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 c ...

of 4 dimensions or higher, a double pyramid or duopyramid or fusil is a

polytope In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...

constructed by 2

orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

polytopes with edges connecting all pairs of vertices between the two. The term fusil is used by Norman Johnson as a rhombic-shape. The term ''duopyramid'' was used by George Olshevsky, as the dual of a

duoprism In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, wher ...

Polygonal forms

The lowest dimensional forms are 4 dimensional and connect two polygons. A ''p''-''q'' duopyramid or ''p''-''q'' fusil, represented by a composite

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

+ , and Coxeter-Dynkin diagram . The regular

16-cell In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mi ...

can be seen as a 4-4 duopyramid or 4-4 fusil, , symmetry , order 128. A ''p-q duopyramid'' or ''p-q'' fusil has

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...

symmetry 'p'',2,''q'' order 4pq. When ''p'' and ''q'' are identical, the symmetry in

Coxeter notation In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram ...

is doubled as or ''p'',2⁺,2''q'' order 8''p''². Edges exist on all pairs of vertices between the ''p''-gon and ''q''-gon. The 1-skeleton of a ''p''-''q'' duopyramid represents edges of each ''p'' and ''q'' polygon and ''pq''

complete bipartite graph In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory ...

between them.

Geometry

A ''p''-''q'' duopyramid can be seen as two regular planar polygons of ''p'' and ''q'' sides with the same center and orthogonal orientations in 4 dimensions. Along with the ''p'' and ''q'' edges of the two polygons, all permutations of vertices in one polygon to vertices in the other form edges. All faces are triangular, with one edge of one polygon connected to one vertex of the other polygon. The ''p'' and ''q'' sided polygons are ''hollow'', passing through the polytope center and not defining faces. Cells are tetrahedra constructed as all permutations of edge pairs between each polygon. It can be understood by analogy to the relation of the 3D

prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentary ...

s and their dual

bipyramid A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices. The "-gonal" in the name of a bipyramid does not ...

s with Schläfli symbol + , and a

rhombus In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...

in 2D as + . A bipyramid can be seen as a 3D degenerated duopyramid, by adding an edge across the

digon In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visua ...

on the inner axis, and adding intersecting interior triangles and tetrahedra connecting that new edge to p-gon vertices and edges. Other nonuniform polychora can be called duopyramids by the same construction, as two orthogonal and co-centered polygons, connected with edges with all combinations of vertex pairs between the polygons. The symmetry will be the product of the symmetry of the two polygons. So a ''rectangle-rectangle duopyramid'' would be topologically identical to the uniform ''4-4 duopyramid'', but a lower symmetry ,2,2 order 16, possibly doubled to 32 if the two rectangles are identical.

Coordinates

The coordinates of a p-q duopyramid (on a unit

3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...

) can be given as: : (cos(2*πi/p),sin(2*πi/p),0,0), ''i''=1..''p'' : (0,0,cos(2*πj/q),sin(2*πj/q)), ''j''=1..''q'' All pairs of vertices are connected by edges.

Perspective projections

Orthogonal projections

The 2n vertices of a ''n-n duopyramid'' can be orthogonally projected into two regular n-gons with edges between all vertices of each n-gon. The regular

can be seen as a 4-4 duopyramid, being dual to the 4-4 duoprism, which is the

tesseract In geometry, a tesseract is the four-dimensional analogue of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eig ...

. As a 4-4 duopyramid, the 16-cell's symmetry is

,2,4 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...

order 64, and doubled to , order 128 with the 2 central squares interchangeable. The regular 16-cell has a higher symmetry ,3,4 order 384.

Example 6-4 duopyramid

References

{{reflist 4-polytopes