''Mathematical Models: From the Collections of Universities and Museums – Photograph Volume and Commentary'' is a book on the physical models of concepts in mathematics that were constructed in the 19th century and early 20th century and kept as instructional aids at universities. It credits Gerd Fischer as editor, but its photographs of models are also by Fischer. It was originally published by

Vieweg+Teubner Verlag Vieweg is a German surname. Notable people with the surname include: * Alexander Vieweg (born 1986), German javelin thrower * Eduard Vieweg (1797–1869), German publisher and bookseller * Friedrich Vieweg (1761–1835), German publisher and bookse ...

for their bicentennial in 1986, both in German (titled ''Mathematische Modelle. Aus den Sammlungen von Universitäten und Museen. Mit 132 Fotografien. Bildband und Kommentarband'') and (separately) in English translation, in each case as a two-volume set with one volume of photographs and a second volume of mathematical commentary. Springer Spektrum reprinted it in a second edition in 2017, as a single dual-language volume.

Topics

The work consists of 132 full-page photographs of mathematical models, divided into seven categories, and seven chapters of mathematical commentary written by experts in the topic area of each category. These categories are: *Wire and thread models, of

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...

s of various dimensions, and of

hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ...

cylinder A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite ...

s, and related

ruled surface In geometry, a Differential geometry of surfaces, surface in 3-dimensional Euclidean space is ruled (also called a scroll) if through every Point (geometry), point of , there is a straight line that lies on . Examples include the plane (mathemat ...

s, described as "elementary

analytic geometry In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, and als ...

" and explained by Fischer himself. *Plaster and wood models of cubic and quartic

algebraic surface In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...

s, including Cayley's ruled cubic surface, the

Clebsch surface In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular cubic surface, studied by and , all of whose 27 exceptional lines can be defined over the real numbers. The term Klein's icosahedral surf ...

, Fresnel's wave surface, the

Kummer surface In algebraic geometry, a Kummer quartic surface, first studied by , is an irreducible nodal surface of degree 4 in \mathbb^3 with the maximal possible number of 16 double points. Any such surface is the Kummer variety of the Jacobian variet ...

, and the

Roman surface In mathematics, the Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane; ...

, with commentary by W. Barth and H. Knörrer. *Wire and plaster models illustrating the

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

and

curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...

of curves and surfaces, including surfaces of revolution,

Dupin cyclide In mathematics, a Dupin cyclide or cyclide of Dupin is any Inversive geometry, geometric inversion of a standard torus, Cylinder (geometry), cylinder or cone, double cone. In particular, these latter are themselves examples of Dupin cyclides. They ...

helicoid The helicoid, also known as helical surface, is a smooth Surface (differential geometry), surface embedded in three-dimensional space. It is the surface traced by an infinite line that is simultaneously being rotated and lifted along its Rotation ...

s, and

minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...

s including the

Enneper surface In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by: \begin x &= \tfrac u \left(1 - \tfracu^2 + v^2\right), \\ y &= \tfrac v \left(1 - \tfracv^2 + u^2\righ ...

, with commentary by M. P. do Carmo, G. Fischer, U. Pinkall, H. and Reckziegel. * Surfaces of constant width including the surface of rotation of the

Reuleaux triangle A Reuleaux triangle is a circular triangle, curved triangle with curve of constant width, constant width, the simplest and best known curve of constant width other than the circle. It is formed from the intersection of three circle, circular dis ...

and the Meissner bodies, described by J. Böhm. *

Uniform star polyhedra In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, ...

, described by E. Quaisser. *Models of the

projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, paral ...

, including the Roman surface (again), the cross-cap, and

Boy's surface In geometry, Boy's surface is an immersion of the real projective plane in three-dimensional space. It was discovered in 1901 by the German mathematician Werner Boy, who had been tasked by his doctoral thesis advisor David Hilbert to prove th ...

, with commentary by U. Pinkall that includes its realization by

Roger Apéry Roger Apéry (; 14 November 1916, Rouen – 18 December 1994, Caen) was a Greek-French mathematician most remembered for Apéry's theorem, which states that is an irrational number. Here, denotes the Riemann zeta function. Biography Apéry wa ...

as a quartic surface (disproving a conjecture of

Heinz Hopf Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of dynamical systems, topology and geometry. Early life and education Hopf was born in Gräbschen, German Empire (now , part of Wrocław, Poland) ...

). * Graphs of functions, both with real and complex variables, including the

Peano surface In mathematics, the Peano surface is the graph of the two-variable function :f(x,y)=(2x^2-y)(y-x^2). It was proposed by Giuseppe Peano in 1899 as a counterexample to a conjectured criterion for the existence of maxima and minima of functions of tw ...

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

s, exponential function and

Weierstrass's elliptic functions In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions is also referred to as ℘-functions and they are usually denoted by the s ...

, with commentary by J. Leiterer.

Audience and reception

This book can be viewed as a supplement to ''Mathematical Models'' by

Martyn Cundy Henry Martyn Cundy (23 December 1913 – 25 February 2005) was a mathematics teacher and professor in Britain and Malawi as well as a singer, musician and poet. He was one of the founders of the School Mathematics Project to reform O level and ...

and A. P. Rollett (1950), on instructions for making mathematical models, which according to reviewer Tony Gardiner "should be in every classroom and on every lecturer's shelf" but in fact sold very slowly. Gardiner writes that the photographs may be useful in undergraduate mathematics lectures, while the commentary is best aimed at mathematics professionals in giving them an understanding of what each model depicts. Gardiner also suggests using the book as a source of inspiration for undergraduate research projects that use its models as starting points and build on the mathematics they depict. Although Gardiner finds the commentary at times overly telegraphic and difficult to understand, reviewer O. Giering, writing about the German-language version of the same commentary, calls it detailed, easy-to-read, and stimulating. By the time of the publication of the second edition, in 2017, reviewer Hans-Peter Schröcker evaluates the visualizations in the book as "anachronistic", superseded by the ability to visualize the same phenomena more easily with modern computer graphics, and he writes that some of the commentary is also "slightly outdated". Nevertheless, he writes that the photos are "beautiful and aesthetically pleasing", writing approvingly that they use color sparingly and aim to let the models speak for themselves rather than dazzling with many color images. And despite the fading strength of its original purpose, he finds the book valuable both for its historical interest and for what it still has to say about visualizing mathematics in a way that is both beautiful and informative.

References

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Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...

, mr=0851009, year=1988 {{citation , last = Gardiner , first = Tony , date = March 1987 , doi = 10.2307/3616334 , issue = 455 , journal =

The Mathematical Gazette ''The Mathematical Gazette'' is a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range. The journ ...

, jstor = 3616334 , page = 94 , title = Review of ''Mathematical Models'' (1st edition) , volume = 71, s2cid = 165554250 {{citation, title=Review of ''Mathematische Modelle'', journal=

zbMATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastru ...

, first=O., last=Giering, zbl=0585.51001 {{citation, title=Review of ''Mathematical Models'' (1st edition), journal=

, first=Hans-Peter, last=Schröcker, zbl=1386.00007 Mathematical tools Mathematics books 1986 non-fiction books 2017 non-fiction books