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 square orthobicupola 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 ...

constructed by two

square cupola In geometry, the square cupola (sometimes called lesser dome) is a cupola with an octagonal In geometry, an octagon () is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasireg ...

s base-to-base.

Construction

The square orthobicupola is started by attaching two

e onto their bases. The resulting polyhedron consisted of eight

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 and ten

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, having eighteen faces in total, as well as thirty-two edges and sixteen vertices. A

convex polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...

in which 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 is a

, and the square orthobicupola is one of them, enumerated as twenty-eighth Johnson solid

J_

. This construction is similar to the next one, the

square gyrobicupola In geometry, the square gyrobicupola is one of the Johnson solids (). Like the square orthobicupola (), it can be obtained by joining two square cupolae () along their bases. The difference is that in this solid, the two halves are rotated 45 de ...

, which is twisted one of the cupolae around 45°.

Properties

The square orthobicupola has surface area

A

of a total sum of its area's faces, eight equilateral triangles and two squares. Its volume

V

is twice that of the square cupola's volume. With the edge length

a

, they are:

\begin
 A &= \left(2 \cdot \sqrt + 10\right)a^2 \approx 13.464a^2, \\
 V &= \left(2+\frac\right)a^3 \approx 3.886a^3.
\end

The square orthobicupola has an

axis of symmetry An axis (: axes) may refer to: Mathematics *A specific line (often a directed line) that plays an important role in some contexts. In particular: ** Coordinate axis of a coordinate system *** ''x''-axis, ''y''-axis, ''z''-axis, common names f ...

(a line passing through the center of two cupolas at their top) that rotates around one-, two-, and third-fourth of a full turn, and is reflected over the plane so the appearance remains symmetrical. The solid is also symmetrical by reflection over three mutually orthogonal planes.

References

External links

* Johnson solids {{Polyhedron-stub