Paper by Erik D. Demaine

Reference:

Erik D. Demaine, Martin L. Demaine, and Anna Lubiw, “Polyhedral Sculptures with Hyperbolic Paraboloids”, in Proceedings of the 2nd Annual Conference of BRIDGES: Mathematical Connections in Art, Music, and Science (BRIDGES'99), Winfield, Kansas, July 30–August 1, 1999, pages 91–100.

Abstract:

This paper describes the results of our experiments with gluing together partial hyperbolic paraboloids, or hypars. We make a paper model of each hypar by folding a polygonal piece of paper along concentric polygons in an alternating fashion. Gluing several hypars together along edges, we obtain a beautiful collection of closed, curved surfaces which we call hyparhedra. Our main examples are given in Figure 6.

We present an algorithm that constructs a hyparhedron given any polyhedron. The surface represents each face of the polyhedron by a “hat” of hypars. For a Platonic solid, the corners of the hypars include the vertices of the polyhedron and its dual, and one can easily reconstruct the input polyhedron from the hyparhedron. More generally, the hyparhedron captures the combinatorial topology of any polyhedron. We also present several possibilities for generalization.

Comments:

To keep the file smaller, the paper does not include the photographs in Figures 5, 6, and 7. Please refer to my hypar webpage to see these photographs.

Here are some photographs from during and after my talk, taken by Carlo Séquin.

Updates:

Ivars Peterson describes these results in his book, Fragments of Infinity: A Kaleidoscope of Math and Art, John Wiley & Sons, Inc., 2001, pages 79-80.

Length:

The paper is 10 pages.

Availability:

The paper is available in PostScript (629k), gzipped PostScript (148k), and PDF (233k).

See information on file formats.

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Related webpages:

Hyperbolic Paraboloids