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,
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.
cool thing people missed about
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
degrees, the hinges collide.
built several working physical models.
Below is a laser cut model.
also in this video)
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
A nice property of hinged polyhedra is that they transform between
several uniform polyhedra.
As it transforms from octahedron
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
phases are compounds
tetrahedra, and coinciding tetrahedra.
The non-self-intersecting stage is goes from rhombicuboctahedron
stage includes two
coinciding octahedra, and the Great
cubicuboctahedron. The intersecting snubs
do not seem to be uniform polyhedra.
The non-self-intersecting stage is goes from Rhombicosidodecahedron
dodecahedron > Icosidodecahedron.
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
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
which you can see them in action.
A "meta hinged octahedron
Another planar tessellations.
has a nice page on hinged tessellations, with surprising
to irregular ones.
Here is an example:
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!
interesting site, it explores what the author
Rossiter calls "Chiral Polyhedral Transformations".
These overcome the limitation that on each vertex, you need
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
in opposite directions.
In this way, you can make a
hinged cube (and lots of other variations):
Recently people are studying
metamaterials, using structures
with several internal degrees of
freedom, so that
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.