Metamorphosis of the Cube

1
Metamorphosisof the Cube

• Erik DemaineMartin DemaineAnna LubiwJoseph
  ORourkeIrena Pashchenko

2

• These foldings and unfoldings
• illustrate two problems.

Unfold a convex polyhedron into a simple polygon
Problem 1.
This problem is solved by the star unfolding.
(Agarwal, Aronov, ORourke, and
Schevon 1997)
3

• These foldings and unfoldings
• illustrate two problems.

Unfold a convex polyhedron into a simple polygon
Problem 1.
This problem is solved by the star unfolding. But
it remains open for cuts along the edges of the
polyhedron.
4

• These foldings and unfoldings
• illustrate two problems.

Fold a simple polygon intoa convex polyhedron
Problem 2.
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 ORourke 1997)
5

• These foldings and unfoldings
• illustrate two problems.

Fold a simple polygon intoa convex polyhedron
Problem 2.
Although Aleksandrovs theorem guarantees
uniqueness, finding the actual convex polyhedron
is an open question. Our examples were done by
hand.
6

• 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 Aleksandrovs theorem,
• ?. ?. ???????????, ???????? ?????????????
• (A. D. Aleksandrov, Convex polyhedra),
• State Press of Technical and Theoretical
  Literature,
• Moscow, 1950, page 195.

7

• 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 Aleksandrovs theorem,
• ?. ?. ???????????, ???????? ?????????????
• (A. D. Aleksandrov, Convex polyhedra),
• State Press of Technical and Theoretical
  Literature,
• Moscow, 1950, page 195.

8

• 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 Aleksandrovs theorem,
• ?. ?. ???????????, ???????? ?????????????
• (A. D. Aleksandrov, Convex polyhedra),
• State Press of Technical and Theoretical
  Literature,
• Moscow, 1950, page 195.

9

• Thanks to
• Glenn Anderson Rick Mabry
• Blair Conrad Michael McCool
• William Cowan Mark Riddell
• Patrick Gilhuly Jeffrey Shallit
• Josée Lajoie
• This work is supported by NSERC and NSF.

10

• Thanks to
• Glenn Anderson Rick Mabry
• Blair Conrad Michael McCool
• William Cowan Dianne Naughton
• Patrick Gilhuly Mark Riddell
• Josée Lajoie Jeffrey Shallit
• This work is supported by NSERC and NSF.