Treks into Intuitive Geometry: The World of Polygons and Polyhedra is a book on geometry, written as a discussion between a teacher and a student in the style of a Socratic dialogue. It was written by Japanese mathematician Jin Akiyama and science writer Kiyoko Matsunaga, and published by Springer-Verlag in 2015 ( ISBN 978-4-431-55841-5), [1] with an expanded second edition in 2024 ( ISBN 978-981-99-8607-1).
The term "intuitive geometry" of the title was used by László Fejes Tóth to refer to results in geometry that are accessible to the general public, and the book concerns topics of this type. [1] [2]
The book has 16 self-contained chapters, [1] each beginning with an illustrative puzzle or real-world application. [3] It includes material on tessellations, polyhedra, and honeycombs, unfoldings of polyhedra and tessellations of unfoldings, cross sections of polyhedra, measuring boxes, gift wrapping, packing problems, wallpaper groups, pentagonal tilings, the Conway criterion for prototiles and Escher-like tilings of the plane by animal-shaped figures, aperiodic tilings including the Penrose tiling, the art gallery theorem, the Euler characteristic, dissection problems and the Dehn invariant, and the Steiner tree problem. [1] [2]
The book is heavily illustrated. And although the results of the book are demonstrated in an accessible way, the book provides sequences of deductions leading to each major claim, and more-complete proofs and references are provided in an appendix. [3]
Although it was initially developed from course material offered to undergraduates at the Tokyo University of Science, [2] the book is aimed at a broad audience, and assumes only a high-school level knowledge of geometry. [1] [2] It could be used to encourage children in mathematics as well as to provide material for teachers and public lecturers. [1] There is enough depth of material to also retain the interest of readers with a more advanced mathematical background. [1] [2]
Reviewer Matthieu Jacquemet writes that the ordering of topics is unintuitive and the dialogue-based format "artificial", but reviewer Tricia Muldoon Brown instead suggests that this format allows the work to flow very smoothly, "more like a novel or a play than a textbook ... with the ease of reading purely for pleasure". [3] Jacquemet assesses the book as "well illustrated and entertaining", [1] and Brown writes that it "is a delightful read". [3]
Reviewer Michael Fox disagrees, finding the dialogue irritating and the book overall "rather disappointing". He cites as problematic the book's cursory treatment of some of its topics, and in particular its treatment of tiling patterns as purely monochromatic, its omission of the frieze groups, and its use of demonstrations by special examples that do not have all the features of the general case. He also complains about idiosyncratic terminology, the use of decimal approximations instead of exact formulas for angles, the small scale of some figures, and an uneven level of difficulty of material. Nevertheless, he writes that "this is an interesting work, with much that cannot be found elsewhere". [2]
In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common.
In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.
In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex.
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences.
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In the mathematical theory of tessellations, the Conway criterion, named for the English mathematician John Horton Conway, is a sufficient rule for when a prototile will tile the plane. It consists of the following requirements: The tile must be a closed topological disk with six consecutive points A, B, C, D, E, and F on the boundary such that:
Jin Akiyama is a Japanese mathematician, known for his appearances on Japanese prime-time television (NHK) presenting magic tricks with mathematical explanations. He is director of the Mathematical Education Research Center at the Tokyo University of Science, and professor emeritus at Tokai University.
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Euler's Gem: The Polyhedron Formula and the Birth of Topology is a book on the formula for the Euler characteristic of convex polyhedra and its connections to the history of topology. It was written by David Richeson and published in 2008 by the Princeton University Press, with a paperback edition in 2012. It won the 2010 Euler Book Prize of the Mathematical Association of America.
Algebra and Tiling: Homomorphisms in the Service of Geometry is a mathematics textbook on the use of group theory to answer questions about tessellations and higher dimensional honeycombs, partitions of the Euclidean plane or higher-dimensional spaces into congruent tiles. It was written by Sherman K. Stein and Sándor Szabó, and published by the Mathematical Association of America as volume 25 of their Carus Mathematical Monographs series in 1994. It won the 1998 Beckenbach Book Prize, and was reprinted in paperback in 2008.
Symmetry in Mechanics: A Gentle, Modern Introduction is an undergraduate textbook on mathematics and mathematical physics, centered on the use of symplectic geometry to solve the Kepler problem. It was written by Stephanie Singer, and published by Birkhäuser in 2001.
Polyominoes: Puzzles, Patterns, Problems, and Packings is a mathematics book on polyominoes, the shapes formed by connecting some number of unit squares edge-to-edge. It was written by Solomon Golomb, and is "universally regarded as a classic in recreational mathematics". The Basic Library List Committee of the Mathematical Association of America has strongly recommended its inclusion in undergraduate mathematics libraries.
Icons of Mathematics: An Exploration of Twenty Key Images is a book on elementary geometry for a popular audience. It was written by Roger B. Nelsen and Claudi Alsina, and published by the Mathematical Association of America in 2011 as volume 45 of their Dolciani Mathematical Expositions book series.
Regular Figures is a book on polyhedra and symmetric patterns, by Hungarian geometer László Fejes Tóth. It was published in 1964 by Pergamon in London and Macmillan in New York.
The Symmetries of Things is a book on mathematical symmetry and the symmetries of geometric objects, aimed at audiences of multiple levels. It was written over the course of many years by John Horton Conway, Heidi Burgiel, and Chaim Goodman-Strauss, and published in 2008 by A K Peters. Its critical reception was mixed, with some reviewers praising it for its accessible and thorough approach to its material and for its many inspiring illustrations, and others complaining about its inconsistent level of difficulty, overuse of neologisms, failure to adequately cite prior work, and technical errors.