# Dual polyhedron

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In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. [1] Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. [2] Starting with any given polyhedron, the dual of its dual is the original polyhedron.

## Contents

Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. For example, the regular polyhedra the (convex) Platonic solids and (star) Kepler–Poinsot polyhedra  form dual pairs, where the regular tetrahedron is self-dual. The dual of an isogonal polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an isohedral polyhedron (one in which any two faces are equivalent [...]), and vice versa. The dual of an isotoxal polyhedron (one in which any two edges are equivalent [...]) is also isotoxal.

Duality is closely related to polar reciprocity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.

## Kinds of duality

There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality.

### Polar reciprocation

In Euclidean space, the dual of a polyhedron ${\displaystyle P}$ is often defined in terms of polar reciprocation about a sphere. Here, each vertex (pole) is associated with a face plane (polar plane or just polar) so that the ray from the center to the vertex is perpendicular to the plane, and the product of the distances from the center to each is equal to the square of the radius. [3]

When the sphere has radius ${\displaystyle r}$ and is centered at the origin (so that it is defined by the equation ${\displaystyle x^{2}+y^{2}+z^{2}=r^{2}}$), then the polar dual of a convex polyhedron ${\displaystyle P}$ is defined as

${\displaystyle P^{\circ }=\{q~{\big |}~q\cdot p\leq r^{2}}$ for all ${\displaystyle p}$ in ${\displaystyle P\},}$

where ${\displaystyle q\cdot p}$ denotes the standard dot product of ${\displaystyle q}$ and ${\displaystyle p}$.

Typically when no sphere is specified in the construction of the dual, then the unit sphere is used, meaning ${\displaystyle r=1}$ in the above definitions. [4]

For each face plane of ${\displaystyle P}$ described by the linear equation

${\displaystyle x_{0}x+y_{0}y+z_{0}z=r^{2},}$

the corresponding vertex of the dual polyhedron ${\displaystyle P^{\circ }}$ will have coordinates ${\displaystyle (x_{0},y_{0},z_{0})}$. Similarly, each vertex of ${\displaystyle P}$ corresponds to a face plane of ${\displaystyle P^{\circ }}$, and each edge line of ${\displaystyle P}$ corresponds to an edge line of ${\displaystyle P^{\circ }}$. The correspondence between the vertices, edges, and faces of ${\displaystyle P}$ and ${\displaystyle P^{\circ }}$ reverses inclusion. For example, if an edge of ${\displaystyle P}$ contains a vertex, the corresponding edge of ${\displaystyle P^{\circ }}$ will be contained in the corresponding face.

For a polyhedron with a center of symmetry, it is common to use a sphere centered on this point, as in the Dorman Luke construction (mentioned below). Failing that, for a polyhedron with a circumscribed sphere, inscribed sphere, or midsphere (one with all edges as tangents), this can be used. However, it is possible to reciprocate a polyhedron about any sphere, and the resulting form of the dual will depend on the size and position of the sphere; as the sphere is varied, so too is the dual form. The choice of center for the sphere is sufficient to define the dual up to similarity.

If a polyhedron in Euclidean space has a face plane, edge line, or vertex lying on the center of the sphere, the corresponding element of its dual will go to infinity. Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required 'plane at infinity'. Some theorists prefer to stick to Euclidean space and say that there is no dual. Meanwhile, Wenninger (1983) found a way to represent these infinite duals, in a manner suitable for making models (of some finite portion).

The concept of duality here is closely related to the duality in projective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra. But for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. [5] Because of the definitional issues for geometric duality of non-convex polyhedra, Grünbaum (2007) argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron.

#### Canonical duals

Any convex polyhedron can be distorted into a canonical form, in which a unit midsphere (or intersphere) exists tangent to every edge, and such that the average position of the points of tangency is the center of the sphere. This form is unique up to congruences.

If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points, and thus will also be canonical. It is the canonical dual, and the two together form a canonical dual compound. [6]

#### Dorman Luke construction

For a uniform polyhedron, each face of the dual polyhedron may be derived from the original polyhedron's corresponding vertex figure by using the Dorman Luke construction. [7]

### Topological duality

Even when a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, and the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are still topologically or abstractly dual.

The vertices and edges of a convex polyhedron form a graph (the 1-skeleton of the polyhedron), embedded on the surface of the polyhedron (a topological sphere). This graph can be projected to form a Schlegel diagram on a flat plane. The graph formed by the vertices and edges of the dual polyhedron is the dual graph of the original graph.

More generally, for any polyhedron whose faces form a closed surface, the vertices and edges of the polyhedron form a graph embedded on this surface, and the vertices and edges of the (abstract) dual polyhedron form the dual graph of the original graph.

An abstract polyhedron is a certain kind of partially ordered set (poset) of elements, such that incidences, or connections, between elements of the set correspond to incidences between elements (faces, edges, vertices) of a polyhedron. Every such poset has a dual poset, formed by reversing all of the order relations. If the poset is visualized as a Hasse diagram, the dual poset can be visualized simply by turning the Hasse diagram upside down.

Every geometric polyhedron corresponds to an abstract polyhedron in this way, and has an abstract dual polyhedron. However, for some types of non-convex geometric polyhedra, the dual polyhedra may not be realizable geometrically.

## Self-dual polyhedra

Topologically, a self-dual polyhedron is one whose dual has exactly the same connectivity between vertices, edges and faces. Abstractly, they have the same Hasse diagram.

A geometrically self-dual polyhedron is not only topologically self-dual, but its polar reciprocal about a certain point, typically its centroid, is a similar figure. For example, the dual of a regular tetrahedron is another regular tetrahedron, reflected through the origin.

Every polygon (that is, a two-dimensional polyhedron) is topologically self-dual, since it has the same number of vertices as edges, and these are switched by duality. But it is not necessarily self-dual (up to rigid motion, for instance). Every polygon has a regular form which is geometrically self-dual about its intersphere: all angles are congruent, as are all edges, so under duality these congruences swap.

Similarly, every topologically self-dual convex polyhedron can be realized by an equivalent geometrically self-dual polyhedron, its canonical polyhedron, reciprocal about the center of the midsphere.

There are infinitely many geometrically self-dual polyhedra. The simplest infinite family are the canonical pyramids of n sides. Another infinite family, elongated pyramids, consists of polyhedra that can be roughly described as a pyramid sitting on top of a prism (with the same number of sides). Adding a frustum (pyramid with the top cut off) below the prism generates another infinite family, and so on.

There are many other convex, self-dual polyhedra. For example, there are 6 different ones with 7 vertices, and 16 with 8 vertices. [8]

A self-dual non-convex icosahedron with hexagonal faces was identified by Brückner in 1900. [9] [10] [11] Other non-convex self-dual polyhedra have been found, under certain definitions of non-convex polyhedra and their duals.

 3 4 5 6
 3 4 5
 3 4 5 6 7

## Dual polytopes and tessellations

Duality can be generalized to n-dimensional space and dual polytopes; in two dimension these are called dual polygons.

The vertices of one polytope correspond to the (n 1)-dimensional elements, or facets, of the other, and the j points that define a (j 1)-dimensional element will correspond to j hyperplanes that intersect to give a (nj)-dimensional element. The dual of an n-dimensional tessellation or honeycomb can be defined similarly.

In general, the facets of a polytope's dual will be the topological duals of the polytope's vertex figures. For the polar reciprocals of the regular and uniform polytopes, the dual facets will be polar reciprocals of the original's vertex figure. For example, in four dimensions, the vertex figure of the 600-cell is the icosahedron; the dual of the 600-cell is the 120-cell, whose facets are dodecahedra, which are the dual of the icosahedron.

### Self-dual polytopes and tessellations

The primary class of self-dual polytopes are regular polytopes with palindromic Schläfli symbols. All regular polygons, {a} are self-dual, polyhedra of the form {a,a}, 4-polytopes of the form {a,b,a}, 5-polytopes of the form {a,b,b,a}, etc.

The self-dual regular polytopes are:

The self-dual (infinite) regular Euclidean honeycombs are:

The self-dual (infinite) regular hyperbolic honeycombs are:

## Related Research Articles

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.

In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.

In elementary geometry, a polytope is a geometric object with flat sides (faces). It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.

In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.

A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

In geometry, the Schläfli symbol is a notation of the form that defines regular polytopes and tessellations.

In geometry, a polytope or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces.

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.

In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.

In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron. That is to say, it touches any given edge at exactly one point. Not every polyhedron has a midsphere, but for every convex polyhedron there is a combinatorially equivalent polyhedron, the canonical polyhedron, that does have a midsphere. The radius of the midsphere is called the midradius.

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space.

In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs.

In geometry, a toroidal polyhedron is a polyhedron which is also a toroid, having a topological genus of 1 or greater. Notable examples include the Császár and Szilassi polyhedra.

In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids.

In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space. It can be defined as the convex hull of a finite set of ideal points. An ideal polyhedron has ideal polygons as its faces, meeting along lines of the hyperbolic space.

## References

### Notes

1. Wenninger (1983), "Basic notions about stellation and duality", p. 1.
2. Cundy & Rollett (1961), 3.2 Duality, pp. 78–79; Wenninger (1983), Pages 3-5. (Note, Wenninger's discussion includes nonconvex polyhedra.)
3. Barvinok (2002), Page 143.
4. See for example Grünbaum & Shephard (2013), and Gailiunas & Sharp (2005). Wenninger (1983) also discusses some issues on the way to deriving his infinite duals.
5. Grünbaum (2007), Theorem 3.1, p. 449.
6. Cundy & Rollett (1961), p.  117; Wenninger (1983), p. 30.
7. 3D Java models at Symmetries of Canonical Self-Dual Polyhedra, based on paper by Gunnar Brinkmann, Brendan D. McKay, Fast generation of planar graphs PDF
8. Anthony M. Cutler and Egon Schulte; "Regular Polyhedra of Index Two", I; Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry April 2011, Volume 52, Issue 1, pp 133–161.
9. N. J. Bridge; "Faceting the Dodecahedron", Acta Crystallographica, Vol. A 30, Part 4 July 1974, Fig. 3c and accompanying text.
10. Brückner, M.; Vielecke und Vielflache: Theorie und Geschichte, Teubner, Leipzig, 1900.

### Bibliography

• Cundy, H. Martyn; Rollett, A. P. (1961), Mathematical Models (2nd ed.), Oxford: Clarendon Press, MR   0124167 .
• Gailiunas, P.; Sharp, J. (2005), "Duality of polyhedra", International Journal of Mathematical Education in Science and Technology, 36 (6): 617–642, doi:10.1080/00207390500064049, S2CID   120818796 .
• Grünbaum, Branko (2003), "Are your polyhedra the same as my polyhedra?", in Aronov, Boris; Basu, Saugata; Pach, János; Sharir, Micha (eds.), Discrete and Computational Geometry: The Goodman–Pollack Festschrift, Algorithms and Combinatorics, vol. 25, Berlin: Springer, pp. 461–488, CiteSeerX  , doi:10.1007/978-3-642-55566-4_21, ISBN   978-3-642-62442-1, MR   2038487 .
• Grünbaum, Branko (2007), "Graphs of polyhedra; polyhedra as graphs", Discrete Mathematics , 307 (3–5): 445–463, doi:10.1016/j.disc.2005.09.037, hdl:, MR   2287486 .
• Grünbaum, Branko; Shephard, G. C. (2013), "Duality of polyhedra", in Senechal, Marjorie (ed.), Shaping Space: Exploring polyhedra in nature, art, and the geometrical imagination, New York: Springer, pp. 211–216, doi:10.1007/978-0-387-92714-5_15, ISBN   978-0-387-92713-8, MR   3077226 .
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN   0-521-54325-8, MR   0730208 .
• Barvinok, Alexander (2002), A course in convexity, Providence: American Mathematical Soc., ISBN   0821829688 .