The Golden Ratio in Arrangements of an Icosahedron, Tetrahedron, and Octahedron

size of icosahedron

show octahedron

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show 12 icosahedra

A regular octahedron

O

is placed within a regular icosahedron

I

so that eight of its 20 faces lie in the faces of

O

.

I

, in turn, is contained within a regular tetrahedron

T

whose four faces contain four of the eight faces of

O

. The vertices of

I

divide the edges of

O

in the golden ratio

ϕ

. If an edge of

I

is extended in the ratio 1:

ϕ

, then its endpoint is on the edge of

T

that divides it in the golden ratio.