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.