Gerard Westendorp

I first learned about hinged tessellations from "The Penguin Book of Curious and Interesting Geometry" By David Wells.

This was probably somewhere around 1995.

I wondered if they could be generalised to polyhedra, and was pleased to find that they could!

I wrote a letter (there was not so much Internet then) to David Wells, including some pictures of the hinged polyhedra I made. David Wells wrote back that he liked this result, and recommended publishing it. I never did that till now, finally, I made this web site.

One cool thing people missed about hinged tessellations is what happens when you continue the hinge motion past the point when the squares start to overlap.

Surprisingly, the tessellation collapses into a single square. It is actually possible to build this as a mechanical construction, by having each square on a different height, and having hinges that connect between different heights. They do not work full circle, after about 270 degrees, the hinges collide.

I built several working physical models. Below is a laser cut model.

Below are the 3 hinged polyhedra I "discovered". I later found out that the octahedral case was discovered by Buckminster Fuller. He called it the Jitterbug. The other 2, I have not seen anywhere else.

A nice property of hinged polyhedra is that they transform between several uniform polyhedra.

The octahedral case.

As it transforms from octahedron to cuboctahedron, it goes through the phase of 'snub octahedron', better known as the icosahedron. So far, this is fairly well known. But now, we progress through the self-intersecting cases, and encounter the great icosahedron! Other phases are compounds of tetrahedra, and coinciding tetrahedra.

The cuboctahedral case.

The non-self-intersecting stage is goes from rhombicuboctahedron > snub cube > cuboctahedron.

The self-intersecting stage includes two coinciding octahedra, and the Great cubicuboctahedron. The intersecting snubs do not seem to be uniform polyhedra.

The icosadodecahedral case.

The non-self-intersecting stage is goes from Rhombicosidodecahedron > Snub dodecahedron > Icosidodecahedron.

The self-intersecting stage includes two coinciding octahedra, and the Great_ditrigonal_icosidodecahedron. On the way also forms a compound of an icosahedron and great dodecahedron. The intersecting snubs are not uniform as far as I can tell.

I constructed several working models of these hinged polyhedra. Below are the most recent ones, made from laser cut pieces. The hinges can be "snapped" on the polyhedron faces, so the assembly is quick.

Here is a Youtube video in which you can see them in action.

A "meta hinged octahedron

Another planar tessellations.

Al
Grant
has a nice page on hinged tessellations, with surprising
generalisations
to irregular ones.

Here is an example:

Here is an example:

Gluing
together hinged
polyhedra.

You can glue together hinged polyhedra to create larger structures, for example cristal lattices. A curious property of such latices they have negative Poisson's ratio: When you stretch them in one direction, they do not, as ordinary materials, contract n the other directions, but expand. This property is called 'auxetic'. Henri Segerman has made a 3D print version of what he called the Jitterbox, Jitterbugs in a lattice. Here is my own example, Glued together hinged cuboctahedra, but addition each square has been sub-divided into a 3X3 collapsible square tessellation. Th red triangles could be shrunk to zero. Now, they form a second internal degree of freedom, the squares can expand, but also the triangles can

You can glue together hinged polyhedra to create larger structures, for example cristal lattices. A curious property of such latices they have negative Poisson's ratio: When you stretch them in one direction, they do not, as ordinary materials, contract n the other directions, but expand. This property is called 'auxetic'. Henri Segerman has made a 3D print version of what he called the Jitterbox, Jitterbugs in a lattice. Here is my own example, Glued together hinged cuboctahedra, but addition each square has been sub-divided into a 3X3 collapsible square tessellation. Th red triangles could be shrunk to zero. Now, they form a second internal degree of freedom, the squares can expand, but also the triangles can

The number of
possibilities seems to be very large!

Antiprism
is an
interesting site, it explores what the author
Adrian
Rossiter calls "Chiral Polyhedral Transformations".

These overcome the limitation that on each vertex, you need an even number of faces meeting to construct a hinged polyhedron. For each face, he creates 2 copies, an inner and outer, and the hinge always connects an inner with an outer. The inner and outer faces move in opposite directions. In this way, you can make a hinged cube (and lots of other variations):

These overcome the limitation that on each vertex, you need an even number of faces meeting to construct a hinged polyhedron. For each face, he creates 2 copies, an inner and outer, and the hinge always connects an inner with an outer. The inner and outer faces move in opposite directions. In this way, you can make a hinged cube (and lots of other variations):

Recently people are studying
hinged
metamaterials, using structures
with several internal degrees of
freedom, so that

Chuck Hoberman, has created many cool hinged structures.

One thing that is missing is some nice theory about all this. They seem to be just cool gadgets, but perhaps they are related to symmetry in some more systematic way.

Chuck Hoberman, has created many cool hinged structures.

One thing that is missing is some nice theory about all this. They seem to be just cool gadgets, but perhaps they are related to symmetry in some more systematic way.