# Some Archimedean Solids in the Icosahedral Lattice

Some Archimedean Solids in the Icosahedral Lattice

This Demonstration considers a set of points of integral linear combinations , where the are six vertices of a regular icosahedron and the coefficients to are integers between and or and , for a total of or points. The coordinates of the are the six permutations of , , and (the golden ratio). The convex hull of is a triacontahedron. Certain choices of linear combinations give the vertices of a dodecahedron, an icosidodecahedron, a truncated dodecahedron, and a truncated icosahedon. Given one vertex on a solid, all the other vertices are points in that are at the same distance as the given one.

L

ie+je+ke+le+me+ne

1

2

3

4

5

6

e

i

i

n

-1

1

-2

2

3=729

6

5=15625

6

e

i

0

1

ϕ=

1+

5

2

L

L

To get a rhombicosidodecahedron or a great rhombicosidodecahedron, coefficients of absolute value 3 or 5 are needed.