Dual polyhedron


Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals also belong to a symmetric class. Thus, 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, having equivalent vertices, is one which is isohedral, having equivalent faces. The dual of an isotoxal polyhedron (having equivalent edges) is also isotoxal.


Kinds of duality


Polar reciprocation

The dual of polyhedron 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 r and is centered at the origin, i.e. defined by equation and P is a convex polyhedron then its polar dual is defined as

where y·x denotes the standard dot product of y and x. Typically when no sphere is specified in the construction of the dual, then the unit sphere is used, meaning r = 1 in the above definition.[4]

For each face of P described by the linear equation

the dual polyhedron P° will have a vertex (x0,y0,z0). Similarly, each vertex of P corresponds to a face of P°, and each edge of P corresponds to an edge of P°. The correspondence between the vertices, edges and faces of P and P° reverses inclusion. For example, if an edge of P contains a vertex, the corresponding edge of P° will be contained in the corresponding face.




Canonical duals



Topological duality




Dorman Luke construction




Dorman Luke's construction then proceeds:

  1. Draw the vertex figure ABCD



Self-dual polyhedra







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.

Family of pyramids




Family of elongated pyramids



Family of diminished trapezohedra






Self-dual compound polyhedra



Dual polytopes and tessellations


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 (n j)-dimensional element. The dual of an n-dimensional tessellation or honeycomb can be defined similarly.


Self-dual polytopes and tessellations


The self-dual regular polytopes are:

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

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

See also



  1. Wenninger (1983), "Basic notions about stellation and duality", p. 1.
  2. Grünbaum (2003)
  4. Barvinok (2002), Page 143.
  6. Grünbaum (2007), Theorem 3.1, p. 449.
  7. Cundy & Rollett (1961), p.  117; Wenninger (1983), p. 30.
  11. Brückner, M.; Velecke und Vielflache: Theorie und Geschichte, Teubner, Leipzig, 1900.


  • 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.
  • 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, 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, 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.