geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the triangular hebesphenorotunda is a

Johnson solid In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...

with 13

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

s, 3

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

s, 3

regular pentagon In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...

s, and 1

regular 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 is de ...

, meaning the total of its faces is 20.

Properties

The triangular hebesphenorotunda is named by , with the prefix ''hebespheno-'' referring to a blunt wedge-like complex formed by three adjacent ''lunes''—a figure where two equilateral triangles are attached at the opposite sides of a square. The suffix (triangular) ''-rotunda'' refers to the complex of three equilateral triangles and three regular pentagons surrounding another equilateral triangle, which bears a structural resemblance to the pentagonal rotunda. Therefore, the triangular hebesphenorotunda has 20 faces: 13

s, 3

s, and 1

. The faces are all

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

s, categorizing the triangular hebesphenorotunda as a

, enumerated the last one

J_

. It is an

elementary polyhedron In geometry, a composite polyhedron is a convex polyhedron that produces other polyhedrons when sliced by a plane. Examples can be found in Johnson solids. Definition and examples A convex polyhedron is said to be ''composite'' if there exists ...

, meaning that it cannot be separated by a plane into two small regular-faced polyhedra. The

surface area The surface area (symbol ''A'') 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 d ...

of a triangular hebesphenorotunda of edge length

a

as:

A = \left(3+\frac\sqrt\right)a^2 \approx 16.389a^2,

and its

volume Volume is a measure of regions in 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) ...

as:

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

Cartesian coordinates

The triangular hebesphenorotunda with edge length

\sqrt - 1

can be constructed by the union of the orbits of the

Cartesian coordinate In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

\begin
 \left( 0,-\frac,\frac \right), \qquad &\left( \tau,\frac,\frac \right) \\
 \left( \tau,-\frac,\frac \right), \qquad &\left(\frac,0,0\right),
\end

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 iden ...

generated by rotation by 120° around the z-axis and the reflection about the yz-plane. Here,

\tau

denotes the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...

References

External links

* {{Johnson solids navigator Elementary polyhedron Johnson solids