Editing Truncated octahedron (section)

== Dissection ==
The truncated octahedron can be dissected into a central [[octahedron]], surrounded by 8 [[triangular cupola]]e on each face, and 6 [[square pyramid]]s above the vertices.<ref>{{cite web|url=http://www.doskey.com/polyhedra/Stewart05.html|title=Adventures Among the Toroids – Chapter 5 – Simplest (R)(A)(Q)(T) Toroids of genus p=1|first=Alex|last=Doskey|website=www.doskey.com}}</ref>

{{multiple image
 | image1 = Excavated truncated octahedron1.png
 | image2 = Excavated truncated octahedron2.png
 | total_width = 400
 | footer = Second and third  genus toroids
}}
Removing the central octahedron and 2 or 4 triangular cupolae creates two [[Stewart toroid]]s, with dihedral and tetrahedral symmetry:

It is possible to slice a [[tesseract]] by a hyperplane so that its sliced cross-section is a truncated octahedron.<ref>{{cite book
 | last1 = Borovik | first1 = Alexandre V.
 | last2 = Borovik | first2 = Anna
 | contribution = Exercise 14.4
 | contribution-url = https://books.google.com/books?id=bVdzDL8KW3wC&pg=PA109
 | doi = 10.1007/978-0-387-79066-4
 | isbn = 978-0-387-79065-7
 | mr = 2561378
 | page = 109
 | publisher = Springer | location = New York
 | series = Universitext
 | title = Mirrors and Reflections
 | title-link = Mirrors and Reflections
 | year = 2010}}</ref>

The [[cell-transitive]] [[bitruncated cubic honeycomb]] can also be seen as the [[Voronoi tessellation]] of the [[Crystal structure|body-centered cubic lattice]]. The truncated octahedron is one of five three-dimensional primary [[Parallelohedron#Zonohedra that tile space|parallelohedra]].