
06480898
b
2016a.00551
Sibley, Thomas Q.
Thinking geometrically. A survey of geometries.
MAA Textbooks. Washington, DC: The Mathematical Association of America (MAA) (ISBN 9781939512086/hbk; 9781614446194/ebook). xxiii, 559~p. (2015).
2015
Washington, DC: The Mathematical Association of America (MAA)
EN
G10
Playfair's axiom
Platonic solids
geodesic dome
cuboctahedron
regular icosahedron
first stellation of a polyhedron
axiomatic system
folium of Descartes
B\'{e}zier curves
pseudosphere
omega triangle
Saccheri quadrilaterals and triangles
orbitstabilizer theorem
frieze pattern
wallpaper pattern
fractal
Koch curve
separation axioms
continuity axiom
spiral of Archimedes
helix
torus
geodesic
Voronoi diagram
art gallery problem
fortress theorem
Zbl 1298.51003
Zbl 0732.51003
Zbl 0326.53001
Zbl 0095.34502
Zbl 0601.05001
This also visually very appealing book offers a wealth of geometric information together with the historical background. The author takes the reader onto a long and engrossing journey to eleven wellselected basic sites of classical and modern geometry. According to the preface the author addresses ``mathematical majors and future secondary teachers'' (= {\it narrow audience}); the reviewer means that the audience should be extended to all persons interested in the foundations of geometry (= {\it extended audience}). {\it Concerning the narrow audience}. The author respects the (Common Core State Standards =) CCSSexpectations and the (National Council of Teachers =) NCTMrecommendations. Geometric intuition and facility in proofs are developed. Visualization by the use of dynamic geometry software is included in many exercises and projects. In the book's blurb the word ``exercise'' is mentioned four times and, indeed, the exercises (together with the answers to selected ones) comprise about 185 pages and the projects about 28 pages of the book's 559 pages. Thus the book contains an extensive collection of exercises. As introduction of each chapter the author presents the historical development of the new geometrical ideas. At the end of each chapter the author exhibits projects and suggested readings (= references). The projects should encourage the student to broaden the ideas discussed in the text; for instance some projects to Chapter 2 (Axiomatic Systems) demand: ``Investigate taxicab geometry, investigate metamathematics, write an essay on the roles of intuition and proof in geometry." For a recently published book dealing with similar topics see also [{\it J. M. Lee}, Axiomatic geometry. Providence, RI: American Mathematical Society (AMS) (2013; Zbl 1298.51003)]. {\it Concerning the extended audience}. These readers benefit from the excellently and interestingly written text. They can pick out exercises or the subchapters on the achievements of such famous mathematicians as Archimedes, Hilbert, G\"odel\dots to name but a few. Also their quotes at the beginning of each chapter deserve attention, for instance Subchapter 2.3 (Models and Metamathematics) is introduced by the following words of Henri Poincar\'e: ``Mathematics is the art of giving the same name to different things''. A substantial part of the book are the 489 very aesthetical figures; ``proofs of theorems in an axiomatic system cannot depend on diagrams, even though diagrams have been part of geometry since the ancient Greeks drew figures in the sand. We need the powerful insight and understanding that diagrams provide. However, the corresponding risk comes with the use of the pictures: We are liable to accept as intuitive a step that does not follow from the given conditions.'' (p. 7071). For sake of completeness we mention that the author rewrote and complemented his book [The geometric viewpoint: a survey of geometries. Reading, MA: AddisonWesley (1998)]. The text is divided into the Preface, 10 chapters, each ends with projects and suggested readings, an Epilogue, 6 Appendices, and, finally, contains Answers to Selected Exercises, Acknowledgements, and Index. Preface: Geometric intuition, the role of proofs, dependence and links between chapters. Chapter 1. Euclidean geometry: Overview and history, Erathosthenes estimates of the circumference of the earth, Euclid's approach to geometry, equality of measure (Eudoxus, method of exhaustion). Parallel lines (``historians suspect that Euclid wasn't completely comfortable with his fifth postulate\dots for he postponed using it until proposition I29''). Threedimensional geometry (Platonic solids\dots). The geometry of a sphere. Buckminster Fuller ([18951985], inventor of the geodesic domes). Chapter 2. Axiomatic systems. Axiomatic systems for Euclidean geometry (compare also the appendices A, B and C), SMSG postulates (and their difference to Euclid's approach). Hilbert's axioms (the five axiom groups, the separation axiom and the equivalent Pasch axiom). Models and metamathematics, G\"odel [19061978]: ``an axiomatic system is consistent if and only if it has a model'', incompleteness theorem. Chapter 3. Analytic geometry (starts with figure 3.0, whose text immediately sheds light on today's significance of analytic geometry for architecture). R. Descartes 15961650, P. de Fermat 16011665, L. Euler 17071783. Analytic model. Conics and local problems (Apollonius of Perga 260190 B.C.E., J. Kepler 15711630, {\it locus problem}, equation of ellipse, hyperbola, and parabola, reflection property of parabola). Parametric equations (introduced in 1748 by L. Euler, {\it cycloid, helix}). Polar coordinates (spiral, Bernoulli's lemniscate). Barycentric coordinates (August M\"obius 1827, center of gravity, trilinear plot in statistics). Other analytic geometries. Curves in (CAD =) computeraided design (B\'{e}zier curves and splines). Higher dimensional analytic geometry. Analytic geometry in ${\Bbb R}^n$ (starts with A. Cayley and others in 1843). ``Gaspard Monge [17461818] developed descriptive geometry\dots''. Chapter 4. NonEuclidean geometry (David Hilbert: ``The most suggestive and notable achievements of the last century is the discovery of nonEuclidean geometry.'' Albert Einstein: ``To this interpretation of geometry I attach great importance for should I have not been acquainted with it, I never would have been able to develop the theory of relativity''). N. Lobachevsky [17931856], J. Bolyai [18021860], reasons why the mathematical community ignored their publications during their lifetimes. G. Riemann [18261866] defined curvature in higher dimensions and envisioned geometries in any number of dimensions with changing curvatures throughout. Models of hyperbolic geometry. C. F. Gauss ([17771855], characterized all constructible regular polygons, proved the fundamental theorem of algebra, determined the orbit of Ceres, developed nonEuclidean geometry, made seminal contributions to differential geometry  curvature, geodesics , extended number theory to complex integers). Giovanni Girolamo Saccheri ([16671733], his book {\it Euclid freed from every flaw} was rescued from obscurity by E. Beltrami, Saccheri quadrilaterals and triangles. Modern mathematics readily accepts hyperbolic geometry and other systems as legitimate alternatives to Euclidean geometry''. Area, distance, and designs (builds the understanding of hyperbolic areas from the SMSG postulates in Appendix B). Spherical and single elliptic geometries, spherical geometry = {\it double elliptic geometry}. Chapter 5. Transformational geometry (F. Klein: ``Geometry is the study of those properties of a set which are preserved under a group of transformations of that set.'') CCSS underline the significance of transformations. Classifying isometries. Klein's definition of geometry (Erlanger Program [1872], from Klein's point of view the group for projective geometry includes the groups of transformations of the Euclidean geometry and some nonEuclidean geometries). Algebraic representation of transformations (Matrices and linear algebra). Similarities and affine transformations. Transformations in higher dimensions; computeraided design. Isometries of the sphere. Inversion and complex plane, M\"obius transformation. Chapter 6. Symmetry. J. H. C. Hessel [17961872], Auguste Bravais [18111863], A. M\"obius, and C. Jordan [18381922] contributed to the classification of all possible types of chemical crystals. Finite plane symmetry groups, {\it cyclic group, dihedral group}. Symmetry in the plane. Frieze patterns (``There are exactly seven groups of symmetries for frieze patterns, up to geometric isometries''). Wallpaper patterns (crystallographic restriction, ``there are exactly seventeen groups of symmetries for wallpaper patterns, up to geometric isometries''). Symmetries in higher dimensions. Finite threedimensional symmetry groups. The crystallographic groups (and their connection with wallpaper patterns). General finite symmetry groups. H. S. M. Coxeter [19072003], book {\it Introduction to geometry} was standard reference for decades. Symmetry in science. Chemical structure (diamond, graphite, salt [NaCl] and potassium chloride [KCl]). Quasicrystals. Symmetry and relativity, the Lorentz transformations of Minkowski space are the symmetries of special relativity. Fractals, selfsimilarity, {\it statistical selfsimilarity}, Hausdorff dimension, {\it fractal dimension}. Chapter 7. Projective geometry (Arthur Cayley: ``Metrical geometry is just a part of [projective] geometry, and [projective] geometry is all geometry"). Renaissance artists worked out the rules of perspective, {\it ideal or vanishing points}, G. Desargues [15931662], Blaise Pascal [16231662] saw the unifying power of what we now call projective methods, Gaspard Monge [17461818] rediscovered the projective ideas, his student J. V. Poncelet [17881867] published the book {\it Treatise of the projective properties of figures} in 1822, A. M\"obius and J. Pl\"ucker created coordinates for projective geometry, ``gradually geometers realized that the synthetic and analytic approaches complemented each other'', A. Cayley and F. Klein showed how to develop Euclidean, hyperbolic, and single elliptic geometries within projective geometry, 20th century brought applications of higher dimensional projective geometry in special relativity, computer graphics, statistical design theory, and photogrammetry). Axiomatic projective geometry (presents 10 axioms for the real projective plane, among them four separation axioms and the continuity axiom). Duality. Perspectivities and projectivities. Analytic projective geometry (linear algebra provides a powerful language for projective geometry). Cross ratio. Conics. Julius Pl\"ucker ([18011868] developed together with A. M\"obius homogeneous coordinates, rivalry with Jacob Steiner [17961863] who favored the synthetic approach. Projective transformations. Subgeometries (A. Cayley constructed Euclidean distance and angle measure from projective geometry, {\it absolute conic}). Hyperbolic geometry as a subgeometry. Single elliptic geometry as a subgeometry. Affine and Euclidean geometries as subgeometries ({\it circular points at infinity}). Projective space (``computer programmers use threedimensional projective space to make perspective views in computeraided design (CAD), ``the Lorentz transformations in the special theory of relativity are transformations of a subgeometry of fourdimensional projective space related to hyperbolic geometry''). Chapter 8. Finite geometries. 1892 Gino Fano: 3dimensional projective space with 15 points. Euler's 36officer problem, 1850 Kirkman's 15schoolgirls problem, currently: crossfertilization geometry, algebra, and combinatorics, errorcorrecting codes. {\it Order of an affine plane}, open problem: which orders of $n$ give affine planes? Projective planes. Design theory (concentrates on balanced incomplete block designs (= BIBD)). Errorcorrecting codes (Hamming distance). Finite analytic geometry (introduces the Galois field $\text{GF}(p)$ and mentions $\text{GF}(p^k)$). Ovals in finite projective planes. Finite analytic spaces. Chapter 9. Differential geometry. Sir Isaac Newton [16421727], foundational textbook {\it Principia Mathematica}, Gottlieb Wilhelm Leibniz [16461710] other founder of calculus, Jakob Bernoulli [16541705], AlexisClaude Clairaut and L. Euler analyzed torsion, C. F. Gauss proved profound theorems, Georg Riemann [18261866] generalized Gauss's results to any number of dimensions, ``Einstein needed a fourdimensional space built from three spatial dimensions and time, all with varying curvature corresponding to the strength of gravitational fields at different points''. Curves and curvature. Surfaces and curvature. Eugenio Beltrami [18351900] realized that a surface with constant negative Gaussian curvature gave a model of (part of) hyperbolic geometry. Geodesics and geometry of surfaces. Arc length on surfaces. Higher dimensions (Riemann: $n$dimensional manifold). Chapter 10. Discrete geometry (Snow's ``ghost map''). Epidemiologists use Voronoi diagrams, ``computational geometry arose in response to the need of efficient algorithms''. Distances between points (``in 1946 Paul Erd\"os posed the general question `What is the minimum number of distances $n$ points determine in $d$dimensional Euclidean space?'''). Triangulations (of a polygon). The art gallery problem. Tilings (monohedral plane tiling). Voronoi diagrams. Fortress theorem. Tilings (following figures gives a monohedral tiling of the plane: any centrally symmetric hexagon, any quadrilateral, any pentagon with two adjacent supplementary angles, any centrally symmetric orthogonal octagon; ``given the unlimited ways to modify the starting tile, there is no way to classify all possible tiles giving monohedral tilings''). Tilings in space: the incorrect claim of Aristotle [proof in exercise 10.3.14]. Chapter 11. Epilogue. Henri Poincar\'e: ``It is by logic we prove, it is by intuition we invent. Logic, therefore, remains barren unless fertilized by intuition.'' ``Geometric thinking lies between the purely formal reasoning of logic and algebra and the concrete insight of physical space'', ``our understanding of geometry had broadened enormously in the past 200 years and continues expanding without loosing touch with its historical and physical roots. As Benoit Mandelbrot said: `Intuition can be changed and refined and modified\dots''' Appendix A. Definitions, postulates, common notions, and propositions from Book I of Euclid's Elements. Appendix B. SMSG axioms for Euclidean geometry. Appendix C. Hilbert's axioms for Euclidean plane geometry. Appendix D. Linear algebra summary. Appendix E. Multivariable calculus summary. Appendix F. Elements of proofs (direct proof, proof by contradiction, induction proof).
Rolf Riesinger (Wien)