geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the triangular hebesphenorotunda is one of the

Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johns ...

s (). . It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the

Platonic Plato's influence on Western culture was so profound that several different concepts are linked by being called Platonic or Platonist, for accepting some assumptions of Platonism, but which do not imply acceptance of that philosophy as a whole. It ...

and Archimedean solids. However, it does have a strong relationship to the

icosidodecahedron In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 ...

, an Archimedean solid. Most evident is the cluster of three

pentagon In geometry, a pentagon (from the Greek language, Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is ...

s and four

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...

s on one side of the solid. If these faces are aligned with a congruent patch of faces on the icosidodecahedron, then the

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A ''regular hexagon'' h ...

al face will lie in the plane midway between two opposing triangular faces of the icosidodecahedron. The triangular hebesphenorotunda also has clusters of faces that can be aligned with corresponding faces of the

rhombicosidodecahedron In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces. It has 20 regular triangular faces, 30 square faces, 12 regular ...

: the three ''lunes'', each ''lune'' consisting of a square and two antipodal triangles adjacent to the square. The faces around each vertex can also be aligned with the corresponding faces of various diminished icosahedra. Johnson uses the prefix ''hebespheno-'' to refer to a blunt wedge-like complex formed by three adjacent ''lunes'', a ''lune'' being a

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

with equilateral triangles attached on opposite sides. The suffix (triangular) ''-rotunda'' refers to the complex of three equilateral triangles and three regular pentagons surrounding another equilateral triangle, which bears structural resemblance to the

pentagonal rotunda In geometry, the pentagonal rotunda is one of the Johnson solids (). It can be seen as half of an icosidodecahedron, or as half of a pentagonal orthobirotunda. It has a total of 17 faces. Formulae The following formulae for volume, surface a ...

. The triangular hebesphenorotunda is the only Johnson solid with faces of 3, 4, 5 and 6 sides.

Cartesian coordinates

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...

for the triangular hebesphenorotunda with edge length – 1 are given by the union of the orbits of the points :

\left(0,-\frac,\frac\right),\left(\tau,\frac,\frac\right),

\left(\tau,-\frac,\frac\right),\left(\frac,0,0\right),

under the action of the

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

generated by rotation by 120° around the z-axis and the reflection about the yz-plane. Here, = (sometimes written ''φ'') is the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

. The first point generates the triangle opposite the hexagon, the second point generates the bases of the triangles surrounding the previous triangle, the third point generates the tips of the pentagons opposite the first triangle, and the last point generates the hexagon. One may then calculate the

surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of ...

of a triangular hebesphenorotunda of edge length ''a'' as :

A=\left(3+\frac\sqrt\right)a^2\approx16.38867a^2,

and its

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...

as :

V=\frac\left(15+7\sqrt\right)a^3\approx5.10875a^3.

A second, inverted, triangular hebesphenorotunda can be obtained by negating the second and third coordinates of each point. This second polyhedron will be joined to the first at their common hexagonal face, and the pair will inscribe an icosidodecahedron. If the hexagonal face is scaled by the golden ratio, then the convex hull of the result will be the entire icosidodecahedron.

References

External links

* {{Polyhedron-stub Johnson solids