^{12}\)
    • \(B_2= \zeta^9 + \zeta^{15} + \zeta^8 +\zeta^2\)
    • \(B_3 =\zeta^{10} + \zeta^{11} + \zeta^{7} +\zeta^{6}\)
    • \(B_0+B_2=A_0\), \(B_0B_2= -1\), \(B_0>0\)
    • \(B_0 = \frac{-1 + \sqrt{17} + \sqrt{34 - 2\sqrt{17}}}{4}\), \(B_2 = \frac{-1 + \sqrt{17} - \sqrt{34 - 2\sqrt{17}}}{4}\)
    • \(B_1+B_3=A_1\), \(B_1B_3= -1\), \(B_{1}> 0\)
    • \(B_1 = \frac{-1 - \sqrt{17} + \sqrt{34 + 2\sqrt{17}}}{4}\), \(B_3 = \frac{-1 - \sqrt{17} - \sqrt{34 + 2\sqrt{17}}}{4}\)
  • 이번에는 8로 나눈 나머지에 따라서 분류
    • \(C_0= \zeta^{16}+ \zeta^1\), \(C_4= \zeta^{13} +\zeta^4\), \(C_0 > C_1\)
    • \(C_0+C_4=B_0\), \(C_0C_4=B_1\)
    • \(C_0= \frac{B_0+\sqrt{B_0^2-4B_1}}{2}= \frac{-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+ \sqrt{68+12\sqrt{17}-4{\sqrt{170+38\sqrt{17}}}} }{8}\)
    • \(C_4= \frac{B_0 - \sqrt{B_0^2-4B_1}}{2}\)
  • 이제 마무리
    • \(\zeta =\frac{{C_0} + \sqrt{{C_0}^2 - 4}}{2}\)
    • \(\cos \frac{2\pi}{17}= \frac{-1+\sqrt{17}+\sqrt{34-2\sqrt{17}}+ \sqrt{68+12\sqrt{17}-4{\sqrt{170+38\sqrt{17}}}} }{16}\)



가우스합과의 관계

  • 참고로 위에서 \(A_0-A_1\) 은 가우스합 임을 알 수 있음.
    • \(\{3, 10, 5, 11, 14, 7, 12, 6\}\) 는 \(\pmod {17}\) 에 대하여 이차비잉여
    • \(\{9, 13, 15, 16, 8, 4, 2, 1\}\)는 \(\pmod {17}\) 에 대하여 이차잉여
    • 따라서 \(A_{0}A_{1}\)를 계산하는 대신에 \(A_0-A_1=\sqrt{17}\) 를 활용할 수도 있음.


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매스매티카 파일 및 계산 리소스



사전형태의 자료


관련논문

  • Garcia, Stephan Ramon, Trevor Hyde, and Bob Lutz. ‘Gauss’ Hidden Menagerie: From Cyclotomy to Supercharacters’. arXiv:1501.07507 [math], 29 January 2015. http://arxiv.org/abs/1501.07507.

관련도서

동영상

노트

말뭉치

  1. Then, on 30 March 1796, the 19 year old Gauss discovered that it was possible to construct the regular heptadecagon (17-gon).[1]
  2. One of the nicest actual constructions of the 17-gon is Richmond's (1893), as reproduced in Stewart's "Galois Theory".[1]
  3. Gauss was clearly fond of this discovery, and there's a story that he asked to have a heptadecagon carved on his tombstone, like the sphere incribed in a cylinder on Archimedes' tombstone.[1]
  4. On the other hand, if proximity to the actual remains is not important, then the heptadecagon on the monument to Gauss in his native town of Brunswick, or even the figure above, may suffice.[1]
  5. Gauss's "heptadecagon', a 17-sided polygon that showed the relationship between geometry and algebra.[2]
  6. But what’s widely considered his first important discovery is his construction of a 17-sided polygon called a heptadecagon, using only a ruler and a compass.[2]
  7. A regular heptadecagon is represented by the Schläfli symbol {17}.[3]
  8. Constructing a regular heptadecagon thus involves finding the cosine of 2 π / 17 {\displaystyle 2\pi /17} in terms of square roots, which involves an equation of degree 17—a Fermat prime.[3]
  9. The explicit construction of a heptadecagon was given by Herbert William Richmond in 1893.[3]
  10. These 4 symmetries can be seen in 4 distinct symmetries on the heptadecagon.[3]
  11. Gauss proved in 1796 (when he was 19 years old) that the heptadecagon is constructible with a compass and straightedge.[4]
  12. The first explicit construction of a heptadecagon was given by Erchinger in about 1800.[4]
  13. You now have points and of a heptadecagon.[4]
  14. Connect the adjacent points for to 17, forming the heptadecagon.[4]
  15. A regular heptadecagon, or 17 sided polygon, was known to have existed by mathematicians for many years, but creating one proved to be a greater challenge.[5]
  16. A regular heptadecagon was first created by 19-year-old Carl Friedrich Gauss in 1796 in a groundbreaking proof.[5]
  17. The proof in regards to this 17-gon's construction marked the first major breakthrough in polygon construction in over 2,000 years.[5]
  18. Apart from a heptadecagon, there are also heptadecagrams, which are 17 sided star polygons.[5]
  19. A regular heptadecagon has internal angles each measuring 158.823529411765 degrees.[6]
  20. The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796.[6]
  21. Gauß proved in 1796 (when he was 19 years old) that the heptadecagon is Constructible with a Compass and Straightedge.[7]
  22. The following elegant construction for the heptadecagon (Yates 1949, Coxeter 1969, Stewart 1977, Wells 1992) was first given by Richmond (1893).[7]
  23. The following animation of a heptadecagon editing.[8]
  24. Go to google.com and search about Gauss' 17-gon construction.[9]
  25. He described in his Disquitiones Arithmeticae, a major work on number theory, how to construct a regular 17-gon with Euclidean tools.[10]
  26. The eighteen year old Gauss began his scientific diary with his construction of the regular 17-gon.[11]

소스

  1. ↑ 1.0 1.1 1.2 1.3 Constructing the Heptadecagon
  2. ↑ 2.0 2.1 Johann Carl Friedrich Gauß Changed History With His 17-Sided Shape
  3. ↑ 3.0 3.1 3.2 3.3 Heptadecagon
  4. ↑ 4.0 4.1 4.2 4.3 Heptadecagon -- from Wolfram MathWorld
  5. ↑ 5.0 5.1 5.2 5.3 How to Create a Regular Heptadecagon using Math!
  6. ↑ 6.0 6.1 Academic Kids
  7. ↑ 7.0 7.1 Heptadecagon
  8. ↑ Heptadecagon facts for kids
  9. ↑ Math Garden: How to construct a regular polygon with 15 sides
  10. ↑ Euclid's Elements, Book IV, Proposition 16
  11. ↑ Mathematical Association of America

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LEMMA': 'heptadecagon'}]
  • [{'LOWER': '17'}, {'OP': '*'}, {'LEMMA': 'gon'}]
  • [{'LEMMA': 'heptakaidecagon'}]
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