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Dual polyhedron

In geometry, a dual polyhedron of a given polyhedron is obtained by interchanging its vertices and faces, such that the vertices of the dual correspond to the faces of the original and vice versa, with edges connecting pairs of vertices in the dual if the corresponding faces in the original share an edge.^[1] This construction, often realized geometrically by placing vertices at the centroids of the original faces and connecting them with edges that cross the original edges perpendicularly, preserves the number of edges while swapping the counts of vertices and faces.^[1] The dual of a dual polyhedron is the original polyhedron, establishing a reciprocal relationship.^[2] Among the five Platonic solids, duality pairs the cube with the octahedron, the dodecahedron with the icosahedron, and leaves the tetrahedron self-dual, meaning it is isomorphic to its own dual.^[2] For instance, the cube's six square faces yield six vertices in its dual octahedron, while the octahedron's eight triangular faces produce eight vertices in the cube.^[2] This duality extends to other classes of polyhedra, such as Archimedean solids, where duals are Catalan solids with identical edge counts but interchanged vertex and face numbers.^[3] Key properties include the equality of the volume-to-surface-area ratios between a polyhedron and its dual when normalized appropriately, and the relation R r = \rho^2, where R is the circumradius, r the inradius, and \rho the midradius of the original.^[4] Duality plays a fundamental role in polyhedral combinatorics, facilitating analyses of symmetry, Euler characteristics, and geometric realizations.^[1]

Core Concepts

Definition

In geometry, a dual polyhedron of a given polyhedron, referred to as the primal polyhedron, is defined such that there exists an anti-isomorphism between their face lattices, establishing a one-to-one correspondence where each vertex of the primal corresponds to a face of the dual, and each face of the primal corresponds to a vertex of the dual.^[5] This correspondence preserves the incidence structure combinatorially: the degree of each face in the dual equals the number of edges incident to the corresponding vertex in the primal, while the valence of each vertex in the dual equals the number of sides of the corresponding face in the primal.^[5] This combinatorial duality can be realized geometrically for convex polyhedra, but exists abstractly for any polyhedron with a valid face lattice.^[4] For the duality to be well-defined, the primal polyhedron must be convex, ensuring that the geometric realization aligns with the combinatorial interchange without ambiguities in face orientations or intersections.^[6] Alternatively, the polyhedron must be simple, meaning it is topologically equivalent to a sphere (genus 0), which guarantees a consistent embedding in three-dimensional space.^[5] This duality preserves key topological invariants, notably Euler's formula: for both the primal and dual, the Euler characteristic satisfies V - E + F = 2, where V is the number of vertices, E the number of edges, and F the number of faces, reflecting their shared spherical topology.^[5] Geometrically, such duals can be realized via polar reciprocation with respect to a sphere centered at an interior point of the primal.^[5]

Basic Properties

One fundamental property of dual polyhedra is the equality in the number of edges. The primal polyhedron and its dual share the exact same number of edges, as each edge in the primal connects two vertices and bounds two faces, corresponding directly to an edge in the dual that connects the dual faces associated with those primal vertices.^[4] A key structural relation is the interchange between faces and vertices. Specifically, the number of faces of the primal polyhedron equals the number of vertices of the dual, and conversely, the number of vertices of the primal equals the number of faces of the dual. This swap arises inherently from the duality construction, where each face of the primal becomes a vertex in the dual and vice versa.^[4] Duality also establishes a correspondence between vertex degrees and face valences. The degree of a vertex in the primal polyhedron, which is the number of edges incident to it, equals the number of sides of the corresponding face in the dual. Similarly, the number of sides of a face in the primal matches the degree of the corresponding vertex in the dual. This valence-degree duality ensures that local connectivity patterns are preserved in a reversed manner.^[4] For a regular polyhedron with Schläfli symbol {p, q}—where p is the number of sides per face and q is the number of faces meeting at each vertex—its dual has the symbol {q, p}. This reversal reflects the face-vertex interchange in the regular context.^[7] Finally, duality preserves regularity. The dual of a regular polyhedron is also regular, maintaining congruent regular polygonal faces and the same vertex figures, albeit interchanged. This property holds for the Platonic solids, where pairs like the cube and octahedron are mutual duals, both exhibiting full symmetry.^[4]

Types of Duality

Geometric Duality

Geometric duality in polyhedra arises primarily through polar reciprocation, a transformation defined with respect to a reference sphere known as the polar sphere. This operation maps each point inside the sphere to a plane outside it, and each plane outside the sphere to a point inside it, preserving incidence relations between vertices, edges, and faces of the primal polyhedron and its dual. Specifically, for a point a not at the origin O, its polar plane is given by a^\dagger = \{ b \in \mathbb{R}^3 \mid Oa \cdot Ob = 1 \} with respect to the unit sphere, while a plane H not through O maps to its pole H^\dagger such that H = \{ a \in \mathbb{R}^3 \mid OH^\dagger \cdot Oa = 1 \}. The vertices of the dual polyhedron correspond to the polar planes of the primal's faces, and the faces of the dual lie in the polar planes of the primal's vertices, establishing a vertex-face correspondence.^[8] To ensure the dual polyhedron is convex, the primal must be positioned such that it lies entirely inside the polar sphere, with the origin O (the center of the sphere) in its interior; this guarantees that the dual, defined as A^* = \{ b \in \mathbb{R}^3 \mid Oa \cdot Ob \leq 1 \ \forall a \in A \}, is convex and bounded. The center of symmetry is typically chosen at the centroid of the primal polyhedron for canonical positioning, aligning the dual symmetrically around the same point and facilitating balanced geometric properties. If the polar sphere serves as a midsphere—tangent to all edges of the primal—the edges of the primal become tangent to the sphere, resulting in a dual polyhedron whose edges are perpendicular to those of the primal, with the sphere serving as the midsphere for both and enhancing symmetry in their edge arrangements.^[8]^[4] The concept of geometric duality through polar reciprocation has historical roots in Johannes Kepler's work, where he explored the reciprocal relationship between the cube and octahedron in his 1619 treatise Harmonices Mundi, laying early groundwork for understanding dual pairs among Platonic solids.^[9]

Topological Duality

Topological duality provides a combinatorial framework for understanding the relationship between a polyhedron and its dual, focusing on abstract incidence structures and graph-theoretic properties rather than spatial geometry. This perspective treats polyhedra as 3-connected planar graphs or more generally as cell complexes on surfaces, where duality interchanges vertices and faces while preserving the overall connectivity and topological invariants. Unlike geometric duality, which relies on metric embeddings, topological duality applies to any polyhedral complex with a well-defined face lattice, enabling analysis of non-realizable or abstract configurations.^[10] The cornerstone of topological duality is the construction of the dual graph. For a polyhedron with graph G, the dual graph G^* has a vertex for each face of the primal polyhedron, and an edge connecting two vertices of G^* if the corresponding primal faces share an edge. This ensures that the degree of each vertex in G^* equals the number of edges bounding the corresponding primal face. For polyhedral graphs—simple, 3-connected planar graphs representing convex polyhedra—the dual graph G^* embeds in the plane and forms the 1-skeleton (edge graph) of the reciprocal polyhedron, maintaining planarity and 3-connectivity.^[11]^[12] This duality extends naturally to polyhedra embedded on orientable surfaces of arbitrary genus g, where it preserves the Euler characteristic \chi = V - E + F = 2 - 2g. The interchange of vertices and faces (with V^* = F and F^* = V) while keeping the edge count fixed (E^* = E) ensures \chi^* = \chi, thus maintaining the surface's topology under homeomorphisms. In the abstract polytope setting, duality generalizes to ranked posets representing incidence structures, where the dual P^* of an n-polytope P is obtained by reversing the partial order on faces; this interchanges the ranks of elements in flags, swapping vertex-like and facet-like incidences without reference to embedding.^[11]^[13] A key distinction from geometric duality is that topological duals exist for non-convex polyhedra or purely abstract polytopes lacking a metric realization in Euclidean space, capturing only the combinatorial type via face lattices and connectivity. For instance, self-dual abstract polytopes, where P \cong P^*, arise from symmetric incidence structures that may not correspond to convex bodies. This abstraction facilitates study in higher ranks or non-Euclidean contexts, emphasizing flags and automorphisms over coordinates.^[13]^[10]

Construction Methods

Polar Reciprocation Process

The polar reciprocation process provides a geometric method to construct the dual of a convex polyhedron by applying a central inversion with respect to a unit sphere centered at the origin, effectively interchanging vertices with faces through point-plane reciprocity. This transformation relies on the inner product in Euclidean space and preserves the combinatorial structure while reversing the roles of vertices, edges, and faces.^[8] The process assumes the primal polyhedron is convex, as non-convex cases may not yield a well-defined dual under this polarity.^[14] To begin, translate and scale the primal polyhedron so that its centroid coincides with the origin and all vertices lie inside the unit sphere, ensuring the origin is in the strict interior.^[8] This setup defines the polarity with respect to the unit sphere, where the radius r = 1 yields a canonical form for the dual, with the dual polyhedron circumscribed about the sphere.^[14] The dual is then the convex body consisting of all points \vec{x} satisfying \vec{x} \cdot \vec{v} \leq 1 for every primal vertex \vec{v}.^[8] Each vertex \vec{v} of the primal polyhedron maps to a supporting plane of the dual, given by the equation \vec{x} \cdot \vec{v} = 1, which is perpendicular to \vec{v} and at a signed distance $1 / \| \vec{v} \| from the origin.^[8] Conversely, each face of the primal polyhedron, defined by its supporting plane equation \vec{x} \cdot \vec{n} = 1 (where \vec{n} is scaled such that the right-hand side is 1), maps to a vertex of the dual located at the position \vec{n}.^[14] This normalization of the face planes ensures the dual vertices lie outside the unit sphere, with coordinates directly derived from the primal face normals adjusted to the constant 1.^[8] The edges of the dual polyhedron correspond to the primal edges via intersections of reciprocal planes: for a primal edge connecting vertices \vec{v}_i and \vec{v}_j, the associated dual edge is the line of intersection between the planes \vec{x} \cdot \vec{v}_i = 1 and \vec{x} \cdot \vec{v}_j = 1, which connects the dual vertices arising from the two faces adjacent to that primal edge.^[14] This intersection lies in the dual and bounds the faces corresponding to \vec{v}_i and \vec{v}_j.^[8] The resulting dual is a polyhedron whose faces are the polars of the primal vertices and whose vertices are the poles of the primal faces, maintaining the overall topology.^[14]

Canonical Duals

The canonical dual of a uniform polyhedron is constructed via polar reciprocation with respect to its midsphere, positioned as the unit sphere centered at the polyhedron's center, such that the edges of the dual are tangent to this sphere at the same points as the primal's edges.^[15] This normalization ensures a standardized geometric form where the primal and dual share the midsphere, forming a canonical dual compound, as described by Coxeter in his analysis of regular polytopes.^[15] For Archimedean solids, the canonical duals are the Catalan solids, which are face-transitive polyhedra with all faces congruent and all edges of equal length.^[3] In these duals, the vertex figures correspond directly to the faces of the original Archimedean solid, preserving the geometric arrangement while inverting the roles of vertices and faces. The full symmetry group of the primal Archimedean solid is inherited by the Catalan dual, maintaining rotational and reflectional equivalences.^[3] A prominent example is the regular tetrahedron, whose canonical dual is itself, as it is self-dual with coinciding inscribed and circumscribed spheres in this positioning. For polyhedra with octahedral (cubic) symmetry, such as the cuboctahedron, the canonical dual is the rhombic dodecahedron, a Catalan solid featuring 12 rhombic faces tangent to the shared unit midsphere. In general, canonical duals of uniform polyhedra exhibit both an inscribed sphere and a circumsphere, though these coincide only in self-dual cases like the tetrahedron.^[15]

Dorman-Luke Construction

The Dorman-Luke construction is a geometric method for determining the shape of the faces in the dual of a uniform polyhedron, relying on the primal polyhedron's vertex figure rather than explicit coordinate calculations or polar reciprocation. Developed by Dorman Luke and detailed in the standard reference on polyhedral models, it provides a practical, hands-on approach to duality that emphasizes the relationship between a vertex configuration and the corresponding dual face.^[16] This technique is particularly suited to Archimedean solids and their duals, the Catalan solids, where the dual faces are irregular polygons derived systematically from regular or semiregular vertex arrangements.^[4] The process begins by selecting a vertex figure from the primal uniform polyhedron, which is a polygon formed by connecting the midpoints of the edges meeting at that vertex; due to uniformity, all vertex figures are congruent and lie in parallel planes. Next, inscribe a circumcircle around this vertex figure, as the figure is always cyclic for uniform polyhedra. Then, at each vertex of the figure, construct the tangent line to the circumcircle; these tangents intersect to form the boundary of the dual face, which becomes a tangential polygon inscribed in its own incircle (the polar counterpart to the primal's circumcircle). Finally, repeating this for each vertex of the primal yields the complete set of dual faces, with dual vertices positioned at the face centers of the primal and dual edges perpendicular to the primal edges.^[16]^[4] This construction offers several advantages as a coordinate-free tool: it facilitates manual drafting or model-building using compass and straightedge, making it accessible for educational or illustrative purposes without algebraic computation, and it naturally produces duals that are tangential polyhedra (with an inscribed sphere tangent to all faces), aligning with the midsphere property of uniform polyhedra.^[16] It also highlights the intrinsic duality between vertex figures and faces, extending conceptually to non-Euclidean settings where spherical projections preserve angular relations. However, the method assumes a convex, uniform primal polyhedron with a well-defined center and cyclic vertex figures, limiting its direct application to non-uniform or star polyhedra, and the resulting dual is determined only up to similarity (scaling).^[4] Historically, Luke, an avid polyhedral modeler from West Palm Beach, Florida, contributed this and related techniques to the literature on geometric constructions, with the method first systematically presented as a practical alternative to more analytic duality approaches in mid-20th-century polyhedra studies.^[16] For instance, applying it to the regular tetrahedron yields its self-dual form, confirming the construction's consistency with known cases.^[4]

Special Cases

Self-Dual Polyhedra

A self-dual polyhedron is defined as a polyhedron that is combinatorially isomorphic to its own dual, meaning there exists a bijection between its vertices and faces that preserves the incidence structure, effectively making the duality mapping an automorphism of the polyhedron. This isomorphism requires that the number of vertices equals the number of faces, denoted as V = F.^[17] For convex self-dual polyhedra, Euler's formula V - E + F = 2 implies a necessary condition on the edge count. Substituting V = F yields $2V - E = 2, so E = 2V - 2. Since V is an integer greater than or equal to 4, E must be even. This condition distinguishes self-dual polyhedra from general polyhedra, where the number of edges can be odd, as in the case of a triangular prism with 9 edges.^[18] The regular tetrahedron provides the simplest example of a self-dual polyhedron, possessing 4 triangular faces and 4 vertices, with the duality mapping interchanging vertices and faces while preserving the regular tetrahedral symmetry. Another basic example is the square pyramid, which has 5 vertices (4 forming the square base and 1 apex) and 5 faces (4 triangular lateral faces and 1 square base); here, the self-duality interchanges the apex vertex with the base face and the base vertices with the triangular faces. More complex examples include the pentagonal pyramid, with 6 vertices and 6 faces, demonstrating how pyramids with regular polygonal bases form an infinite family of self-dual polyhedra when appropriately realized.^[19]^[20] Enumerations of convex self-dual polyhedra reveal a rapid increase in combinatorial types with size. For instance, there is 1 such polyhedron with 4 faces, 1 with 5 faces, 2 with 6 faces, 6 with 7 faces, and 16 with 8 faces. These counts highlight the diversity even among small self-dual polyhedra, though explicit classifications become computationally intensive beyond low face counts.^[21] In geometric realizations, self-dual polyhedra often admit a midsphere—a sphere tangent to all edges—enabling a tangential configuration where edges touch the sphere at their midpoints. This property facilitates symmetric embeddings and is evident in examples like Kirkman's icosahedron, a self-dual polyhedron with 20 faces where all 38 edges are tangent to a common midsphere of radius 12 centered at the origin. Such realizations underscore the interplay between combinatorial self-duality and geometric tangency conditions.^[22]

Regular and Uniform Dual Pairs

Among regular polyhedra, the Platonic solids exhibit particularly symmetric dual relationships, where each solid's dual is another Platonic solid. The regular tetrahedron is self-dual, as its dual is congruent to itself, with vertices positioned at the centroids of its four triangular faces. The cube and regular octahedron form a dual pair, with the octahedron's vertices at the cube's face centers and vice versa; similarly, the regular dodecahedron and regular icosahedron are duals, interchanging their 12 pentagonal and 20 triangular faces, respectively.^[23] These pairings arise from the reciprocal nature of their Schläfli symbols, where the roles of vertices and faces are swapped.^[4] The dual Platonic solids share identical symmetry groups, preserving the full rotational and reflectional structure of the original. The tetrahedron and its dual possess the tetrahedral symmetry group T_d of order 24 (with rotational subgroup A_4 of order 12). The cube-octahedron pair has octahedral symmetry O_h of order 48 (rotational subgroup O of order 24), while the dodecahedron-icosahedron pair exhibits icosahedral symmetry I_h of order 120 (rotational subgroup A_5 of order 60).^[24] Notably, all Platonic dual pairs have the same number of edges: 6 for the tetrahedron, 12 for the cube-octahedron, and 30 for the dodecahedron-icosahedron. In visualization, these pairs often form interpenetrating compounds where one solid's vertices lie at the other's face centers, creating complementary spatial fillings, such as the cube-octahedron compound that embeds harmoniously within a common bounding sphere. Extending to uniform polyhedra, the 13 Archimedean solids—vertex-transitive polyhedra with regular polygonal faces—have duals known as the Catalan solids, which are face-transitive with congruent irregular faces. For instance, the truncated tetrahedron, with its mix of triangular and hexagonal faces, is dual to the triakis tetrahedron, featuring 12 identical isosceles triangular faces; both share 18 edges and tetrahedral symmetry T_d. Similarly, the cuboctahedron is dual to the rhombic dodecahedron, with 24 edges each and octahedral symmetry O_h, where the rhombic faces correspond to the cuboctahedron's vertices. These uniform dual pairs maintain the same edge count and symmetry group as their Archimedean counterparts, enabling analogous compound formations that highlight their geometric complementarity, such as the cuboctahedron-rhombic dodecahedron compound.^[25]^[3]

Generalizations

Dual Polytopes

In the context of n-dimensional geometry, a dual polytope is defined combinatorially as an abstract polytope whose face lattice is the order-reverse of the primal polytope's face lattice, interchanging vertices with facets, edges with (n-2)-faces, and so on, while preserving the incidence relations up to reversal.^[26] This duality ensures that the dual is also an n-dimensional polytope, maintaining the same rank n. Geometrically, for a convex n-polytope P in Euclidean n-space containing the origin in its strict interior, the polar dual P* is realized as the set of points y such that the dot product x · y ≤ 1 for all x in P; here, the vertices of P* correspond to the supporting hyperplanes (facets) of P, and the facets of P* correspond to the vertices of P.^[8] The properties of dual polytopes include the bijection between k-faces of P and (n-k-1)-faces of P*, which implies that the f-vector (recording the number of faces of each dimension) of the dual is the reverse of the primal's boundary f-vector. Consequently, the Euler characteristic of the boundary complex, given by the alternating sum ∑_{k=0}^{n-1} (-1)^k f_k, is preserved for the dual, equaling 1 + (-1)^{n-1} regardless of the specific polytope; this value alternates between 0 and 2 depending on the parity of n, reflecting the topological equivalence to an (n-1)-sphere.^[8] Combinatorial types of polytopes can thus be realized geometrically through this reciprocal polarity with respect to the unit hypersphere centered at the origin, ensuring that every convex polytope admits a dual realization in Euclidean n-space.^[26] Representative examples illustrate these concepts: the n-simplex is self-dual, as its face lattice is symmetric under reversal, with the 4-dimensional simplex (pentachoron) serving as a specific case where vertices and facets both number 5. In contrast, the 4-dimensional hypercube (tesseract), with 16 vertices and 8 cubic facets, is dual to the 4-dimensional cross-polytope (16-cell), which has 8 vertices and 16 tetrahedral facets. These pairs highlight how duality interchanges the roles of vertices and facets while preserving overall combinatorial structure.^[8]

Dual Tessellations

In dual tessellations, the cells of the dual structure correspond one-to-one with the vertices of the primal tessellation, while the vertices of the dual correspond to the cells of the primal. Edges in the dual connect pairs of vertices if the associated primal cells share a face, and faces in the dual arise from primal edges. This reciprocal mapping extends the combinatorial duality of finite polyhedra to infinite space-filling arrangements in Euclidean or hyperbolic geometries, reversing incidence relations while preserving the overall topology.^[27] In three-dimensional Euclidean space, the tetrahedral-octahedral honeycomb, a uniform honeycomb consisting of regular tetrahedra and octahedra (obtained by alternation of the cubic honeycomb), has as its dual the rhombic dodecahedral honeycomb, consisting of rhombic dodecahedra with four meeting at each edge. Honeycomb duals in Euclidean 3D preserve space-filling density, as both primal and dual occupy the full volume without overlaps or voids, maintaining equivalent packing efficiency.^[28] Hyperbolic tessellations exhibit duality through the interchange of cell and vertex configurations, where a honeycomb with Schläfli symbol {p,q,r} has dual {r,q,p}, reversing the branching order from cell faces to vertex figures. For uniform hyperbolic honeycombs described by Wythoff symbols of the form | 2 p q, the dual interchanges p and q to yield | 2 q p, transforming the vertex arrangement of the primal into the cell type of the dual. This construction generates infinite families of dual pairs in hyperbolic 3-space, such as the icosahedral {3,5,3} dual to itself and the dodecahedral {5,3,5}, both self-dual but illustrating the general reversal.^[27] Properties of dual tessellations include the interchange of vertex figures and cells: the vertex figure of the primal, describing local neighborhood around a vertex, becomes a cell of the dual. The coordination number—the number of cells meeting at a vertex in the dual—equals the cell valence of the primal, defined as the number of faces converging at a vertex within a primal cell. These invariances ensure that dual pairs share isomorphic symmetry groups and equivalent Euler characteristics per unit volume in their respective geometries.^[28] A representative example in Euclidean 3D is the Voronoi tessellation, which acts as the dual to the Delaunay triangulation. The Voronoi diagram divides space into polyhedral cells, each comprising points nearest to a given site, while the Delaunay triangulation connects sites via tetrahedra such that no other site lies inside the circumsphere of any tetrahedron; their duality manifests as a dimension-complementary correspondence, with Voronoi vertices matching Delaunay tetrahedra and vice versa. This pair underpins applications in spatial partitioning and lattice analysis.^[29]

Self-Dual Polytopes and Tessellations

A self-dual polytope is defined as a polytope that is combinatorially isomorphic to its polar dual, meaning there exists a combinatorial equivalence between the face lattice of the polytope and that of its dual.^[30] This isomorphism implies that the f-vector of the polytope, which records the number of faces of each dimension, is palindromic: the number of k-dimensional faces equals the number of (n-1-k)-dimensional faces in an n-dimensional polytope.^[30] In three dimensions, this property specifically requires that the number of vertices equals the number of faces, a condition that generalizes to equal counts of k-faces and their complementary ranks in higher dimensions.^[30] In four dimensions, notable examples of self-dual polytopes include the 24-cell, also known as the octaplex, which is a regular 4-polytope with 24 octahedral cells and is combinatorially equivalent to its dual.^[31] Another example is the 5-cell, or pentachoron, a regular 4-simplex that is self-dual due to the inherent symmetry of simplices in any dimension.^[32] These structures highlight how self-duality arises in regular polytopes where the Schläfli symbol is palindromic, preserving the combinatorial structure under duality. Self-duality extends to tessellations, where infinite arrangements of polytopes tile space without gaps or overlaps. The hypercubic honeycomb, with Schläfli symbol {4,3^{n-2},4} in n dimensions, is self-dual because its cells are hypercubes and its vertex figures are dual cross-polytopes, resulting in a symmetric dual tessellation identical to itself.^[33] In two dimensions, the square tiling {4,4} serves as a self-dual example, where the tiling by squares is combinatorially equivalent to its dual, which is again a square tiling.^[34] Such tessellations demonstrate self-duality in Euclidean space across dimensions, with the palindromic symmetry ensuring the dual operation yields the same arrangement. Enumerating self-dual polytopes presents significant challenges, particularly in higher dimensions, where relatively few convex examples are fully classified beyond simplices and certain uniform polytopes. While infinite families exist in hyperbolic space, such as regular honeycombs with palindromic Schläfli symbols like {p,q,p} under appropriate conditions for hyperbolicity, convex self-dual polytopes in Euclidean space remain limited, with known finite instances up to eight dimensions often connected to exceptional structures like those associated with the E_8 root system.^[30] These enumeration difficulties stem from the need to match combinatorial types across dual pairs, compounded by the exponential growth in possible face configurations in higher dimensions.^[30]

References

[1]
[PDF] The Platonic Solids - Whitman College
May 12, 2016 · That is, 1p1,q1l is the dual Platonic solid to 1p2,q2l and vice versa. Definition 2.6 (Dual Polyhedra). For every polyhedron, there exists a ...
[2]
Duals of Regular Polyhedra - Brown Math
The five regular polyhedra fall into three groups: two dual pairs and one polyhedron that is dual to itself.Missing: definition geometry
[3]
[PDF] A note on the rank of self-dual polyhedra. - WPI
A polyhedron P is said to be self-dual if there is an isomorphism δ : P → P∗, where P∗ denotes the dual of P. We may regard δ as a permutation of the ...
[4]
[PDF] An Introduction to Convex Polytopes - UNI-Lj
convex polytopes. The highlights of the book are three main theorems in the ... Two polytopes are said to be dual (and each is said to be a dual of the.
[5]
[PDF] Branko Grünbaum: - Are your polyhedra the same as my polyhedra ...
For a given geometric polyhedron the construction of a dual polyhedron is most often carried out by applying to its faces and vertices a polarity. (that is ...
[6]
Dual Polyhedron -- from Wolfram MathWorld
A dual polyhedron is another polyhedron where faces and vertices occupy complementary locations, and vertices of one correspond to faces of the other.
[7]
Schläfli Symbol -- from Wolfram MathWorld
The symbol has the particularly nice property that its reversal gives the symbol of the dual polyhedron. The following tables gives Schläfli symbols for several ...Missing: duality | Show results with:duality
[8]
[PDF] Chapter 6 Polar Duality, Polyhedra and Polytopes - UPenn CIS
Remark: Some authors call “convex” polyhedra and. “convex” polytopes what we have simply called polyhedra and polytopes. Since Definition 6.2.1 implies that ...
[9]
https://www.degruyterbrill.com/document/doi/10.1515/9781400838561.51/html
[10]
Abstract Regular Polytopes
Abstract regular polytopes stand at the end of more than two millennia of geometrical research, which began with regular polygons and polyhedra.
[11]
Dual Graph -- from Wolfram MathWorld
On the other hand, polyhedral graphs have unique dual graphs. The dual graph G^* of a polyhedral graph G has graph vertices each of which corresponds to a face ...
[12]
Polyhedral Graph -- from Wolfram MathWorld
An n -polyhedral graph (sometimes called a c -net) is a 3-connected simple planar graph on n nodes. Every convex polyhedron can be represented in the plane ...
[13]
[PDF] Abstract Regular Polytopes - Assets - Cambridge University Press
McMullen, Peter, 1955–. Abstract regular polytopes / Peter McMullen, Egon Schulte. p. cm. – (Encyclopedia of mathematics and its applications). Includes ...
[14]
https://arxiv.org/pdf/1403.2125.pdf
[15]
Midsphere -- from Wolfram MathWorld
Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 16, 1973. Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England ...
[16]
Catalan Solid -- from Wolfram MathWorld
The dual polyhedra of the Archimedean solids, given in the following table. They are known as Catalan solids in honor of the Belgian mathematician who first ...
[17]
[PDF] MATHEMATICAL MODELS -
Dorman Luke, of West Palm Beach, Florida, Mr. A. R. Pargeter of Southampton, and Mr. R. F. Wheeler of Hull, to whom much new material is due. Little of the ...
[18]
Self-dual polyhedra of given degree sequence - ResearchGate
Aug 10, 2025 · A self-dual polyhedron on p. vertices has pfaces and 2p−2edges (straightforward consequence of Euler's formula). In [10] we considered the ...<|control11|><|separator|>
[19]
Self-Dual Polyhedron -- from Wolfram MathWorld
A polyhedron that is dual to itself. For example, the tetrahedron is self-dual. Naturally, the skeleton of a self-dual polyhedron is a self-dual graph.Missing: definition | Show results with:definition
[20]
[PDF] Dual Models: One Shape to Make Them All - The Bridges Archive
The (topological) dual P0 of a polyhedron P has, by definition, a vertex for each face of P and an edge. between the vertices that correspond to faces of P ...
[21]
Symmetries of Canonical Self-Dual Polyhedra
A self-dual polyhedron has the same number of faces as vertices, so the table shows the counts of all canonical self-dual polyhedra with up to 16 faces ...
[22]
[PDF] A Polyhedron Full of Surprises
A familiar example involving the regular dodecahedron and its dual, the regular icosahedron, is shown in FIGURE 4. ... Coxeter, Regular Polytopes, 3rd ed., Dover, ...
[23]
Platonic Solid -- from Wolfram MathWorld
The Platonic solids, also called the regular solids or regular polyhedra, are convex polyhedra with equivalent faces composed of congruent convex regular ...
[24]
Polyhedral Group -- from Wolfram MathWorld
One of the symmetry groups of the Platonic solids. There are three polyhedral groups: the tetrahedral group of order 12, the octahedral group of order 24, and ...
[25]
Archimedean Dual -- from Wolfram MathWorld
The Archimedean duals are the 13 duals of the 13 Archimedean solids, sometimes called the Catalan solids. They are summarized in the following table and ...
[26]
What is a dual of a convex polytope?
A dual polytope has the same dimension as $ P$ . The duality theorem states that every convex polytope admits a dual.Missing: n- | Show results with:n-
[27]
[1511.02851] Visualizing Hyperbolic Honeycombs - arXiv
Nov 8, 2015 · We explore visual representations of tilings corresponding to Schläfli symbols. In three dimensions, we call these tilings "honeycombs".
[28]
https://store.doverpublications.com/products/9780486614809
[29]
None
### Summary of Voronoi Diagram and Delaunay Triangulation in 3D, and Polyhedral Tessellations/Honeycombs
[30]
[PDF] Self-duality of Polytopes and its Relations to Vertex Enumeration ...
Abstract We study the complexity of determining whether a polytope given by its vertices or facets is combinatorially isomorphic to its polar dual. We.
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The Self-Dual 24-Cell - Brown Math
Having 24 octahedral cells, it is called the 24-cell (see illustration on the following page). Around each vertex of such an object there will be six octahedra.Missing: 5- | Show results with:5-
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Regular polychoron numbers - OeisWiki
The 5-cell and 24-cell are self-dual, the 16-cell is the dual of the 8-cell, and the 600- and 120-cells are dual to each other. The number of cells, faces ...
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{5,3,5} Honeycomb | Visual Insight - AMS Blogs
Feb 1, 2014 · The cubic honeycomb in Euclidean space is also self-dual, and it has Schläfli symbol {4,3,4}. Coxeter diagrams are a notation similar to but ...
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Square tiling - Polytope Wiki - Miraheze
4 squares join at each vertex of this tiling. It is the only regular tiling of the plane to be self-dual. It is also the 2D hypercubic honeycomb. Square tiling.