3D view of a pentagon hexacontahedron (
animation )
Mesh of the pentagon hexacontahedron
The pentagon hexacontahedron is a convex polyhedron , which is composed of 60 pentagons and is one of the Catalan solids . It is dual to the beveled dodecahedron and has 92 corners and 150 edges.
The following pictures show two pentagon hexacontahedra that are mirror images of each other.
Emergence By connecting the center points of five edges that meet in each corner of the beveled dodecahedron, a chordal pentagon is created , the circumference of which is also the incircle of the tangent pentagon , the boundary surface of the pentagon hexacontahedron. With this special type, all face angles are the same (≈ 153 °) 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; be the golden number .
t
{\ displaystyle t}
ζ
{\ displaystyle \ zeta}
Φ
{\ displaystyle \ Phi}
t
{\ displaystyle t}
real solution to the cubic equation .
8th
t
3
+
8th
t
2
-
Φ
2
=
0
{\ displaystyle 8t ^ {3} + 8t ^ {2} - \ Phi ^ {2} = 0}
t
=
cos
ζ
=
1
12
(
44
+
12
Φ
(
9
+
81
Φ
-
15th
)
3
+
44
+
12
Φ
(
9
-
81
Φ
-
15th
)
3
-
4th
)
{\ displaystyle t = \ cos \, \ zeta = {\ frac {1} {12}} \ left ({\ sqrt [{3}] {44 + 12 \ Phi \, (9 + {\ sqrt {81 \ Phi -15}})}} + {\ sqrt [{3}] {44 + 12 \ Phi \, (9 - {\ sqrt {81 \ Phi -15}})}} - 4 \ right)}
Let be the edge length of the beveled dodecahedron, then the resulting side lengths of the tangent pentagon are given by
d
{\ displaystyle d}
a
=
d
(
1
+
2
t
)
2
(
1
-
2
t
2
)
2
+
2
t
{\ displaystyle a = {\ frac {d \, (1 + 2t)} {2 \, (1-2t ^ {2}) {\ sqrt {2 + 2t}}}}}
b
=
d
2
+
2
t
{\ displaystyle b = {\ frac {d} {\ sqrt {2 + 2t}}}}
The two side lengths are thus in the following relationship:
2
a
(
1
-
2
t
2
)
=
b
(
1
+
2
t
)
{\ displaystyle 2a \ left (1-2t ^ {2} \ right) = b \ left (1 + 2t \ right)}
Related polyhedra
Formulas
For the polyhedron
Sizes of a pentagon hexacontahedron with edge length a or b
Volume 3 ≈ 189.84b 3
V
=
40
a
3
(
1
+
t
)
(
2
+
3
t
)
(
1
-
2
t
2
)
2
(
1
+
2
t
)
3
1
-
2
t
=
5
b
3
(
1
+
t
)
(
2
+
3
t
)
(
1
-
2
t
2
)
1
-
2
t
{\ displaystyle V = {\ frac {40a ^ {3} (1 + t) (2 + 3t) (1-2t ^ {2}) ^ {2}} {(1 + 2t) ^ {3} {\ sqrt {1-2t}}}} = {\ frac {5b ^ {3} (1 + t) (2 + 3t)} {(1-2t ^ {2}) {\ sqrt {1-2t}}} }}
Surface area 2 ≈ 162.73b 2
A.
O
=
120
a
2
(
2
+
3
t
)
(
1
-
2
t
2
)
(
1
+
2
t
)
2
1
-
t
2
=
30th
b
2
(
2
+
3
t
)
(
1
-
2
t
2
)
1
-
t
2
{\ displaystyle A_ {O} = {\ frac {120a ^ {2} (2 + 3t) (1-2t ^ {2})} {(1 + 2t) ^ {2}}} {\ sqrt {1- t ^ {2}}} = {\ frac {30b ^ {2} (2 + 3t)} {(1-2t ^ {2})}} {\ sqrt {1-t ^ {2}}}}
Edge ball radius
r
=
a
(
1
-
2
t
2
)
1
-
4th
t
2
2
(
1
+
t
)
(
1
-
2
t
)
=
b
1
+
t
2
(
1
-
2
t
)
{\ displaystyle r = {\ frac {a \, (1-2t ^ {2})} {1-4t ^ {2}}} {\ sqrt {2 \, (1 + t) (1-2t)} } = b \, {\ sqrt {\ frac {1 + t} {2 \, (1-2t)}}}}
Inc sphere radius
ρ
=
a
(
1
-
2
t
2
)
1
+
2
t
1
+
t
(
1
-
t
)
(
1
-
2
t
)
=
b
2
1
+
t
(
1
-
t
)
(
1
-
2
t
)
{\ displaystyle \ rho = {\ frac {a \, (1-2t ^ {2})} {1 + 2t}} {\ sqrt {\ frac {1 + t} {(1-t) (1-2t )}}} = {\ frac {b} {2}} \, {\ sqrt {\ frac {1 + t} {(1-t) (1-2t)}}}}
Face angle
cos
α
=
t
t
-
1
{\ displaystyle \ cos \, \ alpha = {\ frac {t} {t-1}}}
For the boundary surfaces
Sizes of the tangent pentagon
Area
A.
=
2
a
2
(
2
+
3
t
)
(
1
-
2
t
2
)
(
1
+
2
t
)
2
1
-
t
2
=
b
2
(
2
+
3
t
)
2
(
1
-
2
t
2
)
1
-
t
2
{\ displaystyle A = {\ frac {2a ^ {2} (2 + 3t) (1-2t ^ {2})} {(1 + 2t) ^ {2}}} {\ sqrt {1-t ^ { 2}}} = {\ frac {b ^ {2} (2 + 3t)} {2 \, (1-2t ^ {2})}} {\ sqrt {1-t ^ {2}}}}
Inscribed radius
r
=
a
(
1
-
2
t
2
)
1
+
2
t
1
+
t
1
-
t
=
b
2
1
+
t
1
-
t
{\ displaystyle r = {\ frac {a \, (1-2t ^ {2})} {1 + 2t}} {\ sqrt {\ frac {1 + t} {1-t}}} = {\ frac {b} {2}} \, {\ sqrt {\ frac {1 + t} {1-t}}}}
diagonal
‖
b
{\ displaystyle \ | \, b}
e
=
2
a
(
1
-
2
t
2
)
=
b
(
1
+
2
t
)
{\ displaystyle e = 2a \, (1-2t ^ {2}) = b \, (1 + 2t)}
Obtuse angles (4)
cos
α
=
-
t
{\ displaystyle \ cos \, \ alpha = -t}
Acute angle (1)
cos
β
=
1
-
2
(
1
-
2
t
2
)
2
{\ displaystyle \ cos \, \ beta = 1-2 \, (1-2t ^ {2}) ^ {2}}
application
Sizes in the boundary surfaces of the pentagon hexacontahedron
A process is patented in the USA in which 92 of the 332 dimples of a golf ball lie on the grid points of a pentagon hexacontahedron.
Notes and individual references
↑ t ≈ 0.47157563
↑ With a the longer of the two sides was called.
↑ These formulas apply to the case .
2
a
(
1
-
2
t
2
)
=
b
(
1
+
2
t
)
{\ displaystyle 2a \ left (1-2t ^ {2} \ right) = b \ left (1 + 2t \ right)}
↑ a b c d e f g h i This formula also applies to the pentagonikositetrahedron and the pentagon dodecahedron , provided that the corresponding values for b (short side length), n (number of boundary surfaces) and t (cosine of the smaller central angle) are used and furthermore note that O = n · A and V = 1/3 · O · ρ.
↑ Pentagonal hexecontahedron dimple pattern on golf balls (English)
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;">
![t = \ cos \, \ zeta = {\ frac {1} {12}} \ left ({\ sqrt [{3}] {44 + 12 \ Phi \, (9 + {\ sqrt {81 \ Phi -15 }})}} + {\ sqrt [{3}] {44 + 12 \ Phi \, (9 - {\ sqrt {81 \ Phi -15}})}} - 4 \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1dfda361a986ad59bf0cc266ceb0750728932c)
