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 pentagrammic prism is one of an infinite set of nonconvex prisms formed by square sides and two regular

star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, Decagram (geometry)#Related figures, certain notable ones can ...

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 around ...

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 ...

, 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 Face (geometry), faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruence (geometry), congruent. Uniform po ...

. The pentagram face has an ambiguous interior because it is self-intersecting. The central pentagon region can be considered interior or exterior, depending on how the interior is defined. One definition of the interior is the set of points from which a

ray Ray or RAY may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), the bony or horny spine on ray-finned fish Science and mathematics * Half-line (geometry) or ray, half of a line split at an ...

crosses the boundary an odd number of times; this makes the central pentagon exterior, as every ray beginning within it crosses two edges.

Gallery

Pentagrammic dipyramid

, the pentagrammic

dipyramid In geometry, a bipyramid, dipyramid, or double pyramid is a polyhedron formed by fusing two Pyramid (geometry), pyramids together base (geometry), base-to-base. The polygonal base of each pyramid must therefore be the same, and unless otherwise ...

(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 Face (geometry), faces are the same. More specifically, all faces must be not ...

star dipyramids containing

arrangement of edges. It has 10 intersecting

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...

faces. It is topologically identical to the pentagonal dipyramid. Each star dipyramid is the

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual number, a nu ...

of a star polygon based uniform prism.

Related polyhedra

There are two pentagrammic

trapezohedra In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron Remarks: the faces of a deltohedron are deltoids; a (non-twisted) kite or deltoid can be dissected into two isosceles triangles or "deltas" (Δ), ba ...

(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 oppos ...

respectively, each having intersecting

kite A kite is a tethered heavier than air flight, heavier-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. Kites often have ...

-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

Uniform polyhedra Prismatoid polyhedra {{Polyhedron-stub