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 disphenocingulum or pentakis elongated gyrobifastigium 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 that 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.

Cartesian coordinates

Let ''a'' ≈ 0.76713 be the second smallest positive root of the

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...

\begin &256x^ - 512x^ - 1664x^ + 3712x^9 + 1552x^8 - 6592x^7 \\ &\quad + 1248x^6 + 4352x^5 - 2024x^4 - 944x^3 + 672x^2 - 24x - 23 \end

and

h = \sqrt

and

c = \sqrt

. Then,

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 ...

of a disphenocingulum with edge length 2 are given by the union of the orbits of the points :

\left(1,2a,\frac\right),\ \left(1,0,2c+\frac\right),\ \left(1+\frac,0,2c-\frac+\frac\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 reflections about the xz-plane and the yz-plane.

References

External links

* Johnson solids {{Polyhedron-stub