88c:00001 00A05 00A20 01A55 51-00 51-03 53-00 53-03
Barth, Wolf(D-ERL); Böhm, Johannes(DDR-FSU); do Carmo, Manfredo
P.(BR-IMPA); Fischer, Gerd(D-DSLD); Knörrer, Horst(D-BONN);
Leiterer, Jürgen(DDR-ADW); Pinkall, Ulrich(D-MPI); Quaisser,
Erhard(DDR-POT); Reckziegel, Helmut(D-KOLN)
-> Mathematical models. (English)
Translated from the German.
Friedr. Vieweg & Sohn, Braunschweig, 1986. viii+83 pp. $52.00 the two
volume set. ISBN 3-528-08991-1
_________________________________________________________________
From the mid-nineteenth century until the early part of this century,
models of mathematical objects, fashioned in wire, string, wood and
plaster, provided illustrations for theses and research papers. Many
models still exist but the significance of these models has to a great
extent been lost. This volume of commentaries on the photographs of
models in the companion volume gives the background necessary to help
a modern reader appreciate the power of such visual representations of
mathematical phenomena. Many of the objects described in this book are
related to topics of current research interest, and the authors often
make references to modern literature as well as to the articles of the
nineteenth and early twentieth century which these models originally
were intended to illustrate.
Chapter 1 describes wire modes of cubes and hypercubes, and thread
models of hyperboloids and cylindroids. Chapter 2 discusses plaster
models of algebraic surfaces of order three and four, including a
cubic surface containing 27 straight lines and singular cubic surfaces
following the classification by Cayley and Schlafli. There is also a
modern description of these surfaces in terms of Coxeter diagrams.
Examples of quartic surfaces include plaster models of Kummer
surfaces, Steiner's Roman surface, and surfaces with large numbers of
conical points, as well as wooden models of quartics with cubic
symmetry. Chapter 3, on differential geometry, describes wire models
of curves with singular projections, plaster models of quadrics with
curvature lines, cyclides of Dupin, surfaces of revolution with
constant curvature, helicoids, Enneper's surface and other minimal
surfaces. Chapter 4 discusses convex bodies of constant width in two
and three dimensions, and Chapter 5 considers the Kepler-Poinsot
regular star-polyhedra. Chapter 6 presents models of the real
projective plane, from the cross-cap and Steiner's Roman surface, to
the immersion designed by Werner Boy. This chapter also includes
equations for Boy's immersion found recently by J. Apery. Chapter 7
describes function graphs in the theory of real and complex variables,
including the complex exponential and the Weierstrass $\wp$-function.
In summary, this volume of commentaries describes the geometric,
algebraic, and analytic properties of a classical set of models which
originally provided illustrations and insight in numerous mathematical
areas. With this commentary they can provide the same for modern
readers.
Reviewed by T. Banchoff
© Copyright American Mathematical Society 1988, 1997