Heptahedron

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Polyhedron with seven sides

title: "Heptahedron" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["polyhedra"] description: "Polyhedron with seven sides" topic_path: "general/polyhedra" source: "https://en.wikipedia.org/wiki/Heptahedron" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Polyhedron with seven sides ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/f/f3/Chestahedron.png" caption="access-date=8 August 2022}}"] ::

A heptahedron (: heptahedra) is a polyhedron having seven sides, or faces.

A heptahedron can take a large number of different basic forms, or topologies. The most familiar are the hexagonal pyramid and the pentagonal prism. Also notable is the tetrahemihexahedron, which can be seen as a tessellation of the real projective plane. No heptahedra are regular.

Topologically distinct heptahedron

Convex

There are 34 topologically distinct convex heptahedra, excluding mirror images. (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.)

An example of each type is depicted below, along with the number of sides on each of the faces. The images are ordered by descending number of six-sided faces (if any), followed by descending number of five-sided faces (if any), and so on.

::data[format=table] | 110px|center]] Diminished cube | [[File:Heptahedron32.svg|110px|center]] | [[File:Heptahedron33.svg|110px|center]] | [[File:Heptahedron34.svg|110px|center]] | |---|---|---|---| ::

Concave

::data[format=table] | [[File:Heptahedron concave 01.svg|110px|center]][[File:Heptahedron concave 02.svg|110px|center]] | [[File:Heptahedron concave 09.svg|110px|center]][[File:Heptahedron concave 10.svg|110px|center]] | |---|---| ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/1/1c/Szilassi_polyhedron.svg" caption="The Szilassi polyhedron"] ::

One particularly interesting example is the Szilassi polyhedron, a Toroidal polyhedron with 7 non-convex six sided faces.{{citation |last = Szilassi |first = Lajos |journal = Structural Topology |pages = 69–80 |title = Regular toroids |url = http://www-iri.upc.es/people/ros/StructuralTopology/ST13/st13-06-a3-ocr.pdf |volume = 13 |year = 1986

References

References

  1. Frank Chester. "The Geometry of the Chestahedron".
  2. (5 April 2015). "Counting polyhedra".
  3. Weisstein, Eric W.. "Heptahedron".
  4. https://www.researchgate.net/publication/296814860_Regularization_of_heptahedra_using_geometric_element_transformation_method

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polyhedra