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 ...

, the pentagrammic prism is one of an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two

pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...

s. It is a special case of a right prism with a pentagram as base, which in general has rectangular non-base faces. Topologically it is the same as a convex

pentagonal prism In geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with seven faces, fifteen edges, and ten vertices. As a semiregular (or uniform) polyhedron If faces are all regular, the pentagonal prism is ...

. It is the 78th model in the

list of uniform polyhedra In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive ( transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are c ...

, as the first representative of uniform star prisms, along with the

pentagrammic antiprism In geometry, the pentagrammic antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams. It has 12 faces, 20 edges and 10 vertices. This polyhedron is iden ...

, which is the 79th model.

Geometry

It has 7 faces, 15 edges and 10 vertices. This polyhedron is identified with the indexed name U₇₈ as a

uniform polyhedron In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also ...

. The triangle face has an ambiguous interior because it is self-intersecting. The central pentagon region can be considered a angel or exterior depending on how the interior is defined. One definition of the interior is the set of points that have a ray that crosses the boundary an odd number of times to escape the diameter

Gallery

Pentagrammic dipyramid

, the pentagrammic dipyramid (or bipyramid) is first of the infinite set of

face-transitive In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congrue ...

star dipyramids containing star polygon arrangement of edges. It has 10 intersecting isosceles triangle faces. It is topologically identical to the

pentagonal dipyramid In geometry, the pentagonal bipyramid (or dipyramid) is third of the infinite set of face-transitive bipyramids, and the 13th Johnson solid (). Each bipyramid is the dual of a uniform prism. Although it is face-transitive, it is not a Plat ...

. Each star dipyramid is the dual of a star polygon based uniform prism.

Related polyhedra

There are two pentagrammic

trapezohedra In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a hi ...

(or deltohedra), being dual to the

and

pentagrammic crossed antiprism In geometry, the pentagrammic crossed-antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams. It differs from the pentagrammic antiprism by having oppo ...

respectively, each having intersecting

kite A kite is a tethered heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create lift and drag forces. A kite consists of wings, tethers and anchors. Kites often have a bridle and tail to guide the fac ...

-shaped faces (convex or concave), and a total of 12 vertices:

References

External links

* * * * *http://www.mathconsult.ch/showroom/unipoly/78.html *http://bulatov.org/polyhedra/uniform/u03.html
Paper model of pentagrammic prism
*https://web.archive.org/web/20050313234702/http://www.math.technion.ac.il/~rl/kaleido/data/03.html *https://web.archive.org/web/20060211140715/http://www.ac-noumea.nc/maths/amc/polyhedr/no_conv5_.htm

Prismatoid polyhedra Pyramids and bipyramids {{Polyhedron-stub