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! R H Hardin and N J A Sloane Oct 10 1994 The standard (or regular) version of the snub cube has 24 vertices which form the best packing of 24 points on the sphere in 3-D. However, these 24 points form only a spherical 3-design. A slight modification produces a spherical 7-design, so we call this a better snub cube! The standard snub cube can be obtained by drawing 6 swastikas on the faces of a cube (all with the same orientation) - see below. The improved version is obtained by slightly shrinking each swastika, pushing each one away from the center. (This is more easily visualized if the swastikas are drawn on a sphere instead of a cube. There is a swastika centered at each coordinate axis. 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.