page_title: Jessen's icosahedron
section_title: Construction
hierarchical_section_title: Jessen's icosahedron / Construction
caption_attribution_description: English: A regular icosahedron and a non-convex variant, resembling but different from Jessen's icosahedron
context_page_description: Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, is a non-convex polyhedron with the same number of vertices, edges and faces as the regular icosahedron. Its faces meet only in right angles, even though they cannot all be made parallel to the coordinate planes. It is named for Børge Jessen who investigated it in 1967.
context_section_description: The vertices of Jessen's icosahedron may be chosen to have as their coordinates the 12 points given by the cyclic permutations of the coordinates . With this coordinate representation, the short edges of the icosahedron (the ones with convex angles) have length , and the long (reflex) edges have length . The faces of the icosahedron are equilateral triangles with the short side length, and isosceles triangles with one long edge and two short edges.
  A similar shape can be formed by keeping the vertices of a regular icosahedron in their original positions and replacing certain pairs of equilateral-triangle faces by pairs of isosceles triangles, and this shape has also sometimes incorrectly been called Jessen's icosahedron.
  However, the resulting polyhedron does not have right-angled dihedrals. The vertices of Jessen's icosahedron are perturbed from these positions in order to give all the dihedrals right angles.
caption: Regular icosahedron and its non-convex variant, which differs from Jessen's icosahedron in having obtuse dihedral angles instead of right angles
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