# Jessen's icosahedron

Jessen's icosahedron | |
---|---|

Rank | 3 |

Elements | |

Faces | 8 equilateral triangles, 12 isosceles triangles |

Edges | 24+6 |

Vertices | 12 |

Measures (edge lengths 4, ) | |

Circumradius | |

Central density | 1 |

Related polytopes | |

Army | Pyrike |

Convex hull | Pyritohedral icosahedron |

Abstract & topological properties | |

Flag count | 120 |

Euler characteristic | 2 |

Orientable | Yes |

Genus | 0 |

Skeleton | Icosahedral graph |

Properties | |

Symmetry | B_{3}/2, order 24 |

Flag orbits | 5 |

Convex | No |

Nature | Tame |

**Jessen's icosahedron** or **Jessen's orthogonal icosahedron** is a concave isogonal icosahedron with pyritohedral symmetry.^{[1]} It is abstractly equivalent to the regular icosahedron. All edges have a dihedral angle of 90 degrees, but the six longer edges are concave.

Jessen's icosahedron is a well-known example of a "shaky polyhedron". That is, if the faces are made of a rigid material and the edges are hinges, the solid is flexible, but only infinitesimally so.

If the longer edges are realized with rigid bars and the shorter edges strings, the figure supports its own weight, forming a tensegrity structure known as the *six-bar spherical tensegrity*. Some research has been done on applying this structure in robotics.

A shape similar to Jessen's icosahedron can be produced by symmetrically "indenting" six edges of a regular icosahedron, but the longer-to-shorter edge ratio of the resulting figure is different (equal to the golden ratio in this case). This shape is sometimes mistakenly labeled as Jessen's icosahedron, but it is no longer shaky and does not form a tensegrity structure.

## Vertex coordinates[edit | edit source]

The vertex coordinates of Jessen's icosahedorn are given by even permutations of

- .

These coordinates give edges of length 4 and .

It is an integral polytope.

## External links[edit | edit source]

- Wikipedia contributors. "Jessen's icosahedron".
- Weisstein, Eric W. "Jessen's Orthogonal Icosahedron" at MathWorld.
- McCooey, David. "Jessen's Orthogonal Icosahedron"
- Starck, Maurice. "Jessen's orthogonal icosahedron" on The Polyhedra World.