''Mathematical Models'' is a book on the construction of physical models of mathematical objects for educational purposes. It was written 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 an ...

and A. P. Rollett, and published by the

Clarendon Press Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books ...

in 1951, with a second edition in 1961. Tarquin Publications published a third edition in 1981. The

vertex configuration In geometry, a vertex configurationCrystallography ...

of a

uniform polyhedron In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fa ...

, a generalization of the

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

that describes the pattern of polygons surrounding each

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet * Vertex (computer graphics), a data structure that describes the positio ...

, was devised in this book as a way to name the

Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...

s, and has sometimes been called the ''Cundy–Rollett symbol'' as a nod to this origin.

Topics

The first edition of the book had five chapters, including its introduction which discusses model-making in general and the different media and tools with which one can construct models. The media used for the constructions described in the book include "paper, cardboard, plywood, plastics, wire, string, and sheet metal". The second chapter concerns plane geometry, and includes material on the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

, the

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

dissection problem In geometry, a dissection problem is the problem of partitioning a geometric figure (such as a polytope or ball) into smaller pieces that may be rearranged into a new figure of equal content. In this context, the partitioning is called simply a ...

s, the

mathematics of paper folding The discipline of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability (whether the model can be flattened without damaging it), and the use of paper f ...

tessellation A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...

s, and

plane curve In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic pla ...

s, which are constructed by stitching, by graphical methods, and by mechanical devices. The third chapter, and the largest part of the book, concerns

polyhedron model A polyhedron model is a physical construction of a polyhedron, constructed from cardboard, plastic board, wood board or other panel material, or, less commonly, solid material. Since there are 75 uniform polyhedra, including the five regular con ...

s, made from cardboard or plexiglass. It includes information about the

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...

s, their stellations and

duals ''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers. Track listing :* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, P ...

uniform polyhedron compound In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts tran ...

s, and

deltahedra In geometry, a deltahedron (plural ''deltahedra'') is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek upper case delta (Δ), which has the shape of an equilateral triangle. There are infinitely many del ...

. The fourth chapter is on additional topics in

solid geometry In mathematics, solid geometry or stereometry is the traditional name for the geometry of Three-dimensional space, three-dimensional, Euclidean spaces (i.e., 3D geometry). Stereometry deals with the measurements of volumes of various solid fig ...

and curved surfaces, particularly

quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...

s but also including topological

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

s such as the

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...

Möbius strip In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Augu ...

and

Klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...

, and physical models helping to visualize the map coloring problem on these surfaces. Also included are

sphere packing In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing p ...

s. The models in this chapter are constructed as the boundaries of solid objects, via two-dimensional paper cross-sections, and by

string figure A string figure is a design formed by manipulating string on, around, and using one's fingers or sometimes between the fingers of multiple people. String figures may also involve the use of the mouth, wrist, and feet. They may consist of sing ...

s. The fifth chapter, and the final one of the first edition, includes mechanical apparatus including

harmonograph A harmonograph is a mechanical apparatus that employs pendulums to create a geometric image. The drawings created typically are Lissajous curves or related drawings of greater complexity. The devices, which began to appear in the mid-19th century ...

s and

mechanical linkage A mechanical linkage is an assembly of systems connected to manage forces and movement. The movement of a body, or link, is studied using geometry so the link is considered to be rigid. The connections between links are modeled as providing i ...

s, the

bean machine The Galton board, also known as the Galton box or quincunx or bean machine, is a device invented by Sir Francis Galton to demonstrate the central limit theorem, in particular that with sufficient sample size the binomial distribution approximat ...

and its demonstration of the

central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselv ...

, and analogue computation using

hydrostatics Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body " fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imm ...

. The second edition expands this chapter, and adds another chapter on computational devices such as the

differential analyser The differential analyser is a mechanical analogue computer designed to solve differential equations by integration, using wheel-and-disc mechanisms to perform the integration. It was one of the first advanced computing devices to be used operat ...

Vannevar Bush Vannevar Bush ( ; March 11, 1890 – June 28, 1974) was an American engineer, inventor and science administrator, who during World War II headed the U.S. Office of Scientific Research and Development (OSRD), through which almost all wartime ...

. Much of the material on polytopes was based on the book ''

Regular Polytopes In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...

'' by

H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

, and some of the other material has been drawn from resources previously published in 1945 by the

National Council of Teachers of Mathematics Founded in 1920, The National Council of Teachers of Mathematics (NCTM) is a professional organization for schoolteachers of mathematics in the United States. One of its goals is to improve the standards of mathematics in education. NCTM holds an ...

Audience and reception

At the time they wrote the book, Cundy and Rollett were

sixth form In the education systems of England, Northern Ireland, Wales, Jamaica, Trinidad and Tobago and some other Commonwealth countries, sixth form represents the final two years of secondary education, ages 16 to 18. Pupils typically prepare for A-l ...

teachers in the UK, and they intended the book to be used by mathematics students and teachers for educational activities at that level. However, it may also be enjoyed by a general audience of mathematics enthusiasts. Reviewer Michael Goldberg notes some minor errors in the book's historical credits and its notation, and writes that for American audiences some of the British terminology may be unfamiliar, but concludes that it could still be valuable for students and teachers. Stanley Ogilvy complains about the inconsistent level of rigor of the mathematical descriptions, with some proofs given and others omitted, for no clear reason, but calls this issue minor and in general calls the book's presentation excellent.

Dirk ter Haar Dirk ter Haar FRSE FIP DSc (; Oosterwolde, 19 April 1919 – Drachten, 3 September 2002) was an Anglo-Dutch physicist. Life Dirk ter Haar was born at Oosterwolde in the province Friesland in the north of the Netherlands on 19 April 1919. He stu ...

is more enthusiastic, recommending it to anyone interested in mathematics, and suggesting that it should be required for mathematics classrooms. Similarly, B. J. F. Dorrington recommends it to all mathematical libraries, and The Basic Library List Committee of the

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

has given it their strong recommendation for inclusion in undergraduate mathematics libraries. By the time of its second edition, H. S. M. Coxeter states that ''Mathematical Models'' had become "well known".

References

{{reflist, refs= {{citation, title=Mathematical Models (3rd ed.; listing with no review), work=MAA Reviews, publisher=Mathematical Association of America, accessdate=2020-09-09, url=https://www.maa.org/press/maa-reviews/mathematical-models {{citation, last=Goldberg, first=M., journal=

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=0049560, title=Review of 1st ed. {{citation, last=Müller, first=H. R., journal=

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

, language=German, title=Review of 1st ed., zbl=0047.38807; 2nd ed., {{zbl, 0095.38001 {{citation, last=ter Haar, first=D., authorlink=Dirk ter Haar, date=March 1953, issue=3, journal=

The Scientific Monthly ''The Scientific Monthly'' was a science magazine published from 1915 to 1957. Psychologist James McKeen Cattell, the former publisher and editor of ''The Popular Science Monthly'', was the original founder and editor. In 1958, ''The Scientific Mo ...

, jstor=20668, pages=188–189, title=Briefly reviewed (review of 1st ed.), volume=76 {{citation, last=Stone, first=Abraham, date=April 1953, issue=4, journal=

Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...

, jstor=24944205, page=110, title=Review of 1st ed., volume=188 {{citation, last=Dorrington, first=B. J. F., date=September 1953, doi=10.2307/3608314, issue=321, journal=

The Mathematical Gazette ''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...

, jstor=3608314, page=223, title=Review of 1st ed., volume=37 {{citation, last=Ogilvy, first=C. Stanley, authorlink=C. Stanley Ogilvy, date=November 1959, issue=7, journal=

The Mathematics Teacher Founded in 1920, The National Council of Teachers of Mathematics (NCTM) is a professional organization for schoolteachers of mathematics in the United States. One of its goals is to improve the standards of mathematics in education. NCTM holds an ...

, jstor=27956015, pages=577–578, title=Review of 1st ed., volume=52 {{citation, last=Coxeter, first=H. S. M., authorlink=Harold Scott MacDonald Coxeter, date=December 1962, doi=10.2307/3611791, issue=358, journal=

, jstor=3611791, page=331, title=Review of 2nd ed., volume=46 {{citation , last = Popko , first = Edward S. , contribution = 6.4.1 Cundy–Rollett Symbols , contribution-url = https://books.google.com/books?id=HjTSBQAAQBAJ&pg=PA164 , doi = 10.1201/b12253-22 , isbn = 978-1-4665-0429-5 , location = Boca Raton, Florida , mr = 2952780 , publisher = CRC Press , title = Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere , year = 2012 Mathematical tools Mathematics books 1952 non-fiction books