Some Archimedean Solids in the Icosahedral Lattice

size of L

1

2

convex hull (or points)

icosahedron

dodecahedron

icosidodecahedron

truncated dodecahedron

truncated icosahedon

manual choice

i

-2

-1

0

1

2

j

-2

-1

0

1

2

k

-2

-1

0

1

2

l

-2

-1

0

1

2

m

-2

-1

0

1

2

n

-2

-1

0

1

2

show red point

This Demonstration considers a set of points

L

of integral linear combinations

i

e

1

+j

e

2

+k

e

3

+l

e

4

+m

e

5

+n

e

6

, where the

e

i

are six vertices of a regular icosahedron and the coefficients

i

to

n

are integers between

-1

and

1

or

-2

and

2

, for a total of

6

3

=729

or

6

5

=15625

points. The coordinates of the

e

i

are the six permutations of

0

,

1

, and

ϕ=

1+

5

2

(the golden ratio). The convex hull of

L

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

L

that are at the same distance as the given one.

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