Reshaping Convex Polyhedra / by Joseph O'Rourke, Costin Vîlcu.

Format:

Book

Author/Creator:

O'Rourke, Joseph, 1928- author.

Vîlcu, Costin, author.

Language:

English

Subjects (All):

Convex geometry.

Discrete geometry.

Convex and Discrete Geometry.

Local Subjects:

Convex and Discrete Geometry.

Physical Description:

1 online resource (245 pages)

Edition:

1st ed. 2024.

Place of Publication:

Cham : Springer Nature Switzerland : Imprint: Springer, 2024.

Summary:

The focus of this monograph is converting—reshaping—one 3D convex polyhedron to another via an operation the authors call “tailoring.” A convex polyhedron is a gem-like shape composed of flat facets, the focus of study since Plato and Euclid. The tailoring operation snips off a corner (a “vertex”) of a polyhedron and sutures closed the hole. This is akin to Johannes Kepler’s “vertex truncation,” but differs in that the hole left by a truncated vertex is filled with new surface, whereas tailoring zips the hole closed. A powerful “gluing” theorem of A.D. Alexandrov from 1950 guarantees that, after closing the hole, the result is a new convex polyhedron. Given two convex polyhedra P, and Q inside P, repeated tailoring allows P to be reshaped to Q. Rescaling any Q to fit inside P, the result is universal: any P can be reshaped to any Q. This is one of the main theorems in Part I, with unexpected theoretical consequences. Part II carries out a systematic study of “vertex-merging,” a technique that can be viewed as a type of inverse operation to tailoring. Here the start is P which is gradually enlarged as much as possible, by inserting new surface along slits. In a sense, repeated vertex-merging reshapes P to be closer to planarity. One endpoint of such a process leads to P being cut up and “pasted” inside a cylinder. Then rolling the cylinder on a plane achieves an unfolding of P. The underlying subtext is a question posed by Geoffrey Shephard in 1975 and already implied by drawings by Albrecht Dürer in the 15th century: whether every convex polyhedron can be unfolded to a planar “net.” Toward this end, the authors initiate an exploration of convexity on convex polyhedra, a topic rarely studied in the literature but with considerable promise for future development. This monograph uncovers new research directions and reveals connections among several, apparently distant, topics in geometry: Alexandrov’s Gluing Theorem, shortest paths and cut loci, Cauchy’s Arm Lemma, domes, quasigeodesics, convexity, and algorithms throughout. The interplay between these topics and the way the main ideas develop throughout the book could make the “journey” worthwhile for students and researchers in geometry, even if not directly interested in specific topics. Parts of the material will be of interest and accessible even to undergraduates. Although the proof difficulty varies from simple to quite intricate, with some proofs spanning several chapters, many examples and 125 figures help ease the exposition and illustrate the concepts.

Contents:

I. Tailoring for Every Body

1. Introduction to Part I

2. Preliminaries

3. Domes and Pyramids

4. Tailoring via Sculpting

5. Pyramid Seal Graph

6. Algorithms for Tailoring via Sculpting

7. Crests

8. Tailoring via Flattening

9. Applications of Tailoring

II. Vertex-Merging and Convexity

10. Introduction to Part II

11. Vertex-Merging Reductions and Slit Graphs

12. Planar Spiral Slit Tree

13. Convexity on Convex Polyhedra

14. Minimal-length Enclosing Polygon

15. Spiral Tree on Polyhedron

16. Unfolding via Slit Trees

17. Vertices on Quasigeodesics

18. Conclusions

Bibliography

References

Index.

ISBN:

9783031475115

3031475119