- Source: Elongated triangular gyrobicupola
In geometry, the elongated triangular gyrobicupola is a polyhedron constructed by attaching two regular triangular cupolas to the base of a regular hexagonal prism, with one of them rotated in
60
∘
{\displaystyle 60^{\circ }}
. It is an example of Johnson solid.
Construction
The elongated triangular gyrobicupola is similarly can be constructed as the elongated triangular orthobicupola, started from a hexagonal prism by attaching two regular triangular cupolae onto its base, covering its hexagonal faces. This construction process is known as elongation, giving the resulting polyhedron has 8 equilateral triangles and 12 squares. The difference between those two polyhedrons is one of two triangular cupolas in the elongated triangular gyrobicupola is rotated in
60
∘
{\displaystyle 60^{\circ }}
. A convex polyhedron in which all faces are regular is Johnson solid, and the elongated triangular gyrobicupola is one among them, enumerated as 36th Johnson solid
J
36
{\displaystyle J_{36}}
.
Properties
An elongated triangular gyrobicupola with a given edge length
a
{\displaystyle a}
has a surface area by adding the area of all regular faces:
(
12
+
2
3
)
a
2
≈
15.464
a
2
.
{\displaystyle \left(12+2{\sqrt {3}}\right)a^{2}\approx 15.464a^{2}.}
Its volume can be calculated by cutting it off into two triangular cupolae and a hexagonal prism with regular faces, and then adding their volumes up:
(
5
2
3
+
3
3
2
)
a
3
≈
4.955
a
3
.
{\displaystyle \left({\frac {5{\sqrt {2}}}{3}}+{\frac {3{\sqrt {3}}}{2}}\right)a^{3}\approx 4.955a^{3}.}
Its three-dimensional symmetry groups is the prismatic symmetry, the dihedral group
D
3
d
{\displaystyle D_{3d}}
of order 12. Its dihedral angle can be calculated by adding the angle of the triangular cupola and hexagonal prism. The dihedral angle of a hexagonal prism between two adjacent squares is the internal angle of a regular hexagon
120
∘
=
2
π
/
3
{\displaystyle 120^{\circ }=2\pi /3}
, and that between its base and square face is
π
/
2
=
90
∘
{\displaystyle \pi /2=90^{\circ }}
. The dihedral angle of a regular triangular cupola between each triangle and the hexagon is approximately
70.5
∘
{\displaystyle 70.5^{\circ }}
, that between each square and the hexagon is
54.7
∘
{\displaystyle 54.7^{\circ }}
, and that between square and triangle is
125.3
∘
{\displaystyle 125.3^{\circ }}
. The dihedral angle of an elongated triangular orthobicupola between the triangle-to-square and square-to-square, on the edge where the triangular cupola and the prism is attached, is respectively:
π
2
+
70.5
∘
≈
160.5
∘
,
π
2
+
54.7
∘
≈
144.7
∘
.
{\displaystyle {\begin{aligned}{\frac {\pi }{2}}+70.5^{\circ }&\approx 160.5^{\circ },\\{\frac {\pi }{2}}+54.7^{\circ }&\approx 144.7^{\circ }.\end{aligned}}}
Related polyhedra and honeycombs
The elongated triangular gyrobicupola forms space-filling honeycombs with tetrahedra and square pyramids.
References
External links
Weisstein, Eric W., "Elongated triangular gyrobicupola" ("Johnson solid") at MathWorld.
Kata Kunci Pencarian:
- Daftar bentuk matematika
- Elongated triangular gyrobicupola
- Elongated square gyrobicupola
- Johnson solid
- Triangular cupola
- Square gyrobicupola
- Elongated gyrobifastigium
- Icosahedron
- Elongated bicupola
- List of mathematical shapes
- Tetrahedral-octahedral honeycomb
