Pentagonikositetrahedron

3D view of a Pentagonikositetrahedron ( animation )

Net of the Pentagonikositetrahedron

The pentagonikositetrahedron is a chiral polyhedron , which is composed of 24 irregular pentagons and is one of the Catalan solids . It is dual to the beveled hexahedron and has 38 corners and 60 edges.

The following pictures show two pentagonicosite tetrahedra that are mirror images of each other.

Mirror variant 1
Mirror variant 2

Emergence

Construction of the tangent pentagon on the beveled hexahedron

Left and right handed version of a Pentagonikositetrahedron (cardboard models)

By connecting the centers of five edges that meet in each corner of the beveled hexahedron, a chordal pentagon is created , the circumference of which is at the same time an inscribed circle of the tangent pentagon , the boundary surface of the pentagonikosite tetrahedron. In this special type, all face angles are the same (≈ 136 °) and there is a uniform edge sphere radius .

In the following, the term denotes the cosine of the smaller central angle in the aforementioned chordal pentagon. ${\ displaystyle t}$ ${\ displaystyle \ zeta}$

{\ displaystyle t = \ cos \, \ zeta = {\ frac {1} {6}} \ left ({\ sqrt [{3}] {19 + 3 {\ sqrt {33}}}} + {\ sqrt [{3}] {19-3 {\ sqrt {33}}}} - 2 \ right)}

Let be the edge length of the beveled hexahedron, then the resulting side lengths of the tangent pentagon are given by ${\ displaystyle d}$

{\ displaystyle a = {\ frac {d} {2}} {\ sqrt {2 + 2t}}}

{\ displaystyle b = {\ frac {d} {\ sqrt {2 + 2t}}}}

It follows:

{\ displaystyle a = b \, (1 + t)}

Related polyhedra

Dual body: beveled hexahedron
Inscribed cube
Inscribed octahedron

Formulas

For the polyhedron

Sizes of a Pentagonikositetrahedron with edge length a or b
Volume ≈ 12.45a ³ ≈ 35.63b ³	${\ displaystyle V = {\ frac {4a ^ {3} (2 + 3t) {\ sqrt {1-2t}}} {(1 + t) (1-4t ^ {2})}} = {\ frac {2b ^ {3} (1 + t) (2 + 3t)} {(1-2t ^ {2}) {\ sqrt {1-2t}}}}}$
Surface area ≈ 27.19a ² ≈ 54.8b ²	${\ displaystyle A_ {O} = {\ frac {24a ^ {2} (2 + 3t)} {1 + 2t}} {\ sqrt {\ frac {1-t} {1 + t}}} = {\ frac {12b ^ {2} (2 + 3t)} {(1-2t ^ {2})}} {\ sqrt {1-t ^ {2}}}}$
Edge ball radius	${\ displaystyle r = {\ frac {a} {\ sqrt {2 \, (1 + t) (1-2t)}}} = b \, {\ sqrt {\ frac {1 + t} {2 \, (1-2t)}}}}$
Inc sphere radius	${\ displaystyle \ rho = {\ frac {a} {2 {\ sqrt {(1-2t) (1-t ^ {2})}}}} = {\ frac {b} {2}} \, { \ sqrt {\ frac {1 + t} {(1-t) (1-2t)}}}}$
Face angle ≈ 136 ° 18 ′ 33 ″	${\ displaystyle \ cos \, \ alpha = {\ frac {t} {t-1}}}$

For the boundary surfaces

Sizes in the pentagon of the Pentagonikositetrahedron

Sizes of the tangent pentagon
Area	${\ displaystyle A = {\ frac {a ^ {2} (2 + 3t)} {1 + 2t}} {\ sqrt {\ frac {1-t} {1 + t}}} = {\ frac {b ^ {2} (2 + 3t)} {2 \, (1-2t ^ {2})}} {\ sqrt {1-t ^ {2}}}}$
Inscribed radius	${\ displaystyle r = {\ frac {a} {2 {\ sqrt {1-t ^ {2}}}}} = {\ frac {b} {2}} \, {\ sqrt {\ frac {1+ t} {1-t}}}}$
diagonal ${\ displaystyle \ \| \, b}$	${\ displaystyle e = 2a \, (1-2t ^ {2}) = b \, (1 + 2t)}$
Obtuse angles (4) ≈ 114 ° 48 ′ 43 ″	${\ displaystyle \ cos \, \ alpha = -t}$
Acute angle (1) ≈ 80 ° 45 ′ 6 ″	${\ displaystyle \ cos \, \ beta = 1-2t}$

Remarks

↑ t is the only real solution of the cubic equation 4t ³ + 4t ² - 1 = 0. If the number 1 is added to twice the value of t , the Tribonacci constant is obtained, which is the limit of the ratio (= 1.83928675521416 ...) represents two consecutive numbers of this sequence.
↑ With a > the longer of the two sides of the Pentagonikositetraeders be designated.
↑ These formulas only apply to the case b = a: (1 + t) or equivalent to a = b (1 + t)
↑ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ This formula also applies to the pentagon hexacontahedron and the pentagon dodecahedron , provided that the corresponding values are used for b (short side length), n (number of boundary surfaces) and t (cosine of the smaller central angle) and furthermore note that O = n · A and V = 1/3 · O · ρ.

Web links

Commons : Pentagonikositetrahedron - collection of images, videos and audio files

Wiktionary: Pentagonikositetrahedron - explanations of meanings, word origins, synonyms, translations

Eric W. Weisstein : Pentagonikositetrahedron . In: MathWorld (English).

[1] t is the only real solution of the cubic equation 4t ³ + 4t ² - 1 = 0. If the number 1 is added to twice the value of t , the Tribonacci constant is obtained, which is the limit of the ratio (= 1.83928675521416 ...) represents two consecutive numbers of this sequence.

[2] With a > the longer of the two sides of the Pentagonikositetraeders be designated.

[3] These formulas only apply to the case b = a: (1 + t) or equivalent to a = b (1 + t)

[tformel-4] ↑ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ This formula also applies to the pentagon hexacontahedron and the pentagon dodecahedron , provided that the corresponding values are used for b (short side length), n (number of boundary surfaces) and t (cosine of the smaller central angle) and furthermore note that O = n · A and V = 1/3 · O · ρ.