Transformational plane geometry (Q2928599)

This book is designed for a one semester course at the junior undergraduate level and turns especially to future educators in the USA. Before the authors address the reader in the Preface, surprisingly in the Foreword two graduate students in mathematics education present evidence that the content of the book relates directly to two national (US) standards for mathematics; furthermore, the Foreword contains valuable tips for prospective teachers anywhere. More than 100 very aesthetical figures catch the reader's eyes even when only browsing through the book. ``This text provides a concrete visual alternative to Euclid's purely axiomatic approach to plane geometry. Indeed, as indicated in the National Council of Teachers of Mathematics: Principles and Standards of School Mathematics\dots the transformational approach is the methodology of choice for teaching geometry in the secondary schools:\dots.'' Each new concept is introduced synthetically and analytically, and often consolidated by examples. For the plane Euclidean geometry the authors use 5 undefined terms and 10 postulates, that are more than necessary, in order to begin transformational geometry as soon as possible. Some of the chapters contain ``exploratory activities'' in order to motivate the subsequent ideas; hereto the authors use a software which performs geometrical constructions such as \texttt{Geometer's sketchpad}. The arrangement and clarity of the text meets the most demanding pedagogical and mathematical requirements. Highlights of the book are the classification of isometries and similarities of the Euclidean plane. Although the text offers a wonderful first step into transformational plane geometry, the title promises too much: affine and projective transformations of the plane are not even mentioned. In particular the reviewer appreciated the elaborations in Chapter 7 that are accompanied by numerous well-chosen figures.NEWLINENEWLINEThe text is divided into 8 chapters, each begins with an overview and quotations from famous mathematicians and educators, or from the National Council of Teachers of Mathematics, or from the Common Core State Standards of Mathematics. Each subchapter ends with exercises, an Appendix provides hints and answers to selected exercises.NEWLINENEWLINEThe Foreword (written by Maria Christina Bucur and Lindsay Eisenhut) presents transformational geometry for future teachers (the whys, the dos, the don'ts); the relationship between studying and teaching transformational geometry; instructional methodologies and strategies; the methods ``let students build the problem'', ``incorporate student activities into lesson plans'', ``take advantage of all available resources''; pitfalls and common problems faced by new teachers (don't teach too much in one day, don't get sidetracked by teaching how to use technology) conclusions; and references.NEWLINENEWLINEThe Preface contains among others also a very brief historical sketch of geometry: Euclid, René Descartes, Felix Klein (Erlangen Program).NEWLINENEWLINEIn Chapter 1 (Axioms of Euclidean plane geometry), the 5 undefined terms are presented, that are point, line, distance, half plane, and angle measurement. Section 1.1 (Existence and incidence postulates) treats plane and parallel, unique intersection point of non-parallel lines, 1.2 (Distance and ruler postulates) injectivity, surjectivity, and bijectivity; properties of distance function (collinear, non-collinear, between, segment, and congruent); betweenness theorem for points, existence and uniqueness of midpoints, 1.3 (Plane separation postulate) convexity, same side and opposite side of a line, vertex, sides and the interior of an angle, triangles; Pasch's axiom as theorem, 1.4 (Protractor postulate) the angle measure postulate, the angle construction postulate, the angle addition postulate, the concepts congruent angles, right, acute, and obtuse angle; the continuity postulate with the concept supplementary angles; the supplement postulate including the concepts angle bisector, its existence and uniqueness as theorem, the concepts perpendicular and perpendicular bisector of two points, its existence and uniqueness as theorem, and concepts vertical angles, vertical angles theorem, and Section 1.5 (SAS postulate and Euclidean parallel postulate) congruent triangles and transversal.NEWLINENEWLINEIn Chapter 2 (Theorems of Euclidean plane geometry), the authors establish essential theorems of Euclidean plane geometry. In Section 2.1 (Exterior angle theorem), existence and uniqueness of perpendiculars, in 2.2 (Triangle congruence theorems), ASA, the isosceles triangle theorem, the scalene inequality, the triangle inequality, the concepts right triangle, hypotenuse and legs; pointwise characterizations of perpendicular bisector and angle bisector, in 2.3 (Alternate interior angle theorem and angle sum theorem), simple and crossed quadrilaterals, their vertices, sides and diagonals, parallelograms, rhombuses, theorems on properties of parallelograms and transitivity of parallelism, and in 2.4 (Similar triangles), the AA-theorem, the similar triangles theorem, and SAS for similar triangles are considered.NEWLINENEWLINEChapter 3 (Introduction to transformations, isometries, and similarities) contains the sections 3.1 (Transformation) treating the concepts transformation of \({\mathbb R}^2\), identity, and inverse, 3.2 (Isometries and similarities) treating the concepts measure of the directed angle, congruent modulo, angle sum, circle, radius and diameter, concurrent lines and circles, collineation of \({\mathbb R}^2\), similarity of ration \(r\) and showing that similarities are bijective, and 3.3 Appendix with a proof that isometries are surjective.NEWLINENEWLINEChapter 4 (Translations, rotations, and reflections) considers in Section 4.1 (Translations) the concepts vector, component, zero vector, position vector, initial and terminal point, and standard position. Exploratory Activity 1 (The action of a translation) uses the concepts translation by vector \({v}\), scalar product, vector sum and vector difference, linear transformation (homogeneous and additive), that translations are isometries, the concept dilatation, and the theorem which states that translations are dilatations. Exploratory Activity 2 (The action of a rotation), in Section 4.2 (Rotations) deals with rotation about \(C\), center and angle of a rotation, the theorem that rotations are isometries, equations for a rotation about the origin and about a general point (concept: conjugation), the fact that a non-trivial rotation is not a translation, halfturns about a center, and involution, and the propositions: a halfturn is both an involution and a dilatation, and involutory rotations are halfturns. Section 4.3 (Reflections) contains Exploratory Activity 3 (The action of a reflection) using reflection in a line, the equation of a reflection, the theorems: reflections are isometries, and involutions, the set of translations, the set of non-trivial rotations and the set of all reflections are mutually disjoint sets. Trisecting a general angle with straight edge and compass, considered in this section, is a classical unsolvable problem. Interestingly, this problem has a solution when the straight edge and compass are replaced with a reflecting instrument such as a MIRA. A MIRA is a small plastic device used to help students understand the concept ``reflection''. An algorithm for angle trisection with a MIRA is presented. 4.4 Appendix contains Geometer's sketchpad commands required by the exploratory activities.NEWLINENEWLINEChapter 5 (Compositions of translations, rotations, and reflections) presents in Section 5.1 (The three points theorem) that an isometry with distinct fixed points \(P\) and \(Q\) fixes the line \(PQ\) pointwise, an isometry with three non-collinear fixed points is the identity, and two isometries that agree on three non-collinear points are equal. Section 5.2 (Rotations as compositions of two reflections) contains Exploratory Activity 4 (Rotations as compositions of two reflections) with the theorems: every rotation is a composition of two reflections, and a non-trivial rotation that fixes a line is a halfturn, 5.3. (Translations as compositions of two halfturns or two reflections) contains Exploratory Activity 5 (Translations as compositions of two reflections) with the theorems: the composition of two halfturns is a translation, an isometry \(\alpha\) is a translation if and only if \(\alpha\) is a composition of two reflections in parallel lines, and a composition of two reflections is a translation or a rotation, and Section 5.4 (The angle addition theorem) contains Exploratory Activity 6 (Composing rotations whose rotation angle sum is not a multiple of \(360^{\circ}\)). Here, let \(\theta,\phi\in{\mathbb R}\): (i) If \(\theta+\phi\not\in\,0^{\circ}\), a rotation of \(\theta^{\circ}\) followed by a rotation of \(\phi^{\circ}\) is a rotation of \((\theta+\phi)^{\circ}\); (ii) If \(\theta+\phi\in\,0^{\circ}\), a rotation of \(\theta^{\circ}\) followed by a rotation of \(\phi^{\circ}\) is a translation; (iii) A non-trivial rotation of \(\theta^{\circ}\) followed by a translation is a rotation of \(\theta^{\circ}\); (iv) A translation followed by a non-trivial rotation of \(\theta^{\circ}\) is a rotation of \(\theta^{\circ}\); and (v) A translation followed by a translation is a translation. Sectiopn 5.5 (Glide reflections) shows that the composition of three reflections in concurrent or mutually parallel lines is a reflection in some unique line concurrent or mutually parallel to them, using the concepts glide reflection with axis \(c\) and glide vector \(v\). Further, properties and equations of a glide reflection are considered.NEWLINENEWLINEChapter 6 (Classification of isometries) has the following Sections. Section 6.1 (The fundamental theorem and congruence) shows that an isometry that fixes two distinct points is either a reflection or the identity,NEWLINE an isometry that fixes exactly one point is a non-trivial rotation, an isometry with a fixed point is either a rotation or a reflection. Furhter, the fundamental theorem of transformational plane geometry is presented: A transformation \(\alpha\) is an isometry if and only if \(\alpha\) can be expressed as composition of three or fewer reflections. Congruent plane figures are considered. In 6.2 (Classification of isometries), it is shown that an isometry is exactly one of the following types: a reflection, a glide reflection, a rotation, or a non-trivial translation. A composition of four reflections reduces to a composition of two reflections. The concepts even and odd isometry are used here. An isometry \(\alpha\) is even if and only if \(\alpha\) is a translation or a rotation (the identity is a trivial rotation); \(\alpha\) is odd if and only if \(\alpha\) is a reflection or a glide reflection; an even involutory isometry is a halfturn, an odd involutory isometry is a reflection. 6.3 (Orientation and the isometry recognition problem) deals with positive and negative orientation, orientation-preserving and orientation-reversing transformation, the theorem that an isometry \(\alpha\) is orientation-preserving if and only if \(\alpha\) is even, and orientation-reversing if and only if \(\alpha\) is odd. The concepts direct and opposite isometries, the theorem stating that translations and rotations are even, direct, and orientation-preserving, reflections and glide reflections are odd, opposite, and orientation-reversing are also presented. 6.4 (The geometry of conjugation) treats conjugate isometries, algebraic and geometrical properties of conjugation, and the theorem: Let \(\alpha_1\) and \(\alpha_2\) be conjugate isometries, then \(\alpha_1\) and \(\alpha_2\) are of the same isometry type. Conditions that two rotations or two reflections commute are given.NEWLINENEWLINEChapter 7 (Symmetry of plane figures): ``Felix Klein (1849--1925), father of transformational geometry: Geometry is the study of those properties of a set \(S\) that remain invariant when the elements of \(S\) are subjected to the transformations of some transformation group.'' Plane figures with finitely generated symmetry groups fall into one of five general classes: (1) asymmetrical figures (2) figures with bilateral symmetry (3) rosettes with exactly one point of symmetry (the center of a non-trivial rotational symmetry) (4) frieze patterns with translational symmetries in one direction, and (5) wallpaper patterns with translational symmetries in independent directions. Section 7.1 (Groups of isometries) treats the concepts group, abelian, subgroup, generator, cyclic, cyclic of order \(n\), infinite cyclic. It is shown that all translations form an abelian group, and all rotations about the origin form an abelian group. Every finite group \(G\) of isometries has the following properties: (A) the elements of \(G\) are rotations or reflections; (B) all non-trivial rotations of \(G\) have the same center; (C) the set of rotations of \(G\) form a cyclic subgroup; (D) if \(G\) has reflections, the number of reflections equals the number of rotations. 7.2 (Symmetry type) deals with point symmetry, line symmetry, and bilateral symmetry. The set of all symmetries of a plane figure \(F\) is a group, called the symmetry group of \(F\). Examples are the dihedral group \(D_n\) and the cyclic group \(C_n\). Further the concepts generating set of a subgroup, finitely generated, homomorphism, automorphism, inner and outer automorphism, inner isomorphism, and symmetry type are treated. 7.3 (Rosettes) considers rosette groups. A rosette group \(G\) has the following properties: (A) \(G\) contains only rotations and reflections; (B) the set of rotations in \(G\) forms a finite cyclic subgroup; and (C) \(G\) contains at most finitely many reflections. A theorem of Leonardo da Vinci (1452--1519) shows that every finite group of isometries is either \(C_n\) or \(D_n\) for some \(n\geq\,1\). 7.4 (Frieze patterns) deals with frieze groups. There are exactly seven distinct types of frieze patterns. The list of the seven frieze groups and a recognition flowchart for frieze patterns are given. In Section 7.5 (Wallpaper patterns), the authors present the concepts wallpaper group, unit cell, square, or rectangle, or rhombic unit cell, translation lattice and a criterium for a wallpaper pattern. The concepts used here are \(n\)-center of a wallpaper pattern and cristallographic restriction. If \(P\) is an \(n\)-center of a wallpaper pattern, then \(n\in\{2,3,4,6\}\). A wallpaper pattern with a \(4\)-center has no \(3\)- or \(6\)-centers. There are exactly seventeen distinct symmetry types of wallpaper patterns. A list of the seventeen wallpaper groups and a recognition flowchart for wallpaper patterns is given. Furthermore, edge tessellation is considered. A polygon generating an edge tessellation is one of the following eight types: a rectangle, an equilateral triangle, a \(60\)-right triangle, an isosceles right triangle, a \(120\)-isosceles triangle, a \(120\)-rhombus, a \(60\)-\(90\)-\(120\) kite, or a regular hexagon.NEWLINENEWLINEThe section of Chapter 8 (Similarity) have the following content. Section 8.1 (Plane similarities) treats similarity of ration \(r>0\), the three points theorem for similarities, and the concept stretch about \(C\) of ration \(r>0\), and Section 8.2 (Classification of dilatations) treats dilation about \(C\), the center of dilation, an the theorem showing that a dilatation is a translation, a halfturn or a dilation. In 8.3 (Classification of similarities and similarity recognition problem) similar plane figures, stretch rotation, stretch reflection, and the theorem showing that every non-isometric similarity has a fixed point and the theorem stating that a similarity is exactly one of the following: an isometry, a stretch, a stretch rotation, or a stretch reflection are considered. Further, the construction of a non-isometric similarity \(\alpha\) being no dilation and the construction of the fixed point of \(\alpha\) are presented. Equations of direct and opposite similarities are given. Section 8.4 (Conjugation and similarity symmetry) shows that conjugation preserves the similarity type. The concept internal direct product of two subgroups is used . Some facts on similarity symmetries are given.