아이코시안 (icosian) - 편집 역사

2021년 2월 17일 (수) 10:21에 Pythagoras0님의 편집

2021-02-17T10:21:32Z

Pythagoras0: /* 메타데이터 */ 새 문단

2020-12-28T14:00:09Z

메타데이터: 새 문단

2020년 11월 12일 (목) 08:05에 Pythagoras0님의 편집

2020-11-12T08:05:01Z

2014년 7월 9일 (수) 00:35에 Pythagoras0님의 편집

2014-07-09T00:35:56Z

Pythagoras0: /* 아이코시안과 $E_8$ */

2014-07-08T13:45:26Z

아이코시안과 $E_8$

Pythagoras0: /* 테이블 */

2014-07-08T13:41:24Z

테이블

Pythagoras0: /* 테이블 */

2014-07-08T11:54:05Z

테이블

차이 보기

2014년 7월 8일 (화) 11:46에 Pythagoras0님의 편집

2014-07-08T11:46:27Z

2014년 7월 1일 (화) 14:36에 Pythagoras0님의 편집

2014-07-01T14:36:18Z

2014년 7월 1일 (화) 14:14에 Pythagoras0님의 편집

2014-07-01T14:14:42Z

@@ 56번째 줄: / 56번째 줄: @@
 * Wilson, Robert A. "The geometry of the Hall-Janko group as a quaternionic reflection group." Geometriae Dedicata 20.2 (1986): 157-173.
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q17058942 Q17058942]

← 이전 판		2020년 12월 28일 (월) 14:00 판
55번째 줄:		55번째 줄:
	* Moody, R. V., and J. Patera. “Quasicrystals and Icosians.” Journal of Physics. A. Mathematical and General 26, no. 12 (1993): 2829–53.		* Moody, R. V., and J. Patera. “Quasicrystals and Icosians.” Journal of Physics. A. Mathematical and General 26, no. 12 (1993): 2829–53.
	* Wilson, Robert A. "The geometry of the Hall-Janko group as a quaternionic reflection group." Geometriae Dedicata 20.2 (1986): 157-173.		* Wilson, Robert A. "The geometry of the Hall-Janko group as a quaternionic reflection group." Geometriae Dedicata 20.2 (1986): 157-173.
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q17058942 Q17058942]

@@ 5번째 줄: / 5번째 줄: @@
 ==아이코시안 군==
 * 사원수의 부분군으로 크기는 120이며 다음과 같은 벡터의 좌표에 짝치환을 적용하여 얻어지는 원소들로 구성
-* 8개 $\frac{1}{2}(\pm 2,0,0,0)$
+* 8개 <math>\frac{1}{2}(\pm 2,0,0,0)</math>
-* 16개 $\frac{1}{2}(\pm 1,\pm 1,\pm 1,\pm 1)$
+* 16개 <math>\frac{1}{2}(\pm 1,\pm 1,\pm 1,\pm 1)</math>
-* 96개 $\frac{1}{2}(0,\pm 1,\pm \sigma,\pm \tau)$, 여기서 $\sigma=\frac{1-\sqrt{5}}{2},\tau=\frac{1+\sqrt{5}}{2}$
+* 96개 <math>\frac{1}{2}(0,\pm 1,\pm \sigma,\pm \tau)</math>, 여기서 <math>\sigma=\frac{1-\sqrt{5}}{2},\tau=\frac{1+\sqrt{5}}{2}</math>
 ==아이코시안 환==
-* 계수를 $\mathbb{Z}[\sqrt{5}]$에서 갖는 사원수들이 이루는 환 $\mathscr{I}$
+* 계수를 <math>\mathbb{Z}[\sqrt{5}]</math>에서 갖는 사원수들이 이루는 환 <math>\mathscr{I}</math>
 * [[E8]] 격자에 isometric
-==아이코시안과 $E_8$==
+==아이코시안과 <math>E_8</math>==
-* 아이코시안 군 $G$는 120개의 원소를 갖는다
+* 아이코시안 군 <math>G</math>는 120개의 원소를 갖는다
-* $\sigma G:=\{\sigma s|s\in G\}$라 두자. 여기서 $\sigma=\frac{1-\sqrt{5}}{2}$
+* <math>\sigma G:=\{\sigma s|s\in G\}</math>라 두자. 여기서 <math>\sigma=\frac{1-\sqrt{5}}{2}</math>
-* $G\cup \sigma G$의 240개 원소와 [[E8 루트 시스템]] 사이에 일대일대응이 존재하며, 이는 아이코시안 환의 유클리드 norm에 대하여 등장(isometric)이다
+* <math>G\cup \sigma G</math>의 240개 원소와 [[E8 루트 시스템]] 사이에 일대일대응이 존재하며, 이는 아이코시안 환의 유클리드 norm에 대하여 등장(isometric)이다
-* $G\cup \sigma G$의 원소들을 다음 8개의 원소들을 이용하여 정수계수 선형결합으로 쓰면, 아래의 테이블을 얻는다
+* <math>G\cup \sigma G</math>의 원소들을 다음 8개의 원소들을 이용하여 정수계수 선형결합으로 쓰면, 아래의 테이블을 얻는다
-$$
+:<math>
 \{\sigma \{-1,\tau ,-\sigma ,0\},\{-\tau ,1,0,\sigma \},\{0,-\tau ,1,-\sigma \},\{0,-\sigma ,-\tau ,1\},-\{-1,-\sigma ,0,\tau \},\sigma \{-\sigma ,-\tau ,0,-1\},\sigma \{0,-\sigma ,-\tau ,1\},-\{-1,-\tau ,\sigma ,0\}\}
-$$
+</math>
 ===테이블===
 [[파일:아이코시안 (icosian)1.png]]

@@ 12번째 줄: / 12번째 줄: @@
 ==아이코시안 환==
-* 계수를 $\mathbb{Z}[\sqrt{5}]$에서 갖는 사원수들이 이루는 환
+* 계수를 $\mathbb{Z}[\sqrt{5}]$에서 갖는 사원수들이 이루는 환 $\mathscr{I}$
 * [[E8]] 격자에 isometric
@@ 18번째 줄: / 18번째 줄: @@
 ==아이코시안과 $E_8$==
-* 아이코시안 군 $\mathscr{I}$의 원소 120개를 갖는다
+* 아이코시안 군 $G$는 120개의 원소를 갖는다
-* $\sigma \mathscr{I}:=\{\sigma s|s\in \mathscr{I}\}$라 두자. 여기서 $\sigma=\frac{1-\sqrt{5}}{2}$
+* $\sigma G:=\{\sigma s|s\in G\}$라 두자. 여기서 $\sigma=\frac{1-\sqrt{5}}{2}$
-* $\mathscr{I}\cup \sigma \mathscr{I}$의 240개 원소와 [[E8 루트 시스템]] 사이에 일대일대응이 존재하며, 이는 아이코시안 환의 유클리드 norm에 대하여 등장(isometric)이다
+* $G\cup \sigma G$의 240개 원소와 [[E8 루트 시스템]] 사이에 일대일대응이 존재하며, 이는 아이코시안 환의 유클리드 norm에 대하여 등장(isometric)이다
-* $\mathscr{I}\cup \sigma \mathscr{I}$의 원소들을 다음 8개의 원소들을 이용하여 정수계수 선형결합으로 쓰면, 아래의 테이블을 얻는다
+* $G\cup \sigma G$의 원소들을 다음 8개의 원소들을 이용하여 정수계수 선형결합으로 쓰면, 아래의 테이블을 얻는다
 $$
 \{\sigma \{-1,\tau ,-\sigma ,0\},\{-\tau ,1,0,\sigma \},\{0,-\tau ,1,-\sigma \},\{0,-\sigma ,-\tau ,1\},-\{-1,-\sigma ,0,\tau \},\sigma \{-\sigma ,-\tau ,0,-1\},\sigma \{0,-\sigma ,-\tau ,1\},-\{-1,-\tau ,\sigma ,0\}\}

@@ 21번째 줄: / 21번째 줄: @@
 * $\sigma \mathscr{I}:=\{\sigma s|s\in \mathscr{I}\}$라 두자. 여기서 $\sigma=\frac{1-\sqrt{5}}{2}$
 * $\mathscr{I}\cup \sigma \mathscr{I}$의 240개 원소와 [[E8 루트 시스템]] 사이에 일대일대응이 존재하며, 이는 아이코시안 환의 유클리드 norm에 대하여 등장(isometric)이다
 ===테이블===
 [[파일:아이코시안 (icosian)1.png]]

← 이전 판		2014년 7월 8일 (화) 11:46 판
14번째 줄:		14번째 줄:
	* 계수를 $\mathbb{Z}[\sqrt{5}]$에서 갖는 사원수들이 이루는 환		* 계수를 $\mathbb{Z}[\sqrt{5}]$에서 갖는 사원수들이 이루는 환
	* [[E8]] 격자에 isometric		* [[E8]] 격자에 isometric
		+
		+
		+
		+	==아이코시안과 $E_8$==
		+	* 아이코시안 군 $\mathscr{I}$의 원소 120개를 갖는다
		+	* $\sigma \mathscr{I}:=\{\sigma s\|s\in \mathscr{I}\}$라 두자. 여기서 $\sigma=\frac{1-\sqrt{5}}{2}$
		+	* $\mathscr{I}\cup \sigma \mathscr{I}$의 240개 원소와 [[E8 루트 시스템]] 사이에 일대일대응이 존재하며, 이는 아이코시안 환의 유클리드 norm에 대하여 등장(isometric)이다
		+	===테이블===
		+	$$
		+	\begin{array}{cccc}
		+	& \text{icosian} & \text{vector in }E_8 & \text{Dynkin label} \\
		+	\hline
		+	1 & \{1,0,0,0\} & \{1,0,0,0,1,0,0,0\} & \{0,0,1,1,1,0,0,1\} \\
		+	2 & \{0,1,0,0\} & \{0,1,0,0,0,1,0,0\} & \{0,1,1,1,1,1,0,1\} \\
		+	3 & \{0,0,1,0\} & \{0,0,1,0,0,0,1,0\} & \{0,1,2,1,1,1,1,1\} \\
		+	4 & \{0,0,0,1\} & \{0,0,0,1,0,0,0,1\} & \{2,4,6,5,3,2,1,3\} \\
		+	5 & \{-1,0,0,0\} & \{-1,0,0,0,-1,0,0,0\} & \{0,0,-1,-1,-1,0,0,-1\} \\
		+	6 & \{0,-1,0,0\} & \{0,-1,0,0,0,-1,0,0\} & \{0,-1,-1,-1,-1,-1,0,-1\} \\
		+	7 & \{0,0,-1,0\} & \{0,0,-1,0,0,0,-1,0\} & \{0,-1,-2,-1,-1,-1,-1,-1\} \\
		+	8 & \{0,0,0,-1\} & \{0,0,0,-1,0,0,0,-1\} & \{-2,-4,-6,-5,-3,-2,-1,-3\} \\
		+	9 & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,3,5,4,3,2,1,3\} \\
		+	10 & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-3,-5,-4,-3,-2,-1,-3\} \\
		+	11 & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-1,-1,-1,0,0,0,0\} \\
		+	12 & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,4,3,2,1,1,2\} \\
		+	13 & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,3,2,1,0,2\} \\
		+	14 & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,3,4,3,2,2,1,2\} \\
		+	15 & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-2,-3,-2,-1,-1,-1,-1\} \\
		+	16 & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-2,-2,-2,-1,-1,0,-1\} \\
		+	17 & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,1,2,2,1,0,0,1\} \\
		+	18 & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-1,-2,-2,-1,0,0,-1\} \\
		+	19 & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,2,2,2,1,1,0,1\} \\
		+	20 & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,2,1,1,1,1\} \\
		+	21 & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-3,-4,-3,-2,-2,-1,-2\} \\
		+	22 & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-2,-4,-3,-2,-1,-1,-2\} \\
		+	23 & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-2,-3,-3,-2,-1,0,-2\} \\
		+	24 & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \left\{-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,1,1,1,0,0,0,0\} \\
		+	25 & \left\{0,\frac{1}{2},\frac{\sigma }{2},\frac{\tau }{2}\right\} & \{0,0,0,1,0,1,0,0\} & \{0,1,2,2,1,1,0,1\} \\
		+	26 & \left\{0,\frac{\tau }{2},\frac{1}{2},\frac{\sigma }{2}\right\} & \{0,1,0,0,0,0,1,0\} & \{0,1,1,1,1,1,1,1\} \\
		+	27 & \left\{0,\frac{\sigma }{2},\frac{\tau }{2},\frac{1}{2}\right\} & \{0,0,1,0,0,0,0,1\} & \{2,4,6,4,3,2,1,3\} \\
		+	28 & \left\{\frac{1}{2},0,\frac{\tau }{2},\frac{\sigma }{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,2,2,1,1,2\} \\
		+	29 & \left\{\frac{\sigma }{2},0,\frac{1}{2},\frac{\tau }{2}\right\} & \{0,0,1,1,0,0,0,0\} & \{0,1,2,1,0,0,0,1\} \\
		+	30 & \left\{\frac{\tau }{2},0,\frac{\sigma }{2},\frac{1}{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,4,4,3,2,1,2\} \\
		+	31 & \left\{\frac{1}{2},\frac{\sigma }{2},0,\frac{\tau }{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,2,4,3,2,1,0,2\} \\
		+	32 & \left\{\frac{\tau }{2},\frac{1}{2},0,\frac{\sigma }{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,3,3,2,1,2\} \\
		+	33 & \left\{\frac{\sigma }{2},\frac{\tau }{2},0,\frac{1}{2}\right\} & \{0,1,0,1,0,0,0,0\} & \{0,1,1,1,0,0,0,1\} \\
		+	34 & \left\{\frac{1}{2},\frac{\tau }{2},\frac{\sigma }{2},0\right\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-1,-1,0,0,0,0,0\} \\
		+	35 & \left\{\frac{\sigma }{2},\frac{1}{2},\frac{\tau }{2},0\right\} & \{0,1,1,0,0,0,0,0\} & \{0,1,1,0,0,0,0,1\} \\
		+	36 & \left\{\frac{\tau }{2},\frac{\sigma }{2},\frac{1}{2},0\right\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,4,3,3,2,1,2\} \\
		+	37 & \left\{0,-\frac{1}{2},\frac{\sigma }{2},\frac{\tau }{2}\right\} & \{0,-1,0,1,0,0,0,0\} & \{0,0,1,1,0,0,0,0\} \\
		+	38 & \left\{0,\frac{\tau }{2},-\frac{1}{2},\frac{\sigma }{2}\right\} & \{0,1,-1,0,0,0,0,0\} & \{0,0,-1,0,0,0,0,0\} \\
		+	39 & \left\{0,\frac{\sigma }{2},\frac{\tau }{2},-\frac{1}{2}\right\} & \{0,0,1,-1,0,0,0,0\} & \{0,0,0,-1,0,0,0,0\} \\
		+	40 & \left\{-\frac{1}{2},0,\frac{\tau }{2},\frac{\sigma }{2}\right\} & \left\{-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,2,1,1,1,1,1\} \\
		+	41 & \left\{\frac{\sigma }{2},0,-\frac{1}{2},\frac{\tau }{2}\right\} & \{0,0,0,1,0,0,-1,0\} & \{0,0,0,0,-1,-1,-1,0\} \\
		+	42 & \left\{\frac{\tau }{2},0,\frac{\sigma }{2},-\frac{1}{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-2,-2,-1,0,0,0,-1\} \\
		+	43 & \left\{-\frac{1}{2},\frac{\sigma }{2},0,\frac{\tau }{2}\right\} & \left\{-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,2,1,1,0,1\} \\
		+	44 & \left\{\frac{\tau }{2},-\frac{1}{2},0,\frac{\sigma }{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,1,2,2,2,1,1,1\} \\
		+	45 & \left\{\frac{\sigma }{2},\frac{\tau }{2},0,-\frac{1}{2}\right\} & \{0,1,0,0,0,0,0,-1\} & \{-2,-3,-5,-4,-3,-2,-1,-2\} \\
		+	46 & \left\{-\frac{1}{2},\frac{\tau }{2},\frac{\sigma }{2},0\right\} & \left\{-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-1,-2,-1,-1,0,0,-1\} \\
		+	47 & \left\{\frac{\sigma }{2},-\frac{1}{2},\frac{\tau }{2},0\right\} & \{0,0,1,0,0,-1,0,0\} & \{0,0,0,-1,-1,-1,0,0\} \\
		+	48 & \left\{\frac{\tau }{2},\frac{\sigma }{2},-\frac{1}{2},0\right\} & \left\{\frac{1}{2},-\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,1,2,2,2,1,0,1\} \\
		+	49 & \left\{0,\frac{1}{2},-\frac{\sigma }{2},\frac{\tau }{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,2,1,1,0,2\} \\
		+	50 & \left\{0,\frac{\tau }{2},\frac{1}{2},-\frac{\sigma }{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2}\right\} & \{-1,-1,-1,-1,-1,0,0,0\} \\
		+	51 & \left\{0,-\frac{\sigma }{2},\frac{\tau }{2},\frac{1}{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,2,1,1,1,2\} \\
		+	52 & \left\{\frac{1}{2},0,\frac{\tau }{2},-\frac{\sigma }{2}\right\} & \{1,0,1,0,0,0,0,0\} & \{0,0,1,0,0,0,0,1\} \\
		+	53 & \left\{-\frac{\sigma }{2},0,\frac{1}{2},\frac{\tau }{2}\right\} & \left\{\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,4,3,2,2,1,2\} \\
		+	54 & \left\{\frac{\tau }{2},0,-\frac{\sigma }{2},\frac{1}{2}\right\} & \{1,0,0,0,0,0,0,1\} & \{2,3,5,4,3,2,1,3\} \\
		+	55 & \left\{\frac{1}{2},-\frac{\sigma }{2},0,\frac{\tau }{2}\right\} & \{1,0,0,1,0,0,0,0\} & \{0,0,1,1,0,0,0,1\} \\
		+	56 & \left\{\frac{\tau }{2},\frac{1}{2},0,-\frac{\sigma }{2}\right\} & \{1,0,0,0,0,1,0,0\} & \{0,0,1,1,1,1,0,1\} \\
		+	57 & \left\{-\frac{\sigma }{2},\frac{\tau }{2},0,\frac{1}{2}\right\} & \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,3,2,2,1,2\} \\
		+	58 & \left\{\frac{1}{2},\frac{\tau }{2},-\frac{\sigma }{2},0\right\} & \{1,1,0,0,0,0,0,0\} & \{0,0,0,0,0,0,0,1\} \\
		+	59 & \left\{-\frac{\sigma }{2},\frac{1}{2},\frac{\tau }{2},0\right\} & \left\{\frac{1}{2},\frac{1}{2},\frac{1}{2},-\frac{1}{2},-\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right\} & \{1,2,3,2,2,2,1,2\} \\
		+	60 & \left\{\frac{\tau }{2},-\frac{\sigma }{2},\frac{1}{2},0\right\} & \{1,0,0,0,0,0,1,0\} & \{0,0,1,1,1,1,1,1\}
		+	\end{array}
		+	$$

← 이전 판		2014년 7월 1일 (화) 14:36 판
4번째 줄:		4번째 줄:

	==아이코시안 군==		==아이코시안 군==
−	* 사원수의 부분군으로 크기는 ~~120~~	+	* 사원수의 부분군으로 크기는 120이며 다음과 같은 벡터의 좌표에 짝치환을 적용하여 얻어지는 원소들로 구성
		+	* 8개 $\frac{1}{2}(\pm 2,0,0,0)$
		+	* 16개 $\frac{1}{2}(\pm 1,\pm 1,\pm 1,\pm 1)$
		+	* 96개 $\frac{1}{2}(0,\pm 1,\pm \sigma,\pm \tau)$, 여기서 $\sigma=\frac{1-\sqrt{5}}{2},\tau=\frac{1+\sqrt{5}}{2}$

@@ 14번째 줄: / 14번째 줄: @@
 ==메모==
 * [http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Icosian/ The `Icosian Calculus' By William R. Hamilton]
 * http://mathoverflow.net/questions/117604/a-non-commutative-ring-from-su2
 * http://www.tony5m17h.net/E8H4H4.pdf