Computing the Continuous Discretely : Integer-point Enumeration in Polyhedra

Computing the Continuous Discretely : Integer-point Enumeration in Polyhedra - Undergraduate Texts in Mathematics

Softcover reprint of hardcover 1st ed. 2007

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Publisher's Synopsis

The world is continuous, but the mind is discrete. David Mumford We seek to bridge some critical gaps between various ?elds of mathematics by studying the interplay between the continuous volume and the discrete v- ume of polytopes. Examples of polytopes in three dimensions include crystals, boxes, tetrahedra, and any convex object whose faces are all ?at. It is amusing to see how many problems in combinatorics, number theory, and many other mathematical areas can be recast in the language of polytopes that exist in some Euclidean space. Conversely, the versatile structure of polytopes gives us number-theoretic and combinatorial information that ?ows naturally from their geometry. Fig. 0. 1. Continuous and discrete volume. The discrete volume of a body P can be described intuitively as the number of grid points that lie inside P, given a ?xed grid in Euclidean space. The continuous volume of P has the usual intuitive meaning of volume that we attach to everyday objects we see in the real world. VIII Preface Indeed, the di?erence between the two realizations of volume can be thought of in physical terms as follows. On the one hand, the quant- level grid imposed by the molecular structure of reality gives us a discrete notion of space and hence discrete volume. On the other hand, the N- tonian notion of continuous space gives us the continuous volume.

Book information

ISBN: 9781441921192
Publisher: Springer New York
Imprint: Springer
Pub date:
Edition: Softcover reprint of hardcover 1st ed. 2007
DEWEY: 516.11
DEWEY edition: 22
Language: English
Number of pages: 226
Weight: 379g
Height: 235mm
Width: 155mm
Spine width: 13mm