miscellaneous examples of curious polyhedra

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 a 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 "centre" 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 centres 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.

a near miss polyhedron (Jim McNeill)

A nice assemblage of thirteen square antiprisms:   

In fact the complete toroid needs 13.01 antiprisms!
The error goes unnoticed... but LiveGraphics3D allows you to spot it by suppressing the square faces.

two rhombic zonohedra (Russel Towle)

Zonohedra are convex polyhedra bounded by centrally-symmetrical polygons (thus they have an even number of sides, two by two opposite, parallel and with same length). Nevertheless one may construct non convex polyhedra with such faces; here are two examples with rhombic faces.


a polar zonohedron (convex)

a spirallohedron of order 5 (non convex)

polyhedra to approximate solids (Eric W. Weisstein - MathWorld)

We have seen that spherical polyhedra allow to approximate the sphere; likewise other solids can be approximated with polyhedra, or surfaces in space visualised with polyhedral surfaces.
the torus
There are three types of torus: the classical ring, the closed ring and a third one, less known, where the surface intersects itself in two parts: the apple  (the outside) and the lemon  (the inside).
Reminder: "drag-right vertically" allows you to suppress faces and thus to see the inside of the torus (and especially the lemon  hidden in the apple ).

the sphericon and the oloid
If we cut a bicone with hight equal to the diameter in two going through its axis (the section is a square), we get the sphericon (C. J. Roberts) by rotating one of the halves a quarter of turn around the square's axis. Ian Stewart published an article titled Cone with a Twist.
The same transformation on a cylinder with hight equal to the diameter gives the dual (the cylinder is dual of the bicone).
Similarly on a cone with an apex angle of 60° we can rotate one half by 120° (visitor's suggestion).
Other shapes may be used to get curious solids with this technique.
The oloid is the convex envelope of two orthogonal disks with each center on the other's border (Paul Schatz).



enjoy your breakfast!
A slice of watermelon and a croissant. Be aware! you can't use the Klein's bottle for your coffee; it's not a polyhedron but an unorientable surface with one side (like a Möbius strip) that has no inside or outside!


references:   space filling polyhedra by Guy Inchbald and Eduard Bobik
zonohedra by Russel Towle
torus (MathWorld) by Eric W. Weisstein
The Sphericon  by C. J. Roberts, Roger Kaufman and Steeve Mathias

 summary   October 2004
updated 18-04-2008