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 sphenocorona 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 isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...

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. Johnson uses the prefix ''spheno-'' to refer to a

wedge A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converti ...

-like complex formed by two 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 adj ...

with equilateral triangles attached on opposite sides. Likewise, the suffix ''-corona'' refers to a crownlike complex of 8 equilateral triangles. Joining both complexes together results in the sphenocorona..

Cartesian coordinates

Let ''k'' ≈ 0.85273 be the smallest positive root of the

quartic polynomial In algebra, a quartic function is a function (mathematics), function of the form :f(x)=ax^4+bx^3+cx^2+dx+e, where ''a'' is nonzero, which is defined by a polynomial of Degree of a polynomial, degree four, called a quartic polynomial. A ''qua ...

60x^4-48x^3-100x^2+56x+23.

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 in t ...

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

\left(0,1,2\sqrt\right),\,(2k,1,0),\left(0,1+\frac,\frac\right),\,\left(1,0,-\sqrt\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. 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 arc ...

of a sphenocorona of edge length ''a'' as :

A=\left(2+3\sqrt\right)a^2\approx7.19615a^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). The de ...

as :

\left(\frac\sqrt\right)a^3\approx1.51535a^3.

Variations

The sphenocorona is also the

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

of the isogonal n-gonal double antiprismoid where n is an odd number greater than one, including the

grand antiprism In geometry, the grand antiprism or pentagonal double antiprismoid is a uniform 4-polytope (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300 tetrahedra. It is an anomalous, non-Wythoffian uniform 4-polytope ...

with pairs of trapezoid rather than square faces. : Grand antiprism verf

References

External links

* Johnson solids {{Polyhedron-stub

Cartesian coordinates

Variations

See also

References

External links