Transcript Metamorphosis of the Cube

Metamorphosis
of the Cube
Erik Demaine
Martin Demaine
Anna Lubiw
Joseph O’Rourke
Irena Pashchenko
These foldings and unfoldings
illustrate two problems.
Problem 1. Unfold a convex polyhedron
into a simple polygon
This problem is solved
by the star unfolding.
(Agarwal, Aronov,
O’Rourke, and
Schevon 1997)
These foldings and unfoldings
illustrate two problems.
Problem 1. Unfold a convex polyhedron
into a simple polygon
This problem is solved
by the star unfolding.
But it remains open for
cuts along the edges
of the polyhedron.
These foldings and unfoldings
illustrate two problems.
Problem 2. Fold a simple polygon into
a convex polyhedron
Conditions given by Aleksandrov yield
an algorithm to find all the ways of
gluing pairs of polygon edges together
to form a convex polyhedron.
(Lubiw & O’Rourke 1997)
These foldings and unfoldings
illustrate two problems.
Problem 2. Fold a simple polygon into
a convex polyhedron
Although Aleksandrov’s theorem
guarantees uniqueness, finding the
actual convex polyhedron is an
open question.
Our examples were done by hand.
Animations computed by Mathematica,
(R) Wolfram Research, and rendered by POV-Ray
at the Computer Graphics Lab, U. Waterloo.
The music is “Opening” by Philip Glass, used with
permission from Dunvagen Music Publications.
We thank Therese Biedl for performing the piece.
This video was produced at the Audio Visual
Centre, U. Waterloo, by Dianne Naughton.
The background shows Aleksandrov’s theorem,
А. Д. Александров, Выпуклые Многогранники
(A. D. Aleksandrov, Convex polyhedra),
State Press of Technical and Theoretical Literature,
Moscow, 1950, page 195.
Animations computed by Mathematica,
(R) Wolfram Research, and rendered by POV-Ray.
The music is “Opening” by Philip Glass, used with
permission from Dunvagen Music Publications.
We thank Therese Biedl for performing the piece.
This video was produced at the
Computer Graphics Lab, University of Waterloo.
The background shows Aleksandrov’s theorem,
А. Д. Александров, Выпуклые Многогранники
(A. D. Aleksandrov, Convex polyhedra),
State Press of Technical and Theoretical Literature,
Moscow, 1950, page 195.
Animations computed by Mathematica,
(R) Wolfram Research, and rendered by POV-Ray
at the Computer Graphics Lab, U. Waterloo.
The music is Études by F. Chopin, Op. 25, Nr. 1.
We thank Therese Biedl for performing the piece.
This video was produced at the Audio Visual
Centre, U. Waterloo, by Dianne Naughton.
The background shows Aleksandrov’s theorem,
А. Д. Александров, Выпуклые Многогранники
(A. D. Aleksandrov, Convex polyhedra),
State Press of Technical and Theoretical Literature,
Moscow, 1950, page 195.
Thanks to
Glenn Anderson
Rick Mabry
Blair Conrad
William Cowan
Michael McCool
Mark Riddell
Patrick Gilhuly
Jeffrey Shallit
Josée Lajoie
This work is supported by NSERC and NSF.
Thanks to
Glenn Anderson
Rick Mabry
Blair Conrad
William Cowan
Michael McCool
Dianne Naughton
Patrick Gilhuly
Mark Riddell
Josée Lajoie
Jeffrey Shallit
This work is supported by NSERC and NSF.