Deltoid hexacontahedron

3D view of a deltoid hexacontahedron ( animation )

The deltoidal hexecontahedron (also Deltoidhexakontaeder is called) is a convex polyhedron , composed of 60 deltoid composed and the Catalan bodies counts. It is dual to the rhombicosidodecahedron and has 62 vertices and 120 edges.

Emergence

Construction of the deltoid on the rhombicosidodecahedron

By connecting the centers of four edges that meet in each corner of the rhombicosidodecahedron , a trapezium is created , the circumference of which is also the inscribed circle of the deltoid, the boundary surface of the deltoidal hexacontahedron. In this special type, all face angles are the same (≈ 154 °) and there is a uniform edge sphere radius .

Let be the edge length of the rhombicosidodecahedron, then the resulting side lengths of the deltoid are given by ${\ displaystyle d}$

{\ displaystyle a = \, {\ frac {d} {3}} \, {\ sqrt {25-5 {\ sqrt {5}}}}}

{\ displaystyle b = \, {\ frac {d} {11}} \, {\ sqrt {5 \, (85-31 {\ sqrt {5}})}}}

The side lengths of the deltoid are therefore in the following relationship:

{\ displaystyle 3a \, (7 - {\ sqrt {5}}) = 22b}

Mesh of the deltoid hexacontahedron

Sizes of a deltoidal hexacontahedron with edge length a
volume	${\ displaystyle V = {\ frac {45} {11}} \, a ^ {3} {\ sqrt {\ frac {370 + 164 {\ sqrt {5}}} {25}}}}$
Surface area	${\ displaystyle A_ {O} = {\ frac {9} {11}} \, a ^ {2} {\ sqrt {10 \, (157 + 31 {\ sqrt {5}})}}}$
Inc sphere radius	${\ displaystyle \ rho = {\ frac {3} {2}} \, a \, {\ sqrt {\ frac {135 + 59 {\ sqrt {5}}} {205}}}}$
Edge ball radius	${\ displaystyle r = {\ frac {3} {20}} \, a \, (5 + 3 {\ sqrt {5}})}$
Face angle ≈ 154 ° 7 ′ 17 ″	${\ displaystyle \ cos \, \ alpha = - {\ frac {1} {41}} \, (19 + 8 {\ sqrt {5}})}$
3D edge angle ≈ 153 ° 26 ′ 6 ″	${\ displaystyle \ cos \, \ gamma = - {\ frac {2} {5}} {\ sqrt {5}}}$

Sizes in the dragon square

Sizes of the dragon square
Area	${\ displaystyle A = {\ frac {3} {22}} \, a ^ {2} {\ sqrt {\ frac {157 + 31 {\ sqrt {5}}} {10}}}}$
2. Side length	${\ displaystyle b = {\ frac {3} {22}} \, a \, (7 - {\ sqrt {5}})}$
Short diagonal	${\ displaystyle e = 3a \, {\ sqrt {\ frac {5 - {\ sqrt {5}}} {20}}}}$
Long diagonal	${\ displaystyle f = {\ frac {a} {11}} {\ sqrt {\ frac {470 + 156 {\ sqrt {5}}} {5}}}}$
Inscribed radius	${\ displaystyle r = {\ frac {3} {10}} \, a \, {\ sqrt {\ frac {5 \, (17 + 5 {\ sqrt {5}})} {82}}}}$
Side angle (2) ≈ 86 ° 58 ′ 27 ″	${\ displaystyle \ cos \, \ alpha = {\ frac {1} {10}} \, (5-2 {\ sqrt {5}})}$
Foot angle (1) ≈ 67 ° 46 ′ 59 ″	${\ displaystyle \ cos \, \ beta = {\ frac {1} {40}} \, (9 {\ sqrt {5}} - 5)}$
Head angle (1) ≈ 118 ° 16 ′ 7 ″	${\ displaystyle \ cos \, \ gamma = - {\ frac {1} {20}} \, (5 + 2 {\ sqrt {5}})}$