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This is the metamorphosis of the cube.
I'm professor Erik Demaine in CSAIL at MIT.
And this is joint work with Martin Demaine,
Ana [INAUDIBLE],, Joseph O'Rourke, [? Arina ?] Pachinko.
And we're looking here at different foldings
and unfoldings of surfaces in 3D.
Here we have our friend the cube and a familiar unfolding
of the cube called the cross unfolding.
And it turns out that if you change the creases in the cross
and fold it up a different way, you
can make all sorts of different convex polyhedra.
Here we have a very simple one, a flat doubly covered
But all the others are going to be non-flat.
So we saw the cube, got the quadrilateral.
The next one is a pentahedron, five sides.
And we form exactly the surface of this bilaterally symmetric
On the other hand, if we change the creases in this way,
then we get a tetrahedron.
This is my favorite because the little tab fits exactly
into the pocket at exactly the surface of that tetrahedron,
not a regular tetrahedron, but a tetrahedron nonetheless.
And finally, if we change the creases in this way,
we get an octahedron.
And all of these polyhedra are computed automatically
with an algorithm that given a flat shape
tells you all of the different convex polyhedra
that it can glue up into.
So here we had five different shapes, the cube,
the quadrilateral, the pentahedron, the tetrahedron,
and the octahedron.
But for different polygons, you get different 3D shapes.
Just to give you an idea of the range of unfolding,
unfolding is a lot harder.
If I give you a 3D surface and you
cut it up and try to unfold it, when does it overlap
is a big question.
That was an example of a non-overlapping unfolding.
There's quite a variety in what you could do.
Here's a particularly fun unfolding we called the stop
light unfolding of the cube.
It looks like a stop light.
And here's one example of another 3D convex polyhedron
you can fold it up into, which we call the spaceship
for technical reasons.
And again, the algorithm automatically
computes all the different 3D convex
polyhedra you can glue that flat shape into.
Here we go back to the cube.
And this is ongoing research in computational geometry.