3D view of a hexakis icosahedron (
animation )
The disdyakis triacontahedron (from Greek ἑξάκις hexakis "six" and icosahedron Disdyakistriakontaeder ( Greek δίς dis "twice", δυάκις dyakis "twice" and triacontahedron convex polyhedron , which is composed of 120 irregular triangles composed and belongs to the Catalan bodies . It is dual to the truncated icosidodecahedron and has 62 vertices and 180 edges.
Emergence Rhombic triacontahedron as a base If pyramids with the flank lengths and are placed on the 30 boundary surfaces of a rhombic triacontahedron (edge length ) , a hexakisicosahedron is created, provided the following condition is met:
a
{\ displaystyle a}
b
{\ displaystyle b}
c
(
<
b
)
{\ displaystyle c \, (<b)}
a
10
50
+
10
5
<
b
<
a
10
70
+
2
5
{\ displaystyle {\ frac {a} {10}} {\ sqrt {50 + 10 {\ sqrt {5}}}} \, <b <\, {\ frac {a} {10}} {\ sqrt { 70 + 2 {\ sqrt {5}}}}}
For the aforementioned minimum value of , the pyramids on top have the height 0, so that only the rhombic triacontahedron with the edge length remains.
b
{\ displaystyle b}
a
{\ displaystyle a}
The special hexakisicosahedron with equal surface angles at the edges and is created when is.
a
{\ displaystyle a}
b
{\ displaystyle b}
b
=
a
2
(
3
5
-
5
)
≈
0.854
102
⋅
a
{\ displaystyle b = {\ frac {a} {2}} \, (3 {\ sqrt {5}} - 5) \ approx 0 {,} 854102 \ cdot a}
If b assumes the aforementioned maximum value, the hexakisicosahedron degenerates into a deltoidal hexacontahedron with the edge lengths and .
a
{\ displaystyle a}
b
{\ displaystyle b}
If the maximum value is exceeded , the polyhedron is no longer convex.
b
{\ displaystyle b}
Truncated icosidodecahedron as a base
Construction of the triangle at the truncated icosidodecahedron
By connecting the centers of three edges that meet in each corner of the truncated icosidodecahedron , a triangle is created, the circumference of which is also an inscribed circle of the triangle, the boundary surface of the hexakisicosahedron. In this special type, all face angles are the same (≈ 165 °) and there is a uniform edge sphere radius .
Let be the edge length of the truncated icosidodecahedron, then the resulting side lengths of the triangle are given by
d
{\ displaystyle d}
a
=
2
5
d
15th
(
5
-
5
)
{\ displaystyle a = {\ frac {2} {5}} \, d \, {\ sqrt {15 \, (5 - {\ sqrt {5}})}}}
b
=
3
55
d
15th
(
65
+
19th
5
)
{\ displaystyle b = {\ frac {3} {55}} \, d \, {\ sqrt {15 \, (65 + 19 {\ sqrt {5}})}}}
c
=
d
11
15th
(
85
-
31
5
)
{\ displaystyle c = {\ frac {d} {11}} \, {\ sqrt {15 \, (85-31 {\ sqrt {5}})}}}
Formulas In the following denote the longest edge of the hexakis icosahedron ( ).
a
{\ displaystyle a}
a
>
b
>
c
{\ displaystyle a> b> c}
Regular
The basis is the truncated icosidodecahedron (dual Archimedean solid).
Sizes of a hexakis icosahedron with edge length a
volume
V
=
25th
88
a
3
6th
(
185
+
82
5
)
{\ displaystyle V = {\ frac {25} {88}} a ^ {3} \, {\ sqrt {6 \, (185 + 82 {\ sqrt {5}})}}}
Surface area
A.
O
=
15th
44
a
2
10
(
417
+
107
5
)
{\ displaystyle A_ {O} = {\ frac {15} {44}} a ^ {2} \, {\ sqrt {10 \, (417 + 107 {\ sqrt {5}})}}}
Inc sphere radius
ρ
=
a
4th
15th
(
275
+
119
5
)
241
{\ displaystyle \ rho = {\ frac {a} {4}} \, {\ sqrt {\ frac {15 \, (275 + 119 {\ sqrt {5}})} {241}}}}
Edge ball radius
r
=
a
8th
(
5
+
3
5
)
{\ displaystyle r = {\ frac {a} {8}} \, (5 + 3 {\ sqrt {5}})}
Face angle
cos
α
=
-
1
241
(
179
+
24
5
)
{\ displaystyle \ cos \, \ alpha = - {\ frac {1} {241}} \, (179 + 24 {\ sqrt {5}})}
Sizes of the triangle
Area
A.
=
a
2
352
10
(
417
+
107
5
)
{\ displaystyle A = {\ frac {a ^ {2}} {352}} \, {\ sqrt {10 \, (417 + 107 {\ sqrt {5}})}}}
2. Side length
b
=
3
22nd
a
(
4th
+
5
)
{\ displaystyle b = {\ frac {3} {22}} \, a \, (4 + {\ sqrt {5}})}
3. Side length
c
=
5
44
a
(
7th
-
5
)
{\ displaystyle c = {\ frac {5} {44}} \, a \, (7 - {\ sqrt {5}})}
1. Angle
cos
α
=
1
30th
(
5
-
2
5
)
{\ displaystyle \ cos \, \ alpha = {\ frac {1} {30}} \, (5-2 {\ sqrt {5}})}
2. Angle
cos
β
=
1
20th
(
15th
-
2
5
)
{\ displaystyle \ cos \, \ beta = {\ frac {1} {20}} \, (15-2 {\ sqrt {5}})}
3. Angle
cos
γ
=
1
24
(
9
+
5
5
)
{\ displaystyle \ cos \, \ gamma = {\ frac {1} {24}} \, (9 + 5 {\ sqrt {5}})}
Rhombic The basis is the rhombic triacontahedron (edge length ).
a
{\ displaystyle a}
General
Sizes of a hexakis icosahedron with edge lengths a , b
volume
V
=
2
a
2
(
2
a
5
+
2
5
+
20th
b
2
-
2
a
2
(
5
+
5
)
)
{\ displaystyle V = 2a ^ {2} \ left (2a {\ sqrt {5 + 2 {\ sqrt {5}}}} + {\ sqrt {20b ^ {2} -2a ^ {2} (5+ { \ sqrt {5}}}}) \ right)}
Surface area
A.
O
=
60
a
10
b
2
-
a
2
(
3
+
5
)
10
{\ displaystyle A_ {O} = 60a \, {\ sqrt {\ frac {10b ^ {2} -a ^ {2} (3 + {\ sqrt {5}})} {10}}}}
Pyramid height
k
=
10
b
2
-
a
2
(
5
+
5
)
10
{\ displaystyle k = {\ sqrt {\ frac {10b ^ {2} -a ^ {2} (5 + {\ sqrt {5}})} {10}}}}
Inc sphere radius
ρ
=
a
(
a
50
+
20th
5
+
50
b
2
-
5
a
2
(
5
+
5
)
)
5
10
b
2
-
a
2
(
3
+
5
)
{\ displaystyle \ rho = {\ frac {a \ left (a {\ sqrt {50 + 20 {\ sqrt {5}}}} + {\ sqrt {50b ^ {2} -5a ^ {2} (5+ {\ sqrt {5}}}}) \ right)} {5 {\ sqrt {10b ^ {2} -a ^ {2} (3 + {\ sqrt {5}})}}}}}
Dihedral angle a )
cos
α
1
=
5
b
2
(
1
+
5
)
-
2
a
2
(
3
+
2
5
)
-
2
a
10
b
2
(
5
-
5
)
-
20th
a
2
20th
b
2
-
2
a
2
(
3
+
5
)
{\ displaystyle \ cos \, \ alpha _ {1} = {\ frac {5b ^ {2} (1 + {\ sqrt {5}}) - 2a ^ {2} (3 + 2 {\ sqrt {5} }) - 2a {\ sqrt {10b ^ {2} (5 - {\ sqrt {5}}) - 20a ^ {2}}}} {20b ^ {2} -2a ^ {2} (3 + {\ sqrt {5}})}}}
Dihedral angle b )
cos
α
2
=
2
b
2
5
-
a
2
(
3
+
5
)
10
b
2
-
a
2
(
3
+
5
)
{\ displaystyle \ cos \, \ alpha _ {2} = {\ frac {2b ^ {2} {\ sqrt {5}} - a ^ {2} (3 + {\ sqrt {5}})} {10b ^ {2} -a ^ {2} (3 + {\ sqrt {5}})}}}
Dihedral angle c )
cos
α
3
=
a
2
(
1
-
5
)
+
2
b
2
5
a
2
(
3
+
5
)
-
10
b
2
{\ displaystyle \ cos \, \ alpha _ {3} = {\ frac {a ^ {2} (1 - {\ sqrt {5}}) + 2b ^ {2} {\ sqrt {5}}} {a ^ {2} (3 + {\ sqrt {5}}) - 10b ^ {2}}}}
Sizes of the triangle
Area
A.
=
a
2
10
b
2
-
a
2
(
3
+
5
)
10
{\ displaystyle A = {\ frac {a} {2}} \, {\ sqrt {\ frac {10b ^ {2} -a ^ {2} (3 + {\ sqrt {5}})} {10} }}}
3. Edge length
c
=
5
b
2
-
a
2
5
5
{\ displaystyle c = {\ sqrt {\ frac {5b ^ {2} -a ^ {2} {\ sqrt {5}}} {5}}}}
1. Angle
sin
α
=
a
b
10
b
2
-
a
2
(
3
+
5
)
10
b
2
-
2
a
2
5
{\ displaystyle \ sin \, \ alpha = {\ frac {a} {b}} \, {\ sqrt {\ frac {10b ^ {2} -a ^ {2} (3 + {\ sqrt {5}} )} {10b ^ {2} -2a ^ {2} {\ sqrt {5}}}}}}
2. angle
sin
β
=
10
b
2
-
a
2
(
3
+
5
)
10
b
2
-
2
a
2
5
{\ displaystyle \ sin \, \ beta = {\ sqrt {\ frac {10b ^ {2} -a ^ {2} (3 + {\ sqrt {5}})} {10b ^ {2} -2a ^ { 2} {\ sqrt {5}}}}}}
3. Angle
sin
γ
=
1
b
10
b
2
-
a
2
(
3
+
5
)
10
{\ displaystyle \ sin \, \ gamma = {\ frac {1} {b}} \, {\ sqrt {\ frac {10b ^ {2} -a ^ {2} (3 + {\ sqrt {5}} )} {10}}}}
Special
Sizes of a hexakis icosahedron with edge length a
volume
V
=
8th
a
3
25th
-
10
5
{\ displaystyle V = 8a ^ {3} {\ sqrt {25-10 {\ sqrt {5}}}}}
Surface area
A.
O
=
120
a
2
43
-
19th
5
10
{\ displaystyle A_ {O} = 120a ^ {2} \, {\ sqrt {\ frac {43-19 {\ sqrt {5}}} {10}}}}
Inc sphere radius
ρ
=
a
25th
+
9
5
22nd
{\ displaystyle \ rho = a \, {\ sqrt {\ frac {25 + 9 {\ sqrt {5}}} {22}}}}
Face angle a, b )
cos
α
1
,
2
=
-
1
44
(
31
+
5
5
)
{\ displaystyle \ cos \, \ alpha _ {1, \, 2} = - {\ frac {1} {44}} \, (31 + 5 {\ sqrt {5}})}
Face angle c )
cos
α
3
=
-
1
22nd
(
6th
+
7th
5
)
{\ displaystyle \ cos \, \ alpha _ {3} = - {\ frac {1} {22}} \, (6 + 7 {\ sqrt {5}})}
Sizes of the triangle
Area
A.
=
a
2
43
-
19th
5
10
{\ displaystyle A = a ^ {2} \, {\ sqrt {\ frac {43-19 {\ sqrt {5}}} {10}}}}
2. Side length
b
=
a
2
(
3
5
-
5
)
{\ displaystyle b = {\ frac {a} {2}} \, (3 {\ sqrt {5}} - 5)}
3. Side length
c
=
a
175
-
77
5
10
{\ displaystyle c = a \, {\ sqrt {\ frac {175-77 {\ sqrt {5}}} {10}}}}
1. Angle
sin
α
=
1
5
90
+
38
5
7th
{\ displaystyle \ sin \, \ alpha = {\ frac {1} {5}} {\ sqrt {\ frac {90 + 38 {\ sqrt {5}}} {7}}}}
2. Angle
sin
β
=
30th
-
2
5
35
{\ displaystyle \ sin \, \ beta = {\ sqrt {\ frac {30-2 {\ sqrt {5}}} {35}}}}
3. Angle
sin
γ
=
2
5
4th
-
5
{\ displaystyle \ sin \, \ gamma = {\ frac {2} {5}} {\ sqrt {4 - {\ sqrt {5}}}}}
Web links
<img src="https://de.wikipedia.org//de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
