JBQ's spot on the Wild Wild Web
The musings of a French mathematician living in the heart of the American technology industry

The regular hexahedron
The regular hexahedron is the most common of all polyhedra. It is commonly known as the cube. It has 6 faces, which are all squares. It has 12 edges and 8 vertices.

The regular hexahedron is part of the family of 5 platonic solids, which are the most regular polyhedra. It is also part of the family of prisms (it is made of two identical regular parallel polygons joined by a row of squares).

If you join the centers of the faces of a hexahedron, you'll get an octahedron (mathematically, that means that the octahedron is the dual of the hexahedron). It's also worth mentioning that the regular hexahedron can tile space, and it's the only regular polyhedron with that property (mathematically, there's only one regular honeycomb, where "honeycomb" is the name for a set of polyhedra that fill 3-dimensional space).

The regular hexahedron defines three sets of symmetry axes: one set of 4 3-fold axes that join opposite vertices, one set of 6 2-fold axes that join the middle of opposite edges, and one set of 3 4-fold axes that join the middle of opposite faces.

Hexahedra occur in nature, and especially in chemistry: many ionic crystals, including sodium chloride (i.e. table salt), have their ions arranged according to hexahedra. Other kinds of crystals follow some more complex patterns that also include hexahedra, and that's in fact the case of the diamond lattice.

The regular hexahedron is the shape that is used to build 6-sided dice.

The regular hexahedron can be built with various construction toys, like Zome, Googolplex, Polymorf, Jovo, and even Lego and K'nex - the easiest way to build a regular hexahedron with Lego bricks is to sandwich a 2x2 brick between two 2x2 plates.

The following table lists the various dimensions in a regular hexahedron, assuming that the edge length is 1:

single edge length1
total edge length12
face inradius0.5
face circumradius0.707
polyhedron inradius0.5
polyhedron midradius0.707
polyhedron circumradius0.866
single face area1
total face area6

The following table is the reciprocal of the previous one, it lists the edge length that yields a value of 1 fo the different measurements in a regular hexahedron:

1single edge length
0.083total edge length
2face inradius
1.414face circumradius
2polyhedron inradius
1.414polyhedron midradius
1.155polyhedron circumradius
1single face area
0.408total face area

The following table lists various angles in a regular hexahedron:

angle between vertices (from center)70.5 degrees
angle between edges90 degrees
angle between faces90 degrees
Home page Related articles Posted on May 26 2007