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Dive into the intriguing world of cube metamorphosis as depicted through foldings and unfoldings to solve two problems: unfolding a convex polyhedron into a simple polygon and folding a simple polygon into a convex polyhedron. Witness the innovative solutions, including the star unfolding method and Aleksandrov's conditions for folding, explored through animations created using Mathematica and rendered by POV-Ray. Enjoy the musical accompaniment of Philip Glass's "Opening" as you delve into the visualizations of these geometric transformations. This captivating exploration was produced at the Computer Graphics Lab, University of Waterloo, with acknowledgments to the contributors and supporters from various institutions.
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Metamorphosisof the Cube Erik DemaineMartin DemaineAnna LubiwJoseph O’RourkeIrena Pashchenko
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, O’Rourke, and Schevon 1997)
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.
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 & O’Rourke 1997)
These foldings and unfoldings • illustrate two problems. Fold a simple polygon intoa convex polyhedron Problem 2. 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 Michael McCool • William Cowan Mark Riddell • Patrick Gilhuly Jeffrey Shallit • Josée Lajoie • This work is supported by NSERC and NSF.
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.
