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
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 had not seen
anywhere else. Recently I found an article
by H.F.Verheyen, that mentions them, along with a lot of other
hinge construciton. As time passes, I find more and more people who
have rediscoverd these things.
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
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.
An animation of a 3D printed version:
hinges are printed using a thin
layer of PETG.
On Twitter, @HCO28970306
shows all kinds of hinge constructions, using skillful 3D printing
Here is a Youtube video
which you can see them in action.
"meta hinged octahedron
has a nice page on hinged tessellations, with surprising
to irregular ones.
Here is an example:
by this, I of course had to
make something myself:
hinged Pythagoras tree
people remarked on Twitter that any tiling
composed of rhombi, can
be converted to a
tiling (connect the midpoints
of each rhombus, they form
rectangels that can be hinged) ,@akivaw tweeted that perhaps Penrose
tiles can be made hinged. Yes! So I made this example:
If you take a jitterbug, you can put a structure on top of some of its
faces, so that you can make a hinged tetrahedron, or a hinged cube:
you can extend the hinges
and get this tructaded cube:
the 12 hinges and attached attached triangles to them, so that you get
a hinged icosahedron!:
a tetrahedron that hinges into a gyrobifastigium, one of the 92 Johnson
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 in the
directions, but expand. This property is called 'auxetic'.
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.
possibilities seems to be very large!
interesting site, it explores what the author
Rossiter calls "Chiral Polyhedral Transformations".
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):
turns out this construction also
simplfies the hinges: The construciotn is stable as long as you keep
the hing points together, for example with a simple hook.
people are studying
metamaterials, using structures
with several internal degrees of
freedom, so that
Hoberman, has created many cool
Here is a Youtube
video of a talk I did on hinged
Recently, I learned that a lot of stuff had been going on in the past
that I didn't know about.
Here is a video about Ron
who was doing hinged polyhedra in the sixties!
Many of the things I discovered were in fact rediscoveries. The video
also shows things completely new to me.
He patented hinged tesselations already in 1964.
Here is an animation of a 3D hinged cube array, that generalises the 2D
hinged tesselations in a cool way. Ron Resch built large models of this.
Another person I learned about who was doing things already in the
1960s is Joseph Clinton.
video by Joseph
. He beat me by several
decades in building some of his