# Compound of four cubes

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Compound_of_four_cubes
> Markdown URL: https://mediated.wiki/source/Compound_of_four_cubes.md
> Source: https://en.wikipedia.org/wiki/Compound_of_four_cubes
> Source revision: 1282304459
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

{{Short description|Polyhedral compound}}
{|class="wikitable" style="float:right; margin-left:8px; width:290px"
!bgcolor=#e7dcc3 colspan=2|Compound of four cubes
|-
|align=center colspan=2|280px<br><small>([Animation](/source/%3AFile%3ACompound_of_four_cubes.gif))</small>
|-
|bgcolor=#e7dcc3|Type||Compound
|-
|bgcolor=#e7dcc3|[Convex hull](/source/Convex_hull)||[Chamfered cube](/source/Chamfered_cube)
|-
|bgcolor=#e7dcc3|Polyhedra||4 [cube](/source/cube)s
|-
|bgcolor=#e7dcc3|Faces||32 squares
|-
|bgcolor=#e7dcc3|Edges||48
|-
|bgcolor=#e7dcc3|Vertices||32 <small>(8 + 24)</small>
|-
|bgcolor=#e7dcc3|[Symmetry group](/source/List_of_spherical_symmetry_groups)||[octahedral](/source/Octahedral_symmetry) (''O''<sub>h</sub>)
|-
|bgcolor=#e7dcc3|[Subgroup](/source/Subgroup) restricting to one constituent||3-fold [antiprismatic](/source/Dihedral_symmetry_in_three_dimensions) (''D''<sub>3d</sub>)
|}
The '''compound of four cubes''' or '''Bakos compound'''<ref>[https://demonstrations.wolfram.com/TheBakosCompound/ WOLFRAM Demonstrations Project: The Bakos Compound]</ref> is a [face-transitive](/source/isohedral_figure) [polyhedron compound](/source/polyhedron_compound) of four [cube](/source/cube)s with [octahedral symmetry](/source/octahedral_symmetry). It is the dual of the [compound of four octahedra](/source/compound_of_four_octahedra). Its surface area is 687/77 square lengths of the edge.<ref>{{cite web | url = http://mathworld.wolfram.com/Cube4-Compound.html | website = Math World | publisher = Wolfram | title = Cube 4-Compound | last = Weisstein | first = Eric W. | access-date = 21 August 2021}}</ref>

Its [Cartesian coordinates](/source/Cartesian_coordinates) are (±3, ±3, ±3) and the permutations of (±5, ±1, ±1).

{| <!--table to avoid text floating up-->
|
{{multiple image
 | align = left  | total_width = 400
 | image1 = Compound of four cubes, 2-fold.png
 | image2 = Compound of four cubes, 3-fold.png
 | image3 = Compound of four cubes, front.png
 | footer = Views from 2-fold, 3-fold and 4-fold symmetry axis 
}}
|}

==Extension with fifth cube==

The eight vertices on the 3-fold symmetry axes can be seen as the vertices of a fifth cube of the same size.<ref>The Wolfram page [https://mathworld.wolfram.com/Cube5-Compound.html Cube 5-Compound] shows a small picture of this extension under the name "first cube 4-compound". Also [Grant Sanderson](/source/3Blue1Brown) has used a picture of it to illustrate the term ''symmetry''.</ref>
Referring to the images below, the four old cubes are called colored, and the new one black.
Each colored cube has two opposite vertices on a 3-fold symmetry axis, which are shared with the black cube. (In the picture both 3-fold vertices of the green cube are visible.) The remaining six vertices of each colored cube correspond to the faces of the black cube. This compound shares these properties with the [compound of five cubes](/source/compound_of_five_cubes) (related to the [dodecahedron](/source/regular_dodecahedron)), into which it can be transformed by rotating the colored cubes on their 3-fold axes.

{| <!--table to avoid text floating up-->
|
{{multiple image
 | align = left  | total_width = 500
 | image1 = Compound of four cubes extended.png
 | image2 = Icosahedral to octahedral compound of cubes.gif
 | footer = Extension <small>(see [animation](/source/%3AFile%3ACompound_of_four_cubes_extended.gif))</small> and its transition to the icosahedral compound
}}
|}

==See also==
*[Compound of three octahedra](/source/Compound_of_three_octahedra)
*[Compound of five octahedra](/source/Compound_of_five_octahedra)
*[Compound of ten octahedra](/source/Compound_of_ten_octahedra)
*[Compound of twenty octahedra](/source/Compound_of_twenty_octahedra)
*[Compound of three cubes](/source/Compound_of_three_cubes)
*[Compound of five cubes](/source/Compound_of_five_cubes)
*[Compound of six cubes](/source/Compound_of_six_cubes)
*[Uniform polyhedron compound](/source/Uniform_polyhedron_compound)

== References ==
{{Reflist}}
Category:Polyhedral compounds

{{polyhedron-stub}}

---
Adapted from the Wikipedia article [Compound of four cubes](https://en.wikipedia.org/wiki/Compound_of_four_cubes) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Compound_of_four_cubes?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.