Beveled dodecahedron

3D view of a beveled dodecahedron ( animation )

Section of a room corner of the beveled dodecahedron. The white lines delimit the chordal pentagon (see below).

The beveled dodecahedron (Dodecaedron simum) is a polyhedron ( polyhedron ) that is one of the Archimedean solids . It is composed of 92 surfaces, namely 12 regular pentagons and 80 equilateral triangles , and has 60 corners and 150 edges. Four triangles and one pentagon each form a corner of the room .

The following pictures show two beveled dodecahedra that are mirror images of each other.

Mirror variant 1
Mirror variant 2

The body that is dual to the beveled dodecahedron is the pentagon hexacontahedron .

construction

Transformation of a rhombicosidodecahedron into a beveled dodecahedron

As the name suggests, this polyhedron is created by continually chamfering a dodecahedron , so that at the end twelve (smaller) regular pentagons remain, which are coincident with the original boundary surfaces of the dodecahedron.

If you twist all twelve pentagons of a rhombicosidodecahedron - which are coincident with the boundary surfaces of a circumscribed dodecahedron - by the same specific angle and insert a diagonal into the now distorted squares, a beveled dodecahedron also results.

Formulas

In the following, the term denotes the cosine of the smaller central angle in the chordal pentagon (see graphic above right) with the side lengths and ( denotes the diagonal in the pentagon , denotes the golden number ). ${\ displaystyle t}$ ${\ displaystyle \ zeta}$ ${\ displaystyle a}$ ${\ displaystyle d = \ varphi \, a}$ ${\ displaystyle d}$ ${\ displaystyle \ varphi}$

${\ displaystyle t}$ is the only real solution to the cubic equation . ${\ displaystyle 8t ^ {3} + 8t ^ {2} - \ varphi ^ {2} = 0}$

{\ displaystyle t = \ cos (\ zeta) = {\ frac {1} {12}} \ left ({\ sqrt [{3}] {44 + 12 \ varphi \, (9 + {\ sqrt {81 \ varphi -15}})}} + {\ sqrt [{3}] {44 + 12 \ varphi \, (9 - {\ sqrt {81 \ varphi -15}})}} - 4 \ right)}

Sizes of a beveled dodecahedron with edge length a
volume	${\ displaystyle V = {\ frac {a ^ {3}} {6 {\ sqrt {1-2t}}}} \ left (3 {\ sqrt {10 \, (9t-2 + (4t-1) { \ sqrt {5}})}} + 20 {\ sqrt {2 + 2t}} \ right)}$
Surface area	${\ displaystyle A_ {O} = a ^ {2} \ left (20 {\ sqrt {3}} + 3 {\ sqrt {25 + 10 {\ sqrt {5}}}} \ right)}$
Umkugelradius	${\ displaystyle R = {\ frac {a} {2}} {\ sqrt {\ frac {2-2t} {1-2t}}}}$
Edge ball radius	${\ displaystyle r = {\ frac {a} {2 {\ sqrt {1-2t}}}}}$
1. Diagonal angle ( trine - trigon) ≈ 164 ° 10 ′ 31 ″	${\ displaystyle \ cos \, \ alpha _ {1} = - {\ frac {1} {3}} \ left (1 + 4t \ right)}$
2. Face angle ( Pentagon –Trigon) ≈ 152 ° 55 ′ 48 ″	${\ displaystyle \ cos \, \ alpha _ {2} = {\ frac {1} {\ sqrt {15}}} \ left ((1-2t) {\ sqrt {5 + 2 {\ sqrt {5}} }} - 2 {\ sqrt {(1 + t) (5t + (2t-1) {\ sqrt {5}})}} \ right)}$
Surface-edge-angle (Pentagon – Trigon) ≈ 143 ° 20 ′ 58 ″	${\ displaystyle \ cos \, \ beta = {\ frac {-4t} {\ sqrt {10-2 {\ sqrt {5}}}}}}$
3D edge angle (trine – trine) ≈ 118 ° 8 ′ 12 ″	${\ displaystyle \ cos \, \ gamma = -t}$
Corners solid angle ≈ 1.4355 π	${\ displaystyle \ Omega = \, 3 \ alpha _ {1} +2 \ alpha _ {2} -3 \ pi}$