However, for some types of non-convex geometric polyhedron, the dual polyhedron may not be realized geometrically. Some theorists prefer to stick to Euclidean space and say that there is no dual. Similarly, every topologically self-dual convex polyhedron can be realized by an equivalent geometrically self-dual polyhedron, its canonical polyhedronreciprocal about the center of the midsphere.

Such pairs of polyhedra are still topologically or abstractly dual. The simplest infinite family are the canonical pyramids of n sides.

If multiple symmetry axes are present, they will necessarily intersect at a single point, and this is usually taken to be the centroid. 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.

If the poset is visualized as a Hasse diagramthe dual poset may be visualized simply by turning the Hasse diagram upside down. The concept of duality here is closely related to the duality in projective geometrywhere lines and edges are interchanged.

Failing that, a circumscribed sphere, inscribed sphere, or midsphere one with all edges as tangents is commonly used. The same graph can be projected to form a Schlegel diagram on a flat plane.

Meanwhile, Wenninger found a way to represent these infinite duals, in a manner suitable for making models of some finite portion!

Before beginning the construction, the vertex figure ABCD is obtained by cutting each connected edge at in this case its midpoint.

Adding a frustum pyramid with the top cut off below the prism generates another infinite family, and so on.

For example, there are 6 different ones with 7 vertices, and 16 with 8 vertices. The choice of center for the sphere is sufficient to define the dual up to similarity. Projective polarity works well enough for convex polyhedra. Any convex polyhedron can be distorted into a canonical formin 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.

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. Abstractly, they have the same Hasse diagram.

The polygon EFGH is a face of the dual polyhedron. Every geometric polyhedron corresponds to an abstract polyhedron in this way, and has an abstract dual polyhedron.

Self-dual polyhedra[ edit ] Topologically, a self-dual polyhedron is one whose dual has exactly the same connectivity between vertices, edges and faces.

Mark the points E, F, G, H, where each tangent line meets the adjacent tangent. Draw lines tangent to the circumcircle at each corner A, B, C, D. 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.

Pairs of edges meet on their common intersphere.A graph that is dual to itself.

Wheel graphs are self-dual, as are the examples illustrated above. Naturally, the skeleton of a self-dual polyhedron is a self-dual graph. Since the skeleton of a pyramid is a wheel graph, it follows that pyramids are also self-dual.

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.

Full-Text Paper (PDF): Self-Dual, Self-Petrie Covers of Regular Polyhedra.

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