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 sphenomegacorona 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 16 equilateral triangles and 2 squares as its faces.

Properties

The sphenomegacorona was named by in which he used the prefix ''spheno-'' referring to a wedge-like complex formed by two adjacent ''lunes''—a

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 ...

with

equilateral triangles 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 ...

attached on its opposite sides. The suffix ''-megacorona'' refers to a crownlike complex of 12 triangles, contrasted with the smaller triangular complex that makes the sphenocorona. By joining both complexes, the resulting polyhedron has 16 equilateral triangles and 2 squares, making 18 faces. All of its faces are

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 sphenomegacorona as a

—a

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

polyhedron in which all of the faces are regular polygons—enumerated as the 88th Johnson solid

J_

. It is an elementary polyhedron, meaning 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 sphenomegacorona

A

is the total of polygonal faces' area—16 equilateral triangles and 2 squares. The

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) ...

of a sphenomegacorona is obtained by finding the root of a polynomial, and its decimal expansion—denoted as

\xi

—is given by . With edge length

a

, its surface area and volume can be formulated as:

\begin
 A &= \left(2+4\sqrt\right)a^2 &\approx 8.928a^2, \\
 V &= \xi a^3 &\approx 1.948a^3.
\end

Cartesian coordinates

Let

k \approx 0.59463

be the smallest positive root of the

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

\begin & 1680 x^- 4800 x^ - 3712 x^ + 17216 x^ \\ + & 1568 x^ - 24576 x^ + 2464 x^ + 17248 x^9 \\ - & 3384 x^8 - 5584 x^7 + 2000 x^6+ 240 x^5 \\- & 776 x^4+ 304 x^3 + 200 x^2 - 56 x -23 \end

Then,

Cartesian coordinates 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 ...

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

\begin &\left(0,1,2\sqrt\right),\,(2k,1,0),\,\left(0,\frac+1,\frac\right), \\
&\left(1,0,-\sqrt\right),\,\left(0,\frac+1,\frac\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 reflections about the xz-plane and the yz-plane.

References

External links

* Elementary polyhedron Johnson solids {{Johnson solids navigator