Petrie dual

In topological graph theory, the Petrie dual of a embedded graph (on a 2-manifold with all faces disks) is another embedded graph that has the Petrie polygons of the first embedding as its faces.[1]

The Petrie dual is also called the Petrial, and the Petrie dual of an embedded graph may be denoted .[2] It can be obtained from a signed rotation system or ribbon graph representation of the embedding by twisting every edge of the embedding.

Properties

Like the usual dual graph, repeating the Petrie dual operation twice returns to the original surface embedding. Unlike the usual dual graph (which lies on the same surface as the original embedding but with different vertices and faces) the Petrie dual has the same set of vertices as its original embedding, but typically lies on a different surface.[1]

Surface duality and Petrie duality are two of the six Wilson operations, and together generate the group of these operations.[3]

Regular polyhedra

Appying the Petrie dual to a regular polyhedron produces a regular map.[2]

For example, the Petrie dual of a cube (a bipartite graph with eight vertices and twelve edges, embedded onto a sphere with six square faces) has four hexagonal faces, the equators of the cube. Topologically, it forms an embedding of the same graph onto a torus.[1]

The regular maps obtained in this way are as follows.

  • The petrial tetrahedron, {3,3}π, has 4 vertices, 6 edges, and 3 skew square faces. With an Euler characteristic, χ, of 1, it is topologically identical to the hemi-cube, {4,3}/2.
  • The petrial cube, {4,3}π, has 8 vertices, 12 edges, and 4 skew hexagons, colored red, green, blue and orange here. With an Euler characteristic of 0, it can also be seen in the four hexagonal faces of the hexagonal tiling as type {6,3}(2,0).
  • The petrial octahedron, {3,4}π, has 6 vertices, 12 edges, and 4 skew hexagon faces. It has an Euler characteristic of −2, and has a mapping to the hyperbolic order-4 hexagonal tiling, as type {6,4}3.
  • The petrial dodecahedron, {5,3}π, has 20 vertices, 30 edges, and 6 skew decagonal faces, and Euler characteristic of −4, related to the hyperbolic tiling as type {10,3}5.
  • The petrial icosahedron, {3,5}π, has 12 vertices, 30 edges, and 6 skew decagonal faces, and Euler characteristic of −12, related to the hyperbolic tiling as type {10,5}3.
Regular petrials
Name Petrial tetrahedron
 Petrial cube Petrial octahedron Petrial dodecahedron Petrial icosahedron
Symbol {3,3}π {4,3}π {3,4}π {5,3}π {3,5}π
(v,e,f), χ (4,6,3), χ = 1(8,12,4), χ = 0(6,12,4), χ = −2(20,30,6), χ = −4(12,30,6), χ = −12
Faces 3 skew squares 4 skew hexagons 6 skew decagons
Image
Related
figures
{4,3}3 = {4,3}/2 = {4,3}(2,0)
{6,3}3 = {6,3}(2,0)
{6,4}3 = {6,4}(4,0)		 {10,3}5 {10,5}3

References

  1. 1 2 3 Gorini, Catherine A. (2000), Geometry at Work, MAA Notes, 53, Cambridge University Press, p. 181, ISBN 9780883851647
  2. 1 2 McMullen, Peter; Schulte, Egon (2002), Abstract Regular Polytopes, Encyclopedia of Mathematics and its Applications, 92, Cambridge University Press, p. 192, ISBN 9780521814966
  3. ↑ Jones, G. A.; Thornton, J. S. (1983), "Operations on maps, and outer automorphisms", Journal of Combinatorial Theory, Series B, 35 (2): 93–103, doi:10.1016/0095-8956(83)90065-5, MR 0733017
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