Icosahedron - Related Polyhedra and Polytopes

Related Polyhedra and Polytopes

The icosahedron can be transformed by a truncation sequence into its dual, the dodecahedron:

Family of uniform icosahedral polyhedra
{5,3} t0,1{5,3} t1{5,3} t0,1{3,5} {3,5} t0,2{5,3} t0,1,2{5,3} s{5,3}

As a snub tetrahedron, and alternation of a truncated octahedron it also exists in the tetrahedral and octahedral symmetry families:

Family of uniform tetrahedral polyhedra
{3,3} t0,1{3,3} t1{3,3} t1,2{3,3} t2{3,3} t0,2{3,3} t0,1,2{3,3} s{3,3}
Family of uniform octahedral polyhedra
{4,3} t0,1{4,3} t1{4,3} t0,1{3,4} {3,4} t0,2{4,3} t0,1,2{4,3} s{4,3} h0{4,3} h1,2{4,3}

This polyhedron is topologically related as a part of sequence of regular polyhedra with Schläfli symbols {3,n}, continuing into the hyperbolic plane.


{3,3}

{3,4}

{3,5}

{3,6}

{3,7}

{3,8}

{3,9}

The regular icosahedron, seen as a snub tetrahedron, is a member of a sequence of snubbed polyhedra and tilings with vertex figure (3.3.3.3.n) and Coxeter–Dynkin diagram . These figures and their duals have (n32) rotational symmetry, being in the Euclidean plane for n=6, and hyperbolic plane for any higher n. The series can be considered to begin with n=2, with one set of faces degenerated into digons.

Symmetry 232
+
D3
332
+
T
432
+
O
532
+
I
632
+
P6
732
+
832
+
Order 6 12 24 60
Snub
figure

3.3.3.3.2

3.3.3.3.3

3.3.3.3.4

3.3.3.3.5

3.3.3.3.6

3.3.3.3.7

3.3.3.3.8
Coxeter
Schläfli

s{2,3}

s{3,3}

s{4,3}

s{5,3}

s{6,3}

s{7,3}

s{8,3}
Snub
dual
figure

V3.3.3.3.2

V3.3.3.3.3

V3.3.3.3.4

V3.3.3.3.5

V3.3.3.3.6

V3.3.3.3.7
Coxeter

The icosahedron shares its vertex arrangement with three Kepler–Poinsot solids. The great dodecahedron also has the same edge arrangement.

Picture
Great dodecahedron

Small stellated dodecahedron

Great icosahedron
Coxeter-Dynkin

The icosahedron can tessellate hyperbolic space in the order-3 icosahedral honeycomb, with 3 icosahedra around each edge, 12 icosahedra around each vertex, with Schläfli symbol {3,5,3}. It is one of four regular tessellations in the hyperbolic 3-space.


It is shown here as an edge framework in a Poincaré disk model, with one icosahedron visible in the center.

Read more about this topic:  Icosahedron

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