polyhedral combinatorics Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral comb ...

(a branch of mathematics), a stacked polytope 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 ...

formed from a

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

by repeatedly gluing another simplex onto one of its

facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cut ...

s..

Examples

Every simplex is itself a stacked polytope. In three dimensions, every stacked polytope is a

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

with triangular faces, and several of the

deltahedra In geometry, a deltahedron (plural ''deltahedra'') is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek upper case delta (Δ), which has the shape of an equilateral triangle. There are infinitely many delt ...

(polyhedra with

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...

faces) are stacked polytopes In a stacked polytope, each newly added simplex is only allowed to touch one of the facets of the previous ones. Thus, for instance, the quadaugmented tetrahedron, a shape formed by gluing together five regular tetrahedra around a common line segment is a stacked polytope (it has a small gap between the first and last tetrahedron). However, the similar-looking

pentagonal bipyramid 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 Platoni ...

is not a stacked polytope, because if it is formed by gluing tetrahedra together, the last tetrahedron will be glued to two triangular faces of previous tetrahedra instead of only one. Other non-convex stacked deltahedra include:

Combinatorial structure

The

undirected graph In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' v ...

formed by the vertices and edges of a stacked polytope in ''d'' dimensions is a (''d'' + 1)-tree. More precisely, the graphs of stacked polytopes are exactly the (''d'' + 1)-trees in which every ''d''-vertex

clique A clique ( AusE, CanE, or ), in the social sciences, is a group of individuals who interact with one another and share similar interests. Interacting with cliques is part of normative social development regardless of gender, ethnicity, or popular ...

(complete subgraph) is contained in at most two (''d'' + 1)-vertex cliques.. See in particular p. 420. For instance, the graphs of three-dimensional stacked polyhedra are exactly the

Apollonian network In combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maxima ...

s, the graphs formed from a triangle by repeatedly subdividing a triangular face of the graph into three smaller triangles. One reason for the significance of stacked polytopes is that, among all ''d''-dimensional

simplicial polytope In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only triangular facesPolyhedra, Peter R. Cromwell, 1997. (p.341) and corresponds via Steinitz's ...

s with a given number of vertices, the stacked polytopes have the fewest possible higher-dimensional faces. For three-dimensional simplicial polyhedra the numbers of edges and two-dimensional faces are determined from the number of vertices by

Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for an ...

, regardless of whether the polyhedron is stacked, but this is not true in higher dimensions. Analogously, the simplicial polytopes that maximize the number of higher-dimensional faces for their number of vertices are the

cyclic polytope In mathematics, a cyclic polytope, denoted ''C''(''n'',''d''), is a convex polytope formed as a convex hull of ''n'' distinct points on a rational normal curve in R''d'', where ''n'' is greater than ''d''. These polytopes were studied by Constantin ...

References

{{reflist Polytopes