examples of space filling polyhedra

A polyhedron "fills the space" if it allows to achieve an assembly that fills the entire space (without voids).
Only one regular polyhedron allows for such an assembly: the cube, and among the semi-regular polyhedra there are four: the triangular prism,the hexagonal prism, Lord Kelvin's polyhedron (truncated octahedron) and the rhombic dodecahedron (dual of the cuboctahedron).

The Fedorov's parallelohedra fill the space using only translations. Their faces are two by two opposite and parallel and each edge belongs to a set of parallel edges (four or six edges). There are only five, with their affine transformations: the cube (thus also all the parallelepipeds), the truncated octahedron, the rhombic dodecahedron, the elongated rhombic dodecahedron and the hexagonal prism (not necessarily regular or right).

If one of the above polyhedra can be decomposed into identical polyhedra, obviously the last are also space fillers: for example the cube can be decomposed into three or six square pyramids, into four bipyramids, into three pairs of symmetric tetrahedra ...
 • a regular right hexagonal prism can be decomposed into three regular triangular prisms ...
One can also fill the space using two distinct polyhedra, for example the regular octahedron and tetrahedron (see below), the regular dodecahedron and a curious non convex polyhedron.

three visions of a space filling

regular tetrahedron and octahedron

assembled half-octahedron and tetrahedron ("clog")
The symmetric assembling of two tetrahedra on an octahedron builds a rhombohedron identical to those obtained by assembling two "clogs" along their squared faces; this parallelepiped obviously fills the space.

Thus we get closely related space filling by using either rhombohedra or "clogs" or regular octahedra and tetrahedra.

two convex space-filling polyhedra (Guy Inchbald)

This first hendecahedron (11 faces) has two planes of symmetry.
Four of this polyhedron build an unit which fills the space according to a cubic type lattice.

This second hendecahedron has only one plane of symmetry.
Six of these polyhedra build a "rosette".
Two superposed "rosettes" of opposite directions build an unit which also fills the space according to an hexagonal lattice.

These two curious polyhedra have the same topology and their common canonical form is self dual (with two planes of symmetry).

three non convex space-filling polyhedra (Eduard Bobik)

It is well known that the stellated rhombic dodecahedron fills the space, but it is not obvious that the non convex rhombic dodecahedron above does the same. Its "center" belongs to six faces, two by two opposite and coplanar; this allows to say that its thirteen vertices define only nine faces, three of them being composed by two parallelograms with a common vertex.
Its symmetry group is D3v .
Twelve other copies fit around it (contacts along one face).

This polyhedron with 10 vertices and 16 faces can be constructed by removing two tetrahedra along each of the four equatorial edges of an octahedron.
Its symmetry group is D4h .
Four other copies fit around it (contacts along four faces).

As Eduard explains one can construct this strange polyhedron from a regular octahedron: remove a tetrahedron along each edge, and then distort the faces (push the centers of four faces inside and likewise pull the four others outside); like this you get 14 vertices and 24 faces arranged in four "non flat hexagons".
This polyhedron has a complete tetrahedral symmetry Td .
Four other copies fit around it (contacts along six faces).
Remark: the stellated rhombic dodecahedron (Escher's solid) has eight of these "non flat hexagonal faces" and is on contact with eight other copies.

the rhombic dodecahemioctahedron

Described by Guy Inchbald, this polyhedron can be obtained in two ways by excavation on the cube or on the rhombic dodecahedron.

the tetrahedra which tile the space (a two millennia quest from 350 BC to 2020)

The question goes back to Aristotle: can the regular tetrahedron fill the space? In the 15th century the answer was suspected to be negative, and the proof was provided two centuries later. But if the regular tetrahedron does not fill the space, can other tetrahedra? Ducan Sommerville exhibited the first examples in 1923 The search for other examples was facilitated by two other discoveries:
•  In 1900 Max Dehn defines a number which allows to know if two polyhedra of the same volume can be "scissors-congruent" (one can be cut into a finite number of polyhedra and reassembled with the same parts); the cube and the regular tetrahedron are not: they do not have the same Dehn invariant. In 1965 Jean-Pierre Sydkler proved that the condition is also sufficient. In addition, Hans Debrunner showed in 1980 that a tetrahedron paving the space must have, like the cube, a Dehn invariant equal to 0.
•  In 1976 John H. Conway and Antonia J. Jones seek to identify the "rational tetrahedra" whose measures in degrees of the six dihedral angles are rational numbers. Any tetrahedron with rational dihedral angles has a zero Dehn invariant, so is scissor-congruent to the cube and becomes a candidate for filling the space. Unfortunately they do not have mathematical tools or computers powerful enough to complete their research (solving a complicated polynomial equation).

Using computers Bjorn Poonen and Michael Rubinstein found 59 isolated rational tetrahedra plus two infinite families in the 1990s; It is only in 2020 that they prove, with Kiran S. Kedlaya and Alexander Kolpakov, that they are the only ones. Their classification required sophisticated methods and powerful computers to find special solutions of complicated equations.

59 tetrahedra

At the start of 2021, a group of MIT undergraduates, together with Poonen, proved that one of the isolated rational tetrahedra does not fill space. This is the first example of a tetrahedron which is scissor-congruent to the cube but does not fill the space.

references: space filling polyhedra by Guy Inchbald and Eduard Bobik
Tetrahedron Solutions Finally Proved Decades After Computer Search  Quantamagazine, 02/02/2021
Tetrahedron Undergraduates Hunt for Special Tetrahedra That Fit Together  Quantamagazine, 09/02/2021
"La quête du pavé apériodique unique"  by Jean-Paul Delahaye (in French)  Pour la Science n° 433, November 2013

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