In geometry, the augmented 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 (), and is obtained by adding a

square pyramid In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyramid, ...

to one of the square faces of the sphenocorona. It is the only Johnson solid arising from "cut and paste" manipulations where the components are not all prisms, antiprisms or sections of Platonic or Archimedean solids. . Johnson uses the prefix ''spheno-'' to refer to a wedge-like complex formed by two adjacent ''lunes'', a lune being a square with

equilateral triangles In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...

attached on opposite sides. Likewise, the suffix ''-corona'' refers to a crownlike complex of 8 equilateral triangles. Finally, the descriptor ''augmented'' implies that another polyhedron, in this case a pyramid, is adjointed. Joining both complexes together with the pyramid results in the augmented sphenocorona.

Cartesian coordinates

To calculate

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 ...

for the augmented sphenocorona, one may start by calculating the coordinates of the sphenocorona. 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 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 generated by reflections about the xz-plane and the yz-plane. Calculating the centroid and the normal unit vector of one of the square faces gives the location of its last vertex as :

\left(k+\sqrt,0,k+\sqrt\right).

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 snub square of edge length ''a'' as :

A=\left(1+4\sqrt\right)a^2\approx7.92820a^2,

and its volume as :

V=\left(\frac\sqrt+\frac\right)a^3\approx1.75105a^3.

References

External links

* Johnson solids {{Polyhedron-stub