id: 06480898
dt: b
an: 2016a.00551
au: Sibley, Thomas Q.
ti: Thinking geometrically. A survey of geometries.
so: MAA Textbooks. Washington, DC: The Mathematical Association of America
(MAA) (ISBN 978-1-93951-208-6/hbk; 978-1-61444-619-4/ebook). xxiii,
559~p. (2015).
py: 2015
pu: Washington, DC: The Mathematical Association of America (MAA)
la: EN
cc: G10
ut: Playfair’s axiom; Platonic solids; geodesic dome; cuboctahedron; regular
icosahedron; first stellation of a polyhedron; axiomatic system; folium
of Descartes; Bézier curves; pseudosphere; omega triangle; Saccheri
quadrilaterals and triangles; orbit-stabilizer 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
ci: Zbl 1298.51003; Zbl 0732.51003; Zbl 0326.53001; Zbl 0095.34502; Zbl
0601.05001
li:
ab: 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 well-selected
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 =) CCSS-expectations and the (National Council of Teachers =)
NCTM-recommendations. 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, Gödel\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 Poincaré: “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. 70‒71). For sake
of completeness we mention that the author rewrote and complemented his
book [The geometric viewpoint: a survey of geometries. Reading, MA:
Addison-Wesley (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
I-29”). Three-dimensional geometry (Platonic solids\dots). The
geometry of a sphere. Buckminster Fuller ([1895‒1985], 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, Gödel [1906‒1978]: “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
1596‒1650, P. de Fermat 1601‒1665, L. Euler 1707‒1783. Analytic
model. Conics and local problems (Apollonius of Perga 260‒190 B.C.E.,
J. Kepler 1571‒1630, {\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 Möbius 1827, center of gravity, trilinear plot in statistics).
Other analytic geometries. Curves in (CAD =) computer-aided design
(Bézier curves and splines). Higher dimensional analytic geometry.
Analytic geometry in ${\Bbb R}^n$ (starts with A. Cayley and others in
1843). “Gaspard Monge [1746‒1818] developed descriptive
geometry\dots”. Chapter 4. Non-Euclidean geometry (David Hilbert:
“The most suggestive and notable achievements of the last century is
the discovery of non-Euclidean 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 [1793‒1856], J. Bolyai
[1802‒1860], reasons why the mathematical community ignored their
publications during their lifetimes. G. Riemann [1826‒1866] defined
curvature in higher dimensions and envisioned geometries in any number
of dimensions with changing curvatures throughout. Models of hyperbolic
geometry. C. F. Gauss ([1777‒1855], characterized all constructible
regular polygons, proved the fundamental theorem of algebra, determined
the orbit of Ceres, developed non-Euclidean geometry, made seminal
contributions to differential geometry ‒ curvature, geodesics ‒,
extended number theory to complex integers). Giovanni Girolamo Saccheri
([1667‒1733], 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 non-Euclidean geometries). Algebraic
representation of transformations (Matrices and linear algebra).
Similarities and affine transformations. Transformations in higher
dimensions; computer-aided design. Isometries of the sphere. Inversion
and complex plane, Möbius transformation. Chapter 6. Symmetry. J. H.
C. Hessel [1796‒1872], Auguste Bravais [1811‒1863], A. Möbius, and
C. Jordan [1838‒1922] 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 three-dimensional symmetry groups. The
crystallographic groups (and their connection with wallpaper patterns).
General finite symmetry groups. H. S. M. Coxeter [1907‒2003], 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, self-similarity, {\it statistical
self-similarity}, 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 [1593‒1662], Blaise
Pascal [1623‒1662] saw the unifying power of what we now call
projective methods, Gaspard Monge [1746‒1818] rediscovered the
projective ideas, his student J. V. Poncelet [1788‒1867] published
the book {\it Treatise of the projective properties of figures} in
1822, A. Möbius and J. Plücker 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 Plücker
([1801‒1868] developed together with A. Möbius homogeneous
coordinates, rivalry with Jacob Steiner [1796‒1863] 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 three-dimensional
projective space to make perspective views in computer-aided design
(CAD), “the Lorentz transformations in the special theory of
relativity are transformations of a subgeometry of four-dimensional
projective space related to hyperbolic geometry”). Chapter 8. Finite
geometries. 1892 Gino Fano: 3-dimensional projective space with 15
points. Euler’s 36-officer problem, 1850 Kirkman’s 15-schoolgirls
problem, currently: cross-fertilization geometry, algebra, and
combinatorics, error-correcting 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)). Error-correcting 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
[1642‒1727], foundational textbook {\it Principia Mathematica},
Gottlieb Wilhelm Leibniz [1646‒1710] other founder of calculus, Jakob
Bernoulli [1654‒1705], Alexis-Claude Clairaut and L. Euler analyzed
torsion, C. F. Gauss proved profound theorems, Georg Riemann
[1826‒1866] generalized Gauss’s results to any number of
dimensions, “Einstein needed a four-dimensional 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 [1835‒1900] 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
Erdös 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 Poincaré: “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).
rv: Rolf Riesinger (Wien)