Triangular hebesphenorotunda

In geometry, the triangular hebesphenorotunda is one of the Johnson solids (J₉₂).

Triangular hebesphenorotunda
Type	Johnson J91 - J92 - J1
Faces	13 triangles 3 squares 3 pentagons 1 hexagon
Edges	36
Vertices	18
Vertex configuration	3(33.5) 6(3.4.3.5) 3(3.5.3.5) 2.3(32.4.6)
Symmetry group	C3v
Dual polyhedron	-
Properties	convex
Net

A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.

It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. However, it does have a strong relationship to the icosidodecahedron, an Archimedean solid. Most evident is the cluster of three pentagons and four triangles on one side of the solid. If these faces are aligned with a congruent patch of faces on the icosidodecahedron, then the hexagonal face will lie in the plane midway between two opposing triangular faces of the icosidodecahedron.

The triangular hebesphenorotunda is the only Johnson solid with faces of 3, 4, 5 and 6 sides. The 92 Johnson solids were named and described by Norman Johnson in 1966.

The coordinates of the triangular hebesphenorotunda with edge length 2, and its axis of symmetry aligned to the Z-axis, are:

• The triangle opposite the hexagon:

$\left(0,\ {\frac {2}{\sqrt {3}}},\ {\frac {2\phi ^{2}}{\sqrt {3}}}\right)$, $\left(\pm 1,\ -{\frac {1}{\sqrt {3}}},\ {\frac {2\phi ^{2}}{\sqrt {3}}}\right)$

• The bases of the triangles surrounding the previous triangle:

$\left(\pm 1,\ {\frac {\phi ^{3}}{\sqrt {3}}},\ {\frac {2\phi }{\sqrt {3}}}\right)$, $\left(\pm \phi ^{2},\ -{\frac {1}{\phi {\sqrt {3}}}},\ {\frac {2\phi }{\sqrt {3}}}\right)$, $\left(\pm \phi ,\ -{\frac {\phi +2}{\sqrt {3}}},\ {\frac {2\phi }{\sqrt {3}}}\right)$

• The tips of the pentagons opposite the first triangle:

$\left(\pm \phi ^{2},\ {\frac {\phi ^{2}}{\sqrt {3}}},\ {\frac {2}{\sqrt {3}}}\right)$, $\left(0,\ -{\frac {2\phi ^{2}}{\sqrt {3}}},\ {\frac {2}{\sqrt {3}}}\right)$

• The hexagon:

$\left(\pm 1,\ \pm {\sqrt {3}},\ 0\right)$, $\left(\pm 2,\ 0,\ 0\right)$

where $\phi ={\frac {1+{\sqrt {5}}}{2}}$ is the Golden Ratio.

External links

• Weisstein, Eric W. "Johnson solids". MathWorld.
  • Weisstein, Eric W. "Triangular hebesphenorotunda". MathWorld.