Nets for Regular Spherical Models

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This Demonstration shows some regular spherical models and their nets. A spherical polyhedron is defined by the arcs that are the central projections of the edges of a polyhedron to the sphere. In the case of Wenninger's spherical models, each face of a polyhedron is divided into right-angled triangles that consist of a vertex of the face, midpoint of an edge that adjoins the vertex, and the center of the face. In the case of a triangular face, there are six such triangles. If the polyhedron is regular, these triangles are Möbius triangles. We get Wenninger's model if all three sides of each triangle are included. If only one leg is included, we get a regular model. If the other leg is included, we get the dual model. If the hypotenuse is the only side, we get the spherical model of the convex hull of the preceding two models.


Example 1: tetrahedron, tetrahedron, cube.

Example 2: cube, octahedron, rhombic dodecahedron.

Example 3: dodecahedron, icosahedron, triacontahedron.


Contributed by: Izidor Hafner (October 2014)
Open content licensed under CC BY-NC-SA



Spherical models are patterns formed by arcs of great circles on a sphere. In the case of regular models, the arcs can be combined into Möbius triangles. To get a spherical polyhedron, some arcs from Möbius triangles are removed.


[1] M. J. Wenninger, Spherical Models, New York: Dover, 1999.

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