Deltoidal hexacontahedron

Deltoidal hexacontahedron
Deltoidal hexacontahedron
	; ( rotating model , 3D model )
Type of	catalan body
Properties	convex , isohedral
Combinatorics
60 facets; 120 ribs; 62 peaks	Χ = 2
Verge of	deltoids :;
Vertex configuration	20 (4 3 ); 30 (4 4 ); 12 (4 5 )
Face configuration	V3.4.5.4
Dual Polyhedron	rhomboikosododecahedron
	Scan
Classification
Designations	oD, deD
Symmetry group	I h (icosahedral)

The deltoidal hexacontahedron (from “ deltoid ” and other Greek ἑξήκοντα - “sixty”, ἕδρα - “face”) is a semi-correct polyhedron (Catalan body), dual to the rhombo-dodeca-dodecahedron . Made up of 60 identical convex deltoids .

It has 62 vertices. At 12 vertices (located just as the vertices of the icosahedron ) converge with their smallest angles along 5 faces; at 20 vertices (located just as the vertices of the dodecahedron ) converge with their largest angles on 3 faces; in the remaining 30 vertices (located in the same way as the vertices of the icosododecahedron ) converge with their average largest angles of 4 faces.

12 vertices are located just like the vertices of the icosahedron
The 20 vertices are located just like the vertices of the dodecahedron
30 vertices are located just like the vertices of the icosododecahedron

It has 120 edges - 60 “long” (together forming something like “bloated” skeleton of icosahedron) and 60 “short” (forming “bloated” skeleton of dodecahedron).

The deltoidal hexacontahedron is one of the six Catalan bodies in which there is no Hamiltonian cycle ^[1] ; Hamiltonian path for all six also not.

Metric performance and angles

Edge of the deltoidal hexacontahedron

If the “short” edges of the deltoidal hexekontahedron have a length $b$ ${\ displaystyle b}$ , then its “long” edges have a length

a={\frac {1}{6}}\left(7+{\sqrt {5}}\right)b\approx 1{,}5393447b.

{\ displaystyle a = {\ frac {1} {6}} \ left (7 + {\ sqrt {5}} \ right) b \ approx 1 {,} 5393447b.}

The surface area and volume of the polyhedron are expressed as

S={\sqrt {10\left(437+185{\sqrt {5}}\right)}}\;b^{2}\approx 92{,}2319129b^{2},

{\ displaystyle S = {\ sqrt {10 \ left (437 + 185 {\ sqrt {5}} \ right)}} \; b ^ {2} \ approx 92 {,} 2319129b ^ {2},}

V={\frac {1}{3}}{\sqrt {2\left(14765+6602{\sqrt {5}}\right)}}\;b^{3}\approx 81{,}0041436b^{3}.

{\ displaystyle V = {\ frac {1} {3}} {\ sqrt {2 \ left (14765 + 6602 {\ sqrt {5}} \ right)}} \; b ^ {3} \ approx 81 {, } 0041436b ^ {3}.}

The radius of the inscribed sphere (relating to all faces of the polyhedron at their centers of the inscribed circles ) will be equal to

r={\frac {1}{2}}{\sqrt {{\frac {1}{205}}\left(2855+1269{\sqrt {5}}\right)}}\;b\approx 2{,}6347977b,

{\ displaystyle r = {\ frac {1} {2}} {\ sqrt {{\ frac {1} {205}} \ left (2855 + 1269 {\ sqrt {5}} \ right)}} \; b \ approx 2 {,} 6347977b,}

radius of the semi-written sphere (relating to all edges) -

\rho ={\frac {1}{20}}\left(25+13{\sqrt {5}}\right)b\approx 2{,}7034442b,

{\ displaystyle \ rho = {\ frac {1} {20}} \ left (25 + 13 {\ sqrt {5}} \ right) b \ approx 2 {,} 7034442b,}

radius of a circle inscribed on the face -

r_{\Gamma \mathrm {P} }={\sqrt {\rho ^{2}-r^{2}}}={\frac {1}{2}}{\sqrt {{\frac {1}{410}}\left(317+127{\sqrt {5}}\right)}}\;b\approx 0{,}6053525b,

{\ displaystyle r _ {\ Gamma \ mathrm {P}} = {\ sqrt {\ rho ^ {2} -r ^ {2}} = {\ frac {1} {2}} {\ sqrt {{\ frac {1} {410}} \ left (317 + 127 {\ sqrt {5}} \ right)}} \; b \ approx 0 {,} 6053525b,}

the smaller diagonal of the face (dividing the face into two isosceles triangles ) -

e={\sqrt {{\frac {1}{10}}\left(25+2{\sqrt {5}}\right)}}\;b\approx 1{,}7167451b,

{\ displaystyle e = {\ sqrt {{\ frac {1} {10}} \ left (25 + 2 {\ sqrt {5}} \ right)}} \; b \ approx 1 {,} 7167451b,}

larger diagonal of the face (dividing a face into two equal triangles) -

f={\frac {1}{3}}{\sqrt {{\frac {1}{5}}\left(75+31{\sqrt {5}}\right)}}\;b\approx 1{,}7908292b.

{\ displaystyle f = {\ frac {1} {3}} {\ sqrt {{\ frac {1} {5}} \ left (75 + 31 {\ sqrt {5}} \ right)}} \; b \ approx 1 {,} 7908292b.}

It is impossible to describe a sphere near the deltoidal hexekontahedron — so that it passes through all the vertices.

The largest angle of the face (between the two "short" sides) is $\arccos \left(-{\frac {5+2{\sqrt {5}}}{20}}\right)\approx 118{,}27^{\circ };$ ${\ displaystyle \ arccos \ left (- {\ frac {5 + 2 {\ sqrt {5}}} {20} \ right) \ approx 118 {,} 27 ^ {\ circ};}$ smallest edge angle (between two “long” sides) $\arccos \,{\frac {9{\sqrt {5}}-5}{40}}\approx 67{,}78^{\circ };$ ${\ displaystyle \ arccos \, {\ frac {9 {\ sqrt {5}} - 5} {40}} \ approx 67 {,} 78 ^ {\ circ};}$ two medium-sized angles (between the “short” and “long” sides) $\arccos \,{\frac {5-2{\sqrt {5}}}{10}}\approx 86{,}97^{\circ }.$ ${\ displaystyle \ arccos \, {\ frac {5-2 {\ sqrt {5}}} {10}} \ approx 86 {,} 97 ^ {\ circ}.}$

The dihedral angle for any edge is the same and equal to $\arccos \left(-{\frac {19+8{\sqrt {5}}}{41}}\right)\approx 154{,}12^{\circ }.$ ${\ displaystyle \ arccos \ left (- {\ frac {19 + 8 {\ sqrt {5}}} {41} \ right) \ approx 154 {,} 12 ^ {\ circ}.}$

Notes

↑ Weisstein, Eric W. Catalan body counts (eng.) On the Wolfram MathWorld website.

Links

Weisstein, Eric W. The deltoidal hexacontahedron (English) on the Wolfram MathWorld website.

[1] Weisstein, Eric W. Catalan body counts (eng.) On the Wolfram MathWorld website.

Deltoidal hexacontahedron

Metric performance and angles

Notes

Links

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