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2013 | Buch

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Shaping Space

Exploring Polyhedra in Nature, Art, and the Geometrical Imagination

herausgegeben von: Marjorie Senechal

Verlag: Springer New York

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Molecules, galaxies, art galleries, sculptures, viruses, crystals, architecture, and more: Shaping Space—Exploring Polyhedra in Nature, Art, and the Geometrical Imagination is an exuberant survey of polyhedra and at the same time a hands-on, mind-boggling introduction to one of the oldest and most fascinating branches of mathematics.

Some of the world’s leading geometers present a treasury of ideas, history, and culture to make the beauty of polyhedra accessible to students, teachers, polyhedra hobbyists, and professionals such as architects and designers, painters and sculptors, biologists and chemists, crystallographers, physicists and earth scientists, engineers and model builders, mathematicians and computer scientists.

The creative chapters by more than 25 authors explore almost every imaginable side of polyhedra. From the beauty of natural forms to the monumental constructions made by man, there is something to fascinate every reader. The book is dedicated to the memory of the legendary geometer H. S. M. Coxeter and the multifaceted design scientist Arthur L. Loeb.

Inhaltsverzeichnis

Frontmatter

First Steps

Frontmatter

1. Introduction to the Polyhedron Kingdom

Abstract

What is a polyhedron? The question is short, the answer is long. Although you may never have heard of the Polyhedron Kingdom before, it is nearly as vast and as varied as the animal, mineral, and vegetable kingdoms (and it overlaps all three of them). There are aristocrats and workers, families and individuals, old polyhedra with long and interesting histories and young polyhedra born yesterday or the day before. In this kingdom you can take a walking tour of polyhedral architecture, visit a nature preserve and an art gallery and an artisans’ polyhedra fair. As you stroll along you may even glimpse polyhedral ghosts from four-dimensional space.

Marjorie Senechal

2. Six Recipes for Making Polyhedra

Abstract

This chapter includes six “recipes” for making polyhedra, devised by famous polyhedra31.3pc]Please provide affiliation for “Arthur Vi Hart”. chefs. Some recipes are for beginners, others are intermediate or advanced. You can use these recipes, or devise your own. Building models is fun, and will give you a deeper understanding of the chapters that follow.

Marion Walter, Jean Pedersen, Magnus Wenninger, Doris Schattschneider, Arthur L. Loeb, Erik Demaine, Martin Demaine, Vi Hart

3. Regular and Semiregular Polyhedra

Abstract

The cube, the octahedron, and the tetrahedron obviously have been admired for thousands of years. It is impossible to say who first described them. Certainly the Pythagoreans knew all about them. I understand that a dodecahedron was found in Italy which was apparently made in 500 B.C. or perhaps even earlier, and that icosahedral dice were used by the ancient Egyptians. They can be seen in the British Museum, although there is some doubt about their exact date. All the five so-called Platonic solids are described in the later books of Euclid. Subsequent writers have made it much easier to see that the number of Platonic solids is just five.

H. S. M. Coxeter

4. Milestones in the History of Polyhedra

Abstract

Considering the fact that polyhedra have been studied for so long, it is rather surprising that there has been no exhaustive study of their history. But we are very lucky that the authors of four modern classics on the theory of polyhedra—Brückner, Coxeter, Fejes-Tóth and Grünbaum—were interested in historical information and provided detailed historical notes in their books. I propose to present an outline of the milestones in the history of the subject, putting together the thread of what happened as the theory developed. I will pay special attention to regularity concepts.

Joseph Malkevitch

5. Polyhedra: Surfaces or Solids?

Abstract

What is a polyhedron? Since I am especially interested in the relationship between concepts and images, I decided to approach the subject from that point of view and try to relate mathematical concepts and images. A polyhedron is an image of many, many different concepts, some of them inconsistent with each other.

Arthur L. Loeb

6. Dürer’s Problem

Abstract

In 1525 the German painter and thinker Albrect Dürer published his masterwork on geometry, whose title translates as “On Teaching Measurement with a Compass and Straightedge.”

Joseph O’Rourke

Polyhedra in Nature and Art

Frontmatter

7. Exploring the Polyhedron Kingdom

Abstract

Having paid our respects to the rulers of the Polyhedron Kingdom and their extended families, we’re ready for a walking tour.

Marjorie Senechal

8. Spatial Perception and Creativity

Abstract

I come from Montreal, where I belong to a group called the Structural Topology Research Group. “Structural topology” is an often criticized term, but we are stuck with it.

Janos Baracs

9. Goldberg Polyhedra

Abstract

The regular polyhedra—see Chapter 1—are famous for their history, applications, beauty, and mathematical properties. Though not yet famous, the Goldberg Polyhedra too are notable in all these ways.

George Hart

10. Polyhedra and Crystal Structures

Abstract

I have long been interested in searching for interesting relationships between polyhedra and crystal structures, especially with the application of polyhedra as units for crystal structures. Crystallography uses geometry as a foundation. As a crystal scientist, I am interested in understanding how and why certain crystal structures are built the way they are, particularly from a geometric viewpoint. I am also constantly searching for relationships among the various crystal structures.

Chung Chieh

11. Polyhedral Molecular Geometries

Abstract

H.S.M. Coxeter has said that “the chief reason for studying regular polyhedra is still the same as in the times of the Pythagoreans, namely, that their symmetrical shapes appeal to one’s artistic sense.” The success of modern molecular chemistry affirms the validity of this statement; there is no doubt that aesthetic appeal has contributed to the rapid development of what could be termed polyhedral chemistry. The chemist Earl Muetterties movingly described his attraction to boron hydride chemistry, comparing it to Escher’s devotion to periodic drawings:

Magdolna Hargittai, Istvan Hargittai

12. Form, Function, and Functioning

Abstract

Polyhedra are objects worthy of study and admiration in their own right. They have been inspirations for mathematicians, artists, and architects, and have also served as models for abstract notions about the biological and physical world. The sophistication of such modeling has evolved over the centuries, influencing both physical and mathematical theories. In studying polyhedra we see over and over again ways in which theory is inspired by nature, and ways in which science is inspired by theory.

George Fleck

Polyhedra in the Geometrical Imagination

Frontmatter

13. The Polyhedron Kingdom Tomorrow

Abstract

Now you’ve reached the kingdom’s wild frontier: polyhedron theory at the research level. The natives in these parts speak a slightly different dialect from the artists and scientists you’ve met already, so listen carefully. (If you only grasp a phrase here and there, or even nothing at all, read on anyway: the sights—the chapters that follow—are not to be missed.)

Marjorie Senechal

14. Paneled and Molecular Polyhedra: How Stable Are They?

Abstract

Polyhedral models can be physically constructed in a variety of ways. The inexpensive methods described in Chapter 2 include sticks connected at endpoints, or paper, creased and glued with tape along edges. But the resulting structures are not always sufficiently stable: sticks may slip off their connecting joints, paper bends, and even when sturdy carton is used, not having enough tape may lead to loose paper ends or a flexible polyhedral model.

Ileana Streinu

15. Duality of Polyhedra

Abstract

The expression “mathematical folklore” refers to the results that most mathematicians take for granted, but which may never have been proved to be true. (Indeed, some of them are not true.) It is widely believed that the first person to call attention to this phenomenon was the French mathematician Jean Dieudonné.

Branko Grünbaum, G. C. Shephard

16. Combinatorial Prototiles

Abstract

Tiling problems have been investigated throughout the history of mathematics, leading to a vast literature on the subject. Our present knowledge of tilings of the plane is quite good, although there are of course many open problems even in the comparatively elementary and easily accessible levels.

Egon Schulte

17. Polyhedra Analogues of the Platonic Solids

Abstract

In this chapter we investigate polyhedra in Euclidean 3-space, E ³, without self-intersections and with some local and global properties related to those of the Platonic solids. A polyhedron is the geometric realization of a compact 2-manifold in E ³ such that its 2-faces are (not necessarily convex) plane polygons bounded by finitely many line segments. Adjacent faces and edges are not coplanar. A flag of a polyhedron P is any triple consisting of a vertex, an edge, and a face of P, all mutually incident.

Jörg M. Wills

18. Convex Polyhedra, Dirichlet Tessellations, and Spider Webs

Abstract

Plane pictures of three-dimensional convex polyhedra, plane sections of three-dimensional Dirichlet tessellations, and flat spider webs with tension in all the threads are essentially the same geometric object. At the root of this remarkable coincidence is a single geometric diagram that permits us to offer a unified image of the connections among these and other objects. Some hints of these connections are more than a century old, but others are very recent. We begin with an historical sketch.

Walter Whiteley, Peter F. Ash, Ethan Bolker, Henry Crapo

19. Uniform Polyhedra from Diophantine Equations

Abstract

A simple set of coordinates eases the study of metrical properties of uniform polyhedra. For instance, the six vertices of the regular octahedron {3,4} have Cartesian coordinates (±1,0,0), etc. where “etc.” means “permute the coordinates in all possible ways.” I find it pleasing in such examples that the coordinates are given by systematic choices. Observe further that the coordinates provide all integral solutions to the Diophantine equation

Barry Monson

20. Torus Decompostions of Regular Polytopes in 4-space

Abstract

When a regular polyhedron in ordinary 3-space is inscribed in a sphere, then a decomposition of the sphere into bands perpendicular to an axis of symmetry of the polyhedron determines a corresponding decomposition of the polyhedron. For example, a cube with two horizontal faces can be described as a union of two horizontal squares and a band of four vertical squares, and an octahedron with a horizontal face is a union of two horizontal triangles and a band formed by the six remaining triangles.

Thomas F. Banchoff

21. Tensegrities and Global Rigidity

Abstract

In 1947 a young artist named Kenneth Snelson invented an intriguing structure: a few sticks suspended rigidly in mid air without touching each other. It seemed like a magic trick. He showed this to the entrepreneur, builder, visionary, and self-styled mathematician, R. Buckminster Fuller, who called it a tensegrity because of its “tensional integrity.” Fuller talked about tensegrities and wrote about them extensively. Snelson went on to build a great variety of fascinating tensegrity sculptures all over the world, including the 60-foot work of art at the Hirschhorn Museum in Washington, DC. shown in Figure.

Robert Connelly

23. Ten Problems in Geometry

Abstract

Geometry is a field of knowledge, but it is at the same time an active field of research—our understanding of space, about shapes, about geometric structures develops in a lively dialogue, where problems arise, new questions are asked every day. Some of the problems are settled nearly immediately, some of them need years of careful study by many authors, still others remain as challenges for decades. In this chapter, we describe ten problems waiting to be solved.

Moritz W. Schmitt, Günter M. Ziegler

Backmatter

Titel: Shaping Space
herausgegeben von: Marjorie Senechal
Verlag: Springer New York
Electronic ISBN: 978-0-387-92714-5
Print ISBN: 978-0-387-92713-8
DOI: https://doi.org/10.1007/978-0-387-92714-5