WOLFRAM|DEMONSTRATIONS PROJECT

Exact Coordinates of Golden Rhombic Solids

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pick multiple wireframes
1
2
3
4
5
6
7
8
9
10
11
pick wireframes to be solid
1
2
3
4
5
6
7
8
9
10
11
study relations
icosahedron
prolate rhombohedron
hexecontahedron
rhombic 120-hedron B
rhombic 120-hedron
subdivided double
triacontahedron
double triacontahedron
triacontahedron
oblate rhombohedron
rhombic icosahedron
rhombic dodecahedron
of the second kind
icosahedron
vertex index and its coordinates
vertex indicesfor face
1
{0,τ,1}
2
{0,τ,-1}
3
{0,-τ,-1}
4
{0,-τ,1}
5
{1,0,τ}
6
{-1,0,τ}
7
{-1,0,-τ}
8
{1,0,-τ}
9
{τ,1,0}
10
{τ,-1,0}
11
{-τ,-1,0}
12
{-τ,1,0}
{1,9,2}
{1,2,12}
{1,6,5}
{1,5,9}
{1,12,6}
{2,8,7}
{2,7,12}
{2,9,8}
{3,10,4}
{3,4,11}
{3,7,8}
{3,11,7}
{3,8,10}
{4,5,6}
{4,10,5}
{4,6,11}
{5,10,9}
{6,12,11}
{7,11,12}
{8,9,10}
This Demonstration gives exact coordinates of various golden rhombic solids. Using three golden rectangles, the coordinates of the vertices of the regular icosahedron are
0
,
±1
, or
±τ
, where
τ=
1+
5
2
is the golden ratio. The coordinates of all of the golden rhombic solids are constructed by adding vectors, so their coordinates are of the form
a+bτ
, where
a
and
b
are integers.
The prolate rhombohedron is constructed using the origin and the vertices on one face of the icosahedron as its first four vertices. Twenty rhombohedra constructed in this way form the hexecontahedron, and the other solids are built up from there.