# 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 =729 or =15625 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

i+j+k+l+m+n

e

1

e

2

e

3

e

4

e

5

e

6

e

i

i

n

-1

1

-2

2

6

3

6

5

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.