"콕세터 군 H3"의 두 판 사이의 차이

수학노트
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1번째 줄: 1번째 줄:
 
==개요==
 
==개요==
−
* 다음과 같이 정의되는 콕세터 군
+
* 다음과 같이 정의되는 콕세터 군 <math>H_3</math>
−
$$
+
:<math>
−
\left\langle r_1,r_2,r_3 \mid r_i^2=(r_3r_1)^2=(r_1r_2)^3=(r_2r_3)^5=1\right\rangle
+
\left\langle r_1,r_2,r_3 \mid r_i^2=(r_3r_1)^2=(r_1r_2)^5=(r_2r_3)^3=1\right\rangle
−
$$
+
</math>
 
* 불변량
 
* 불변량
−
$$
+
:<math>
 
\begin{array}{c|ccccc}
 
\begin{array}{c|ccccc}
−
   & \text{rank} & \text{degree} & \text{exponent} & \text{order} & \text{coxeter} \\
+
   & \text{rank} & \text{degree} & \text{exponent} & \text{order} & \text{Coxeter} \\
 
\hline
 
\hline
 
  H_3 & 3 & 2,6,10 & 1,5,9 & 120 & 10
 
  H_3 & 3 & 2,6,10 & 1,5,9 & 120 & 10
 
\end{array}
 
\end{array}
−
$$
+
</math>
 +
 
 +
==푸앵카레 다항식==
 +
* <math>H_3</math>의 푸앵카레 다항식은 다음과 같다
 +
:<math>
 +
\begin{aligned}
 +
P_{W}(q)&=\sum_{w\in W}q^{\ell(w)} \\
 +
&=1+3 q+5 q^2+7 q^3+9 q^4+11 q^5+12 q^6+12 q^7+12 q^8+12 q^9+11 q^{10}+9 q^{11}+7 q^{12}+5 q^{13}+3 q^{14}+q^{15}
 +
\end{aligned}
 +
</math>
 +
 
 +
 
 +
==콕세터 원소==
 +
* 콕세터 다항식, 즉 콕세터 원소의 특성다항식은 다음과 같다
 +
:<math>
 +
-(x+1) \left(x^2- \varphi x +1\right)
 +
</math>
 +
여기서 <math>\varphi=\frac{1+\sqrt{5}}{2}</math>
 +
* 콕세터 다항식의 세 해는 <math>\zeta, \zeta^5,\zeta^9</math>로 주어지며 여기서 <math>\zeta=e^{2\pi i/10}</math>
  
  
 
==루트 시스템==
 
==루트 시스템==
 
* 30개의 원소로 구성
 
* 30개의 원소로 구성
−
* 다음과 같은 세 벡터로 생성
+
* 다음과 같은 세 벡터가 simple system을 이룬다
−
$$
+
:<math>
 
\begin{align}
 
\begin{align}
−
r_1= (1+2 \alpha,1 , -2 \alpha)  \\
+
r_1= \beta(1+2 \alpha,1 , -2 \alpha)  \\
−
r_2= (-1-2 \alpha  , 1 , 2 \alpha)  \\
+
r_2= \beta(-1-2 \alpha  , 1 , 2 \alpha)  \\
−
r_3= (2 \alpha  , -1-2 \alpha  , 1)
+
r_3= \beta(2 \alpha  , -1-2 \alpha  , 1)
 
\end{align}
 
\end{align}
−
$$
+
</math>
−
여기서 $\alpha=\cos \pi/5$
+
여기서 <math>\alpha=\cos \pi/5, \beta=\cos 2\pi/5</math>
  
 +
[[파일:콕세터 군 H32.png]]
  
 
===콕세터 평면으로의 사영===
 
===콕세터 평면으로의 사영===
31번째 줄: 50번째 줄:
  
  
 +
==테이블==
 +
* 원소
 +
:<math>
 +
\begin{array}{ccc}
 +
  & w & \ell(w) \\
 +
\hline
 +
1 & \{\} & 0 \\
 +
2 & \{1\} & 1 \\
 +
3 & \{2\} & 1 \\
 +
4 & \{3\} & 1 \\
 +
5 & \{2,1\} & 2 \\
 +
6 & \{3,1\} & 2 \\
 +
7 & \{1,2\} & 2 \\
 +
8 & \{3,2\} & 2 \\
 +
9 & \{2,3\} & 2 \\
 +
10 & \{1,2,1\} & 3 \\
 +
11 & \{3,2,1\} & 3 \\
 +
12 & \{2,3,1\} & 3 \\
 +
13 & \{2,1,2\} & 3 \\
 +
14 & \{3,1,2\} & 3 \\
 +
15 & \{2,3,2\} & 3 \\
 +
16 & \{1,2,3\} & 3 \\
 +
17 & \{2,1,2,1\} & 4 \\
 +
18 & \{3,1,2,1\} & 4 \\
 +
19 & \{2,3,2,1\} & 4 \\
 +
20 & \{1,2,3,1\} & 4 \\
 +
21 & \{1,2,1,2\} & 4 \\
 +
22 & \{3,2,1,2\} & 4 \\
 +
23 & \{2,3,1,2\} & 4 \\
 +
24 & \{1,2,3,2\} & 4 \\
 +
25 & \{2,1,2,3\} & 4 \\
 +
26 & \{1,2,1,2,1\} & 5 \\
 +
27 & \{3,2,1,2,1\} & 5 \\
 +
28 & \{2,3,1,2,1\} & 5 \\
 +
29 & \{1,2,3,2,1\} & 5 \\
 +
30 & \{2,1,2,3,1\} & 5 \\
 +
31 & \{3,1,2,1,2\} & 5 \\
 +
32 & \{2,3,2,1,2\} & 5 \\
 +
33 & \{1,2,3,1,2\} & 5 \\
 +
34 & \{2,1,2,3,2\} & 5 \\
 +
35 & \{1,2,1,2,3\} & 5 \\
 +
36 & \{3,2,1,2,3\} & 5 \\
 +
37 & \{3,1,2,1,2,1\} & 6 \\
 +
38 & \{2,3,2,1,2,1\} & 6 \\
 +
39 & \{1,2,3,1,2,1\} & 6 \\
 +
40 & \{2,1,2,3,2,1\} & 6 \\
 +
41 & \{1,2,1,2,3,1\} & 6 \\
 +
42 & \{3,2,1,2,3,1\} & 6 \\
 +
43 & \{2,3,1,2,1,2\} & 6 \\
 +
44 & \{1,2,3,2,1,2\} & 6 \\
 +
45 & \{2,1,2,3,1,2\} & 6 \\
 +
46 & \{1,2,1,2,3,2\} & 6 \\
 +
47 & \{3,2,1,2,3,2\} & 6 \\
 +
48 & \{3,1,2,1,2,3\} & 6 \\
 +
49 & \{2,3,1,2,1,2,1\} & 7 \\
 +
50 & \{1,2,3,2,1,2,1\} & 7 \\
 +
51 & \{2,1,2,3,1,2,1\} & 7 \\
 +
52 & \{1,2,1,2,3,2,1\} & 7 \\
 +
53 & \{3,2,1,2,3,2,1\} & 7 \\
 +
54 & \{3,1,2,1,2,3,1\} & 7 \\
 +
55 & \{1,2,3,1,2,1,2\} & 7 \\
 +
56 & \{2,1,2,3,2,1,2\} & 7 \\
 +
57 & \{1,2,1,2,3,1,2\} & 7 \\
 +
58 & \{3,2,1,2,3,1,2\} & 7 \\
 +
59 & \{3,1,2,1,2,3,2\} & 7 \\
 +
60 & \{2,3,1,2,1,2,3\} & 7 \\
 +
61 & \{1,2,3,1,2,1,2,1\} & 8 \\
 +
62 & \{2,1,2,3,2,1,2,1\} & 8 \\
 +
63 & \{1,2,1,2,3,1,2,1\} & 8 \\
 +
64 & \{3,2,1,2,3,1,2,1\} & 8 \\
 +
65 & \{3,1,2,1,2,3,2,1\} & 8 \\
 +
66 & \{2,3,1,2,1,2,3,1\} & 8 \\
 +
67 & \{2,1,2,3,1,2,1,2\} & 8 \\
 +
68 & \{1,2,1,2,3,2,1,2\} & 8 \\
 +
69 & \{3,2,1,2,3,2,1,2\} & 8 \\
 +
70 & \{3,1,2,1,2,3,1,2\} & 8 \\
 +
71 & \{2,3,1,2,1,2,3,2\} & 8 \\
 +
72 & \{1,2,3,1,2,1,2,3\} & 8 \\
 +
73 & \{2,1,2,3,1,2,1,2,1\} & 9 \\
 +
74 & \{1,2,1,2,3,2,1,2,1\} & 9 \\
 +
75 & \{3,2,1,2,3,2,1,2,1\} & 9 \\
 +
76 & \{3,1,2,1,2,3,1,2,1\} & 9 \\
 +
77 & \{2,3,1,2,1,2,3,2,1\} & 9 \\
 +
78 & \{1,2,3,1,2,1,2,3,1\} & 9 \\
 +
79 & \{1,2,1,2,3,1,2,1,2\} & 9 \\
 +
80 & \{3,2,1,2,3,1,2,1,2\} & 9 \\
 +
81 & \{3,1,2,1,2,3,2,1,2\} & 9 \\
 +
82 & \{2,3,1,2,1,2,3,1,2\} & 9 \\
 +
83 & \{1,2,3,1,2,1,2,3,2\} & 9 \\
 +
84 & \{2,1,2,3,1,2,1,2,3\} & 9 \\
 +
85 & \{1,2,1,2,3,1,2,1,2,1\} & 10 \\
 +
86 & \{3,2,1,2,3,1,2,1,2,1\} & 10 \\
 +
87 & \{3,1,2,1,2,3,2,1,2,1\} & 10 \\
 +
88 & \{2,3,1,2,1,2,3,1,2,1\} & 10 \\
 +
89 & \{1,2,3,1,2,1,2,3,2,1\} & 10 \\
 +
90 & \{2,1,2,3,1,2,1,2,3,1\} & 10 \\
 +
91 & \{3,1,2,1,2,3,1,2,1,2\} & 10 \\
 +
92 & \{2,3,1,2,1,2,3,2,1,2\} & 10 \\
 +
93 & \{1,2,3,1,2,1,2,3,1,2\} & 10 \\
 +
94 & \{2,1,2,3,1,2,1,2,3,2\} & 10 \\
 +
95 & \{3,2,1,2,3,1,2,1,2,3\} & 10 \\
 +
96 & \{3,1,2,1,2,3,1,2,1,2,1\} & 11 \\
 +
97 & \{2,3,1,2,1,2,3,2,1,2,1\} & 11 \\
 +
98 & \{1,2,3,1,2,1,2,3,1,2,1\} & 11 \\
 +
99 & \{2,1,2,3,1,2,1,2,3,2,1\} & 11 \\
 +
100 & \{3,2,1,2,3,1,2,1,2,3,1\} & 11 \\
 +
101 & \{2,3,1,2,1,2,3,1,2,1,2\} & 11 \\
 +
102 & \{1,2,3,1,2,1,2,3,2,1,2\} & 11 \\
 +
103 & \{2,1,2,3,1,2,1,2,3,1,2\} & 11 \\
 +
104 & \{3,2,1,2,3,1,2,1,2,3,2\} & 11 \\
 +
105 & \{2,3,1,2,1,2,3,1,2,1,2,1\} & 12 \\
 +
106 & \{1,2,3,1,2,1,2,3,2,1,2,1\} & 12 \\
 +
107 & \{2,1,2,3,1,2,1,2,3,1,2,1\} & 12 \\
 +
108 & \{3,2,1,2,3,1,2,1,2,3,2,1\} & 12 \\
 +
109 & \{1,2,3,1,2,1,2,3,1,2,1,2\} & 12 \\
 +
110 & \{2,1,2,3,1,2,1,2,3,2,1,2\} & 12 \\
 +
111 & \{3,2,1,2,3,1,2,1,2,3,1,2\} & 12 \\
 +
112 & \{1,2,3,1,2,1,2,3,1,2,1,2,1\} & 13 \\
 +
113 & \{2,1,2,3,1,2,1,2,3,2,1,2,1\} & 13 \\
 +
114 & \{3,2,1,2,3,1,2,1,2,3,1,2,1\} & 13 \\
 +
115 & \{2,1,2,3,1,2,1,2,3,1,2,1,2\} & 13 \\
 +
116 & \{3,2,1,2,3,1,2,1,2,3,2,1,2\} & 13 \\
 +
117 & \{2,1,2,3,1,2,1,2,3,1,2,1,2,1\} & 14 \\
 +
118 & \{3,2,1,2,3,1,2,1,2,3,2,1,2,1\} & 14 \\
 +
119 & \{3,2,1,2,3,1,2,1,2,3,1,2,1,2\} & 14 \\
 +
120 & \{3,2,1,2,3,1,2,1,2,3,1,2,1,2,1\} & 15
 +
\end{array}
 +
</math>
 +
 +
==재미있는 사실==
 +
* 2011년 9월 미국수학회보(Notices of the American Mathematical Society)의 표지에 콕세터 평면으로의 사영이 등장, [http://www.ams.org/notices/201108/rnoti-sep-2011-cover1.pdf 링크]
 +
 +
 +
==관련된 항목들==
 +
* [[교대군 A5]]
 +
* [[정이십면체 뫼비우스 변환군]]
 +
* [[콕세터 군 H4]]
 +
 +
 +
==매스매티카 파일 및 계산 리소스==
 +
* https://docs.google.com/file/d/0B8XXo8Tve1cxQURSVzA3QXYzQTA/edit?usp=drivesdk
 +
 +
 +
==사전 형태의 자료==
 +
* http://en.wikipedia.org/wiki/Binary_icosahedral_group
 
[[분류:리군과 리대수]]
 
[[분류:리군과 리대수]]
 +
 +
==메타데이터==
 +
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q4913898 Q4913898]
 +
===Spacy 패턴 목록===
 +
* [{'LOWER': 'binary'}, {'LOWER': 'icosahedral'}, {'LEMMA': 'group'}]

2021년 2월 17일 (수) 02:22 기준 최신판

개요

  • 다음과 같이 정의되는 콕세터 군 \(H_3\)

\[ \left\langle r_1,r_2,r_3 \mid r_i^2=(r_3r_1)^2=(r_1r_2)^5=(r_2r_3)^3=1\right\rangle \]

  • 불변량

\[ \begin{array}{c|ccccc} & \text{rank} & \text{degree} & \text{exponent} & \text{order} & \text{Coxeter} \\ \hline H_3 & 3 & 2,6,10 & 1,5,9 & 120 & 10 \end{array} \]

푸앵카레 다항식

  • \(H_3\)의 푸앵카레 다항식은 다음과 같다

\[ \begin{aligned} P_{W}(q)&=\sum_{w\in W}q^{\ell(w)} \\ &=1+3 q+5 q^2+7 q^3+9 q^4+11 q^5+12 q^6+12 q^7+12 q^8+12 q^9+11 q^{10}+9 q^{11}+7 q^{12}+5 q^{13}+3 q^{14}+q^{15} \end{aligned} \]


콕세터 원소

  • 콕세터 다항식, 즉 콕세터 원소의 특성다항식은 다음과 같다

\[ -(x+1) \left(x^2- \varphi x +1\right) \] 여기서 \(\varphi=\frac{1+\sqrt{5}}{2}\)

  • 콕세터 다항식의 세 해는 \(\zeta, \zeta^5,\zeta^9\)로 주어지며 여기서 \(\zeta=e^{2\pi i/10}\)


루트 시스템

  • 30개의 원소로 구성
  • 다음과 같은 세 벡터가 simple system을 이룬다

\[ \begin{align} r_1= \beta(1+2 \alpha,1 , -2 \alpha) \\ r_2= \beta(-1-2 \alpha , 1 , 2 \alpha) \\ r_3= \beta(2 \alpha , -1-2 \alpha , 1) \end{align} \] 여기서 \(\alpha=\cos \pi/5, \beta=\cos 2\pi/5\)

콕세터 군 H32.png

콕세터 평면으로의 사영

콕세터 군 H31.png


테이블

  • 원소

\[ \begin{array}{ccc} & w & \ell(w) \\ \hline 1 & \{\} & 0 \\ 2 & \{1\} & 1 \\ 3 & \{2\} & 1 \\ 4 & \{3\} & 1 \\ 5 & \{2,1\} & 2 \\ 6 & \{3,1\} & 2 \\ 7 & \{1,2\} & 2 \\ 8 & \{3,2\} & 2 \\ 9 & \{2,3\} & 2 \\ 10 & \{1,2,1\} & 3 \\ 11 & \{3,2,1\} & 3 \\ 12 & \{2,3,1\} & 3 \\ 13 & \{2,1,2\} & 3 \\ 14 & \{3,1,2\} & 3 \\ 15 & \{2,3,2\} & 3 \\ 16 & \{1,2,3\} & 3 \\ 17 & \{2,1,2,1\} & 4 \\ 18 & \{3,1,2,1\} & 4 \\ 19 & \{2,3,2,1\} & 4 \\ 20 & \{1,2,3,1\} & 4 \\ 21 & \{1,2,1,2\} & 4 \\ 22 & \{3,2,1,2\} & 4 \\ 23 & \{2,3,1,2\} & 4 \\ 24 & \{1,2,3,2\} & 4 \\ 25 & \{2,1,2,3\} & 4 \\ 26 & \{1,2,1,2,1\} & 5 \\ 27 & \{3,2,1,2,1\} & 5 \\ 28 & \{2,3,1,2,1\} & 5 \\ 29 & \{1,2,3,2,1\} & 5 \\ 30 & \{2,1,2,3,1\} & 5 \\ 31 & \{3,1,2,1,2\} & 5 \\ 32 & \{2,3,2,1,2\} & 5 \\ 33 & \{1,2,3,1,2\} & 5 \\ 34 & \{2,1,2,3,2\} & 5 \\ 35 & \{1,2,1,2,3\} & 5 \\ 36 & \{3,2,1,2,3\} & 5 \\ 37 & \{3,1,2,1,2,1\} & 6 \\ 38 & \{2,3,2,1,2,1\} & 6 \\ 39 & \{1,2,3,1,2,1\} & 6 \\ 40 & \{2,1,2,3,2,1\} & 6 \\ 41 & \{1,2,1,2,3,1\} & 6 \\ 42 & \{3,2,1,2,3,1\} & 6 \\ 43 & \{2,3,1,2,1,2\} & 6 \\ 44 & \{1,2,3,2,1,2\} & 6 \\ 45 & \{2,1,2,3,1,2\} & 6 \\ 46 & \{1,2,1,2,3,2\} & 6 \\ 47 & \{3,2,1,2,3,2\} & 6 \\ 48 & \{3,1,2,1,2,3\} & 6 \\ 49 & \{2,3,1,2,1,2,1\} & 7 \\ 50 & \{1,2,3,2,1,2,1\} & 7 \\ 51 & \{2,1,2,3,1,2,1\} & 7 \\ 52 & \{1,2,1,2,3,2,1\} & 7 \\ 53 & \{3,2,1,2,3,2,1\} & 7 \\ 54 & \{3,1,2,1,2,3,1\} & 7 \\ 55 & \{1,2,3,1,2,1,2\} & 7 \\ 56 & \{2,1,2,3,2,1,2\} & 7 \\ 57 & \{1,2,1,2,3,1,2\} & 7 \\ 58 & \{3,2,1,2,3,1,2\} & 7 \\ 59 & \{3,1,2,1,2,3,2\} & 7 \\ 60 & \{2,3,1,2,1,2,3\} & 7 \\ 61 & \{1,2,3,1,2,1,2,1\} & 8 \\ 62 & \{2,1,2,3,2,1,2,1\} & 8 \\ 63 & \{1,2,1,2,3,1,2,1\} & 8 \\ 64 & \{3,2,1,2,3,1,2,1\} & 8 \\ 65 & \{3,1,2,1,2,3,2,1\} & 8 \\ 66 & \{2,3,1,2,1,2,3,1\} & 8 \\ 67 & \{2,1,2,3,1,2,1,2\} & 8 \\ 68 & \{1,2,1,2,3,2,1,2\} & 8 \\ 69 & \{3,2,1,2,3,2,1,2\} & 8 \\ 70 & \{3,1,2,1,2,3,1,2\} & 8 \\ 71 & \{2,3,1,2,1,2,3,2\} & 8 \\ 72 & \{1,2,3,1,2,1,2,3\} & 8 \\ 73 & \{2,1,2,3,1,2,1,2,1\} & 9 \\ 74 & \{1,2,1,2,3,2,1,2,1\} & 9 \\ 75 & \{3,2,1,2,3,2,1,2,1\} & 9 \\ 76 & \{3,1,2,1,2,3,1,2,1\} & 9 \\ 77 & \{2,3,1,2,1,2,3,2,1\} & 9 \\ 78 & \{1,2,3,1,2,1,2,3,1\} & 9 \\ 79 & \{1,2,1,2,3,1,2,1,2\} & 9 \\ 80 & \{3,2,1,2,3,1,2,1,2\} & 9 \\ 81 & \{3,1,2,1,2,3,2,1,2\} & 9 \\ 82 & \{2,3,1,2,1,2,3,1,2\} & 9 \\ 83 & \{1,2,3,1,2,1,2,3,2\} & 9 \\ 84 & \{2,1,2,3,1,2,1,2,3\} & 9 \\ 85 & \{1,2,1,2,3,1,2,1,2,1\} & 10 \\ 86 & \{3,2,1,2,3,1,2,1,2,1\} & 10 \\ 87 & \{3,1,2,1,2,3,2,1,2,1\} & 10 \\ 88 & \{2,3,1,2,1,2,3,1,2,1\} & 10 \\ 89 & \{1,2,3,1,2,1,2,3,2,1\} & 10 \\ 90 & \{2,1,2,3,1,2,1,2,3,1\} & 10 \\ 91 & \{3,1,2,1,2,3,1,2,1,2\} & 10 \\ 92 & \{2,3,1,2,1,2,3,2,1,2\} & 10 \\ 93 & \{1,2,3,1,2,1,2,3,1,2\} & 10 \\ 94 & \{2,1,2,3,1,2,1,2,3,2\} & 10 \\ 95 & \{3,2,1,2,3,1,2,1,2,3\} & 10 \\ 96 & \{3,1,2,1,2,3,1,2,1,2,1\} & 11 \\ 97 & \{2,3,1,2,1,2,3,2,1,2,1\} & 11 \\ 98 & \{1,2,3,1,2,1,2,3,1,2,1\} & 11 \\ 99 & \{2,1,2,3,1,2,1,2,3,2,1\} & 11 \\ 100 & \{3,2,1,2,3,1,2,1,2,3,1\} & 11 \\ 101 & \{2,3,1,2,1,2,3,1,2,1,2\} & 11 \\ 102 & \{1,2,3,1,2,1,2,3,2,1,2\} & 11 \\ 103 & \{2,1,2,3,1,2,1,2,3,1,2\} & 11 \\ 104 & \{3,2,1,2,3,1,2,1,2,3,2\} & 11 \\ 105 & \{2,3,1,2,1,2,3,1,2,1,2,1\} & 12 \\ 106 & \{1,2,3,1,2,1,2,3,2,1,2,1\} & 12 \\ 107 & \{2,1,2,3,1,2,1,2,3,1,2,1\} & 12 \\ 108 & \{3,2,1,2,3,1,2,1,2,3,2,1\} & 12 \\ 109 & \{1,2,3,1,2,1,2,3,1,2,1,2\} & 12 \\ 110 & \{2,1,2,3,1,2,1,2,3,2,1,2\} & 12 \\ 111 & \{3,2,1,2,3,1,2,1,2,3,1,2\} & 12 \\ 112 & \{1,2,3,1,2,1,2,3,1,2,1,2,1\} & 13 \\ 113 & \{2,1,2,3,1,2,1,2,3,2,1,2,1\} & 13 \\ 114 & \{3,2,1,2,3,1,2,1,2,3,1,2,1\} & 13 \\ 115 & \{2,1,2,3,1,2,1,2,3,1,2,1,2\} & 13 \\ 116 & \{3,2,1,2,3,1,2,1,2,3,2,1,2\} & 13 \\ 117 & \{2,1,2,3,1,2,1,2,3,1,2,1,2,1\} & 14 \\ 118 & \{3,2,1,2,3,1,2,1,2,3,2,1,2,1\} & 14 \\ 119 & \{3,2,1,2,3,1,2,1,2,3,1,2,1,2\} & 14 \\ 120 & \{3,2,1,2,3,1,2,1,2,3,1,2,1,2,1\} & 15 \end{array} \]

재미있는 사실

  • 2011년 9월 미국수학회보(Notices of the American Mathematical Society)의 표지에 콕세터 평면으로의 사영이 등장, 링크


관련된 항목들


매스매티카 파일 및 계산 리소스


사전 형태의 자료

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'binary'}, {'LOWER': 'icosahedral'}, {'LEMMA': 'group'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=콕세터_군_H3&oldid=51993"