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In
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 pentagonal orthobirotunda is a polyhedron constructed by attaching two
pentagonal rotunda The pentagonal rotunda is a convex polyhedron with regular polygonal faces. These faces comprise ten equilateral triangles, six regular pentagons, and one regular decagon, making a total of seventeen. The pentagonal rotunda is an example of Johns ...
e along their
decagon In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°. Regular decagon A '' regular decagon'' has a ...
al faces, matching like faces. It is an example of
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 ...
.


Construction

The pentagonal orthobirotunda is constructed by attaching two
pentagonal rotunda The pentagonal rotunda is a convex polyhedron with regular polygonal faces. These faces comprise ten equilateral triangles, six regular pentagons, and one regular decagon, making a total of seventeen. The pentagonal rotunda is an example of Johns ...
s to their base, covering decagon faces. The resulting polyhedron has 32 faces, 30 vertices, and 60 edges. This construction is similar to
icosidodecahedron In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (''icosi-'') triangular faces and twelve (''dodeca-'') pentagonal faces. An icosidodecahedron has 30 identical Vertex (geometry), vertices, with two triang ...
(or pentagonal gyrobirotunda), an
Archimedean solid The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They ...
: the difference is one of its rotundas twisted around 36°, making the pentagonal faces connect to the triangular one, a process known as
gyration In geometry, a gyration is a rotation in a discrete subgroup of symmetries of the Euclidean plane such that the subgroup does not also contain a reflection symmetry whose axis passes through the center of rotational symmetry. In the orbifold ...
. 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 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 the
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 ...
. The pentagonal orthobirotunda is one of them, enumerated as the 34th Johnson solid J_ .


Properties

The surface area of an icosidodecahedron A can be determined by calculating the area of all pentagonal faces. The volume of an icosidodecahedron V can be determined by slicing it off into two pentagonal rotunda, after which summing up their volumes. Therefore, its surface area and volume can be formulated as: \begin A &= \left(5\sqrt+3\sqrt\right) a^2 &\approx 29.306a^2 \\ V &= \fraca^3 &\approx 13.836a^3. \end


References


External links

* Johnson solids {{Polyhedron-stub