Snub Cube

Off[General::spell];
\(\(Off[General::spelll];\)
 (*\ 
      1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11\ *) \)

Three Square distances of Snub Cube vertices

Three square distances of neighboured vertices of a Snub Cube
get computed; on each face of the Cube lays a twisted square,
which here gets represented by the absolute length a to walk
along an edge of the cube,the absolute length b to walk into
the cube face,parallel to an other edge of the face and the
absolute length c=2-a-b.The Cube,on which the Snub Cube
gets constructed,has edge-length 2.The three distances
between two Snub Cube vertices on one Cube face and a third
vertex on a neighboured Cube face must be equal.Here are this
three square distances computed,dependent to the values of a,b,c and the square distance dSquare.

Search for the solutions (a,b,c,dSquare) of this equations.

Solve[{a + b + c == 2,  (((b - a))^2) + (c^2) == dSquare,  
        2*(a^2) + (((2 - 2*b))^2) == dSquare,  
        (a^2) + (b^2) + (((b - a))^2) == dSquare}, {a, b, c, 
        dSquare}]

There is one real-valued solution of the three equations.

dSquare = 
        (2/9) (-18 + ((32 (2^(1/3)))/(((-13 + 3 (Sqrt[33])))^(2/3))) + ((4 (2^(2/3)))/(((-13 + 3 (Sqrt[33])))^(1/3))) - (((2 (-13 + 3 (Sqrt[33]))))^(1/3)) + (((2 (-13 + 3 (Sqrt[33]))))^(2/3)));
\(		\(c = 
        1\/18\ \((\(-12\) + \(32\ 2\^\(1/3\)\)\/\((\(-13\) + 3\ \@33)\)\^\(2/3\) + \(16\ 2\^\(2/3\)\)\/\((\(-13\) + 3\ \@33)\)\^\(1/3\) - 
              4\ \((2\ \((\(-13\) + 3\ \@33)\))\)\^\(1/3\) + \((2\ \((\(-13\) + 3\ \@33)\))\)\^\(2/3\))\);\)\)
\(		\(a = 
        1\/18\ \((24 - \(32\ 2\^\(1/3\)\)\/\((\(-13\) + 3\ \@33)\)\^\(2/3\) + \(8\ 2\^\(2/3\)\)\/\((\(-13\) + 3\ \@33)\)\^\(1/3\) - 
              2\ \((2\ \((\(-13\) + 3\ \@33)\))\)\^\(1/3\) - \((2\ \((\(-13\) + 3\ \@33)\))\)\^\(2/3\))\);\)\)
\(		\(b = 
        4\/3 - \(4\ 2\^\(2/3\)\)\/\(3\ \((\(-13\) + 3\ \@33)\)\^\(1/3\)\) + 
          1\/3\ \((2\ \((\(-13\) + 3\ \@33)\))\)\^\(1/3\);\)\)

Geometrical data for the Snub Cube

Computate geometrical data of Snub Cube

e  =  (1/2) (Sqrt[dSquare]);
f = (Sqrt[1 + 2  (e^2)]);
g = 1;
h = (Sqrt[1 + (e^2)]);
k = (Sqrt[(h^2) - (1/3) (e^2)]);

The Snub Cube gets scaled to vertex radius 1.

aSC  =  a/f;
bSC  =  b/f;
cSC  =  c/f;
fSC  =  1;
eSC  =  e/f;
gSC  =  g/f;
hSC  =  h/f;
kSC  =  k/f;

Half edge-length

Simplify[eSC]
N[eSCn, 32]

Distance from vertex to center

Simplify[fSC]
N[fSCn, 32]

Distance from quadrilateral mid-point to center

Simplify[gSC]
N[gSCn, 32]

Distance from edge mid-point to center

Simplify[hSC]
N[hSCn, 32]

Distance from triangle mid-point to center

Simplify[kSC]
N[kSCn, 32]

Coordinates

All vertices have one coordinate with the absolute value of the distance
from quadrilateral mid-point to center; the other two coordinates have
the absolute value of the following two terms.

Simplify[gSC - aSC]
N[gSC - aSC, 32]

Simplify[gSC - bSC]
N[gSC - bSC, 32]