A := .8662468181078205913835980: B := .4225186537611115291185464: C := .2666354015167047203315344: a[1] := [ -A, B, -C ]: a[4] := [ -A, -B, C ]: a[2] := [ -A, -C, -B ]: a[3] := [ -A, C, B ]: a[21] := [ A, C, -B ]: a[24] := [ A, -C, B ]: a[22] := [ A, B, C ]: a[23] := [ A, -B, -C ]: a[6] := [ -B, A, C ]: a[17] := [ B, A, -C ]: a[9] := [ -C, A, -B ]: a[14] := [ C, A, B ]: a[12]:= [ -C, -A, B ]: a[15] := [ C, -A, -B ]: a[7] := [ -B, -A, -C ]: a[20] := [ B, -A, C ]: a[8] := [ -B, -C, A ]: a[19] := [ B, C, A ]: a[11] := [ -C, B, A ]: a[16] := [ C, -B, A ]: a[5] := [ -B, C, -A ]: a[18] := [ B, -C, -A ]: a[13] := [ C, B, -A ]: a[10]:= [ -C, -B, -A ]: .8662468181078205913835980 .4225186537611115291185464 .2666354015167047203315344 .8662468181078205913835980 -.4225186537611115291185464 -.2666354015167047203315344 .8662468181078205913835980 .2666354015167047203315344 -.4225186537611115291185464 .8662468181078205913835980 -.2666354015167047203315344 .4225186537611115291185464 -.8662468181078205913835980 .4225186537611115291185464 -.2666354015167047203315344 -.8662468181078205913835980 -.4225186537611115291185464 .2666354015167047203315344 -.8662468181078205913835980 .2666354015167047203315344 .4225186537611115291185464 -.8662468181078205913835980 -.2666354015167047203315344 -.4225186537611115291185464 .2666354015167047203315344 .8662468181078205913835980 .4225186537611115291185464 -.2666354015167047203315344 .8662468181078205913835980 -.4225186537611115291185464 -.4225186537611115291185464 .8662468181078205913835980 .2666354015167047203315344 .4225186537611115291185464 .8662468181078205913835980 -.2666354015167047203315344 -.2666354015167047203315344 -.8662468181078205913835980 .4225186537611115291185464 .2666354015167047203315344 -.8662468181078205913835980 -.4225186537611115291185464 .4225186537611115291185464 -.8662468181078205913835980 .2666354015167047203315344 -.4225186537611115291185464 -.8662468181078205913835980 -.2666354015167047203315344 .4225186537611115291185464 .2666354015167047203315344 .8662468181078205913835980 -.4225186537611115291185464 -.2666354015167047203315344 .8662468181078205913835980 .2666354015167047203315344 -.4225186537611115291185464 .8662468181078205913835980 -.2666354015167047203315344 .4225186537611115291185464 .8662468181078205913835980 .4225186537611115291185464 -.2666354015167047203315344 -.8662468181078205913835980 -.4225186537611115291185464 .2666354015167047203315344 -.8662468181078205913835980 .2666354015167047203315344 .4225186537611115291185464 -.8662468181078205913835980 -.2666354015167047203315344 -.4225186537611115291185464 -.8662468181078205913835980 A New Spherical 24-point 7-design in 3 Dimensions or A Better Snub Cube!哈丁和NJ J斯隆1994 10月10日,Snub立方体的标准(或规则)版本有24个顶点,在3D-球面上形成24个点的最佳填充,然而,这24个点只形成球面3个设计。标准的Subbe立方体可以通过在立方体的面上画6个十字字(所有的方向都相同)来获得。(如果把万花筒画在一个球上而不是一个立方体上,这是更容易看到的。在每个坐标轴上有一个十字字中心。They each move along that axis , away from the center.) Here are the details. The new (non-regular) snub cube. -------------------------------- The 24 vertices are found as follows. Let a,b,c be the roots of the polynomial 105*Z^3-105*Z^2+21*Z-1 so that approximately a = .7503835498819236124134653 b = .1785220127761020457111678 c = .07109443734197434187536685 and let A = sqrt(a), B = sqrt(b), C = sqrt(c) so that approximately A := .8662468181078205913835980: B := .4225186537611115291185464: C := .2666354015167047203315344: with A^2+B^2+C^2=1. Then the 24 points are P[1] := [ -A, B, -C ]: P[4] := [ -A, -B, C ]: P[2] := [ -A, -C, -B ]: P[3] := [ -A, C, B ]: P[21] := [ A, C, -B ]: P[24] := [ A, -C, B ]: P[22] := [ A, B, C ]: P[23] := [ A, -B, -C ]: P[6] := [ -B, A, C ]: P[17] := [ B, A, -C ]: P[9] := [ -C, A, -B ]: P[14] := [ C, A, B ]: P[12]:= [ -C, -A, B ]: P[15] := [ C, -A, -B ]: P[7] := [ -B, -A, -C ]: P[20] := [ B, -A, C ]: P[8] := [ -B, -C, A ]: P[19] := [ B, C, A ]: P[11] := [ -C, B, A ]: P[16] := [ C, -B, A ]: P[5] := [ -B, C, -A ]: P[18] := [ B, -C, -A ]: P[13] := [ C, B, -A ]: P[10]:= [ -C, -B, -A ]: or in short ( A +-[B C] ) ( A +-[C -B] ) and the points you get from them by cycling, and by negating the 1st and 3rd coords. The group has order 24: on coords we have (1,2,3) (2,3).diag(1,1,-1) diag(-1,1,-1) diag(1,-1,-1) thus: cycle the coords and change the signs of an even number of coords, or apply an odd perm of the coords and change the sign of an odd number of coords # To check this, we apply the Reznick test for t-designs: n:=3; N:=24; t:=6; s:=3; sb:=2; # test a lhs1:=simplify((1/N)*sum( sum( b[k][i]*x[i],i=1..n )^(2*s), k=1..N )); i:='i': k:='k': rhs1:= expand( product( (1+2*j)/(n+2*j) , j=0..s-1 )* sum ( x[i]^2, i=1..n)^s ); i:='i': van:=expand(lhs1-rhs1); # test b lhs2:=simplify(sum( sum( b[k][i]*x[i],i=1..n )^(2*sb+1), k=1..N )); set A^2=a, B^2=b, C^2=c then a,b,c satisfy a^3+b^3+c^3 = 3/7 abc = 1/105 a^2*(b+c) + b^2*(a+c) + c^2*(a+b) = 6/35 so a is a root of 12 3 2 2 1795856326022129150390625 a (105 a - 105 a + 21 a - 1) 6 5 4 3 2 2 (11025 a + 11025 a + 8820 a + 1995 a + 336 a + 21 a + 1) that is, either 105*a^3-105*a^2+21*a-1 (yes) or 11025*a^6+11025*a^5+8820*a^4+1995*a^3+336*a^2+21*a+1 (no) which leads to the above coords The regular snub cube --------------------- The coords look much the same, namely start with (A,B,C), apply any even perm and change any even number of signs, or apply any odd perm and change any odd number of signs. (Again the group has order 24) So the 24 pts are A B C A -B -C A -C B A C -B -A -B C -A B -C -A C B -A -C -B and their cyclic shifts The first 4 points (also the last 4) form a swastika. A -C B *-| | * A B C | | --------- | | A -B -C * | |-* A C -B There are 6 swastikas in all. We take A,B,C to satisfy: A^2 + B^2 + C^2 = 1 Also we want squared edge-length = 2 - 2 A^2 = 2 - 2 (AB + BC + CA) = 2 + 2 C^2 - 4 AB so that A^2 = AB + BC + CA and then we find that A = (1 - B^2) / 2B C = (1-B^2)(1-3B^2)/ 2B(1+B^2) and finally 7 B^6 - 3 B^4 + 5 B^2 - 1 = 0 which has a unique positive real root at B = .4623206278... so A = .8503402075 C = .2513586457 The adjacencies in the snub cube are that A B C is adjacent to 5 other nodes, namely A -C B A C -B C A B B C A B A -C and the squared edge-length is .553843063, so the edge-length is .744206331 This is the best 24-packing in 3-D (theorem of Robinson) but it is a spherical 3-design only, not even a 4-design. Comparing the two bodies: in the new version, the swastikas are further from the center (A is bigger), but the arms (B and C) are shorter.