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