3D view of a triacis icosahedron (
animation )
The Triakis icosahedron is a convex polyhedron , which is composed of 60 isosceles triangles and belongs to the Catalan solids . It is dual to the truncated dodecahedron and has 32 vertices and 90 edges.
Emergence
If pyramids with the flank length are placed on the 20 boundary surfaces of an icosahedron (edge length ) , a triacis icosahedron is created, provided the following condition is met:
a
{\ displaystyle a}
b
{\ displaystyle b}
a
3
3
<
b
<
a
4th
10
-
2
5
{\ displaystyle {\ frac {a} {3}} {\ sqrt {3}} <b <{\ frac {a} {4}} {\ sqrt {10-2 {\ sqrt {5}}}}}
For the aforementioned minimum value of , the pyramids on top have the height 0, so that only the icosahedron with the edge length remains.
b
{\ displaystyle b}
a
{\ displaystyle a}
The special Triakis icosahedron with equal surface angles arises when is.
b
=
a
22nd
(
15th
-
5
)
{\ displaystyle b = {\ frac {a} {22}} \, (15 - {\ sqrt {5}})}
Takes the above At the maximum value, the triacisicosahedron degenerates into a rhombic triacontahedron with the length of the edge .
b
{\ displaystyle b}
b
{\ displaystyle b}
If the maximum value is exceeded , the polyhedron is no longer convex and finally degenerates into an icosahedral star .
b
{\ displaystyle b}
b
=
a
2
(
1
+
5
)
{\ displaystyle b = {\ frac {a} {2}} \ left (1 + {\ sqrt {5}} \ right)}
Formulas
General
Sizes of a triacis kosahedron with edge lengths a , b
volume
V
=
5
12
a
2
(
a
(
3
+
5
)
+
4th
3
b
2
-
a
2
)
{\ displaystyle V = {\ frac {5} {12}} a ^ {2} \ left (a (3 + {\ sqrt {5}}) + 4 {\ sqrt {3b ^ {2} -a ^ { 2}}} \ right)}
Surface area
A.
O
=
15th
a
4th
b
2
-
a
2
{\ displaystyle A_ {O} = 15a {\ sqrt {4b ^ {2} -a ^ {2}}}}
Pyramid height
k
=
1
3
9
b
2
-
3
a
2
{\ displaystyle k = {\ frac {1} {3}} {\ sqrt {9b ^ {2} -3a ^ {2}}}}
Inc sphere radius
ρ
=
a
4th
10
a
+
4th
b
a
+
2
b
+
2
5
{\ displaystyle \ rho = {\ frac {a} {4}} {\ sqrt {{\ frac {10a + 4b} {a + 2b}} + 2 {\ sqrt {5}}}}}
Dihedral angle (over edge a )
cos
α
1
=
(
12
b
2
-
5
a
2
)
5
-
8th
a
3
b
2
-
a
2
9
(
4th
b
2
-
a
2
)
{\ displaystyle \ cos \, \ alpha _ {1} = {\ frac {(12b ^ {2} -5a ^ {2}) {\ sqrt {5}} - 8a {\ sqrt {3b ^ {2} - a ^ {2}}}} {9 (4b ^ {2} -a ^ {2})}}}
Dihedral angle (over edge b )
cos
α
2
=
2
b
2
-
a
2
4th
b
2
-
a
2
{\ displaystyle \ cos \, \ alpha _ {2} = {\ frac {2b ^ {2} -a ^ {2}} {4b ^ {2} -a ^ {2}}}}
Special
Edge sphere in the special triacis icosahedron: the
spherical caps clearly stand out on the individual triangular surfaces. The
incircles are also intersections of the triangles with the edge sphere.
Sizes of a triacis kosahedron with edge length a
volume
V
=
5
44
a
3
(
5
+
7th
5
)
{\ displaystyle V = {\ frac {5} {44}} a ^ {3} (5 + 7 {\ sqrt {5}})}
Surface area
A.
O
=
15th
11
a
2
109
-
30th
5
{\ displaystyle A_ {O} = {\ frac {15} {11}} a ^ {2} {\ sqrt {109-30 {\ sqrt {5}}}}}
2. Side length
b
=
a
22nd
(
15th
-
5
)
{\ displaystyle b = {\ frac {a} {22}} \, (15 - {\ sqrt {5}})}
Pyramid height
k
=
a
66
(
5
5
-
9
)
3
{\ displaystyle k = {\ frac {a} {66}} (5 {\ sqrt {5}} - 9) {\ sqrt {3}}}
Inc sphere radius
ρ
=
a
4th
10
(
33
+
13
5
)
61
{\ displaystyle \ rho = {\ frac {a} {4}} {\ sqrt {\ frac {10 (33 + 13 {\ sqrt {5}})} {61}}}}
Edge ball radius
r
=
a
4th
(
1
+
5
)
{\ displaystyle r = {\ frac {a} {4}} \ left (1 + {\ sqrt {5}} \ right)}
Face angle ≈ 160 ° 36 ′ 45 ″
cos
α
=
-
1
61
(
24
+
15th
5
)
{\ displaystyle \ cos \, \ alpha = - {\ frac {1} {61}} (24 + 15 {\ sqrt {5}})}
Remarks
↑
a
3
3
<
b
<
a
4th
10
-
2
5
{\ displaystyle {\ frac {a} {3}} {\ sqrt {3}} <b <{\ frac {a} {4}} {\ sqrt {10-2 {\ sqrt {5}}}}}
↑
b
=
a
22nd
(
15th
-
5
)
{\ displaystyle b = {\ frac {a} {22}} \, (15 - {\ 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;">