You are here

Geometry: With Geometry Explorer

Michael Hvidsten
Publication Date: 
Number of Pages: 
Hardcover with CDROM
[Reviewed by
P. N. Ruane
, on

Given the number of high quality geometry texts published within the past ten years or so, one could be forgiven for assuming that the subject has undergone some sort of renaissance within mathematics education. In reality, however, geometry still has no coherent presence within school mathematics and it continues to play only a minor role within undergraduate mathematics.

So here is a book whose overall aim is to ‘develop an intuitive feel for geometry whilst also establishing sound understanding of underlying proofs and abstractions’. This, of course, is the common challenge faced by authors of university geometry texts, who necessarily have the two-fold task of laying foundations that should have been provided at school, whilst simultaneously transporting students into the realm of advanced geometric topics. But having taught geometry to undergraduate students, I long ago ceased to be amazed by their lack of basic geometric knowledge, and I have sometimes spent more time going backwards than forwards.

In addressing this dilemma, Michael Hvidsten has invoked the use of new technology in the form of Geometry Explorer (provided with the book). This, of course, is not the only way in which geometric intuition can be strengthened, and it can’t be expected to totally compensate for many previous years of geometric neglect. Nonetheless, although employed in a supplementary fashion, I very much like the use to which it is put. For example, in the very first exercise project, students use it to explore many properties of the Golden Rectangle. Much later on, it is used to investigate the tiling of the hyperbolic plane, and there are many other ‘projects’ that also involve the application of this software. In addition, students may be inclined to experiment with it of their own volition — and the more the better.

Anyway, the book begins with a stimulating chapter on axiomatic methods, with historical commentary running throughout. It considers the strengths and weaknesses of various sets of axioms including those of Euclid, Hilbert and Birkhoff. These are considered in the context of consistency, independence and completeness, followed by some explanation of Gödel’s Incompleteness theorem. Chapters 2 to 6 are respectively devoted to Euclidean Geometry, Analytic Geometry, Constructions, Transformation Geometry and Symmetry.

Chapters 7 and 8 provide a cogent introduction to Non-Euclidean Geometry and Non-Euclidean Transformations and I particularly liked the author’s treatment of all this material. Initial motivation comes from a re-examination of Euclid’s axioms and the history of the parallel postulate. Also, Project 2 (in the very first chapter) requires the use of Geometry Explorer to test the validity of Euclidean properties in hyperbolic geometry, thereby setting the scene for later major investigation of the Poincaré model. The Klein model is also introduced and shown to be structurally equivalent to that of Poincaré. Marvellous stuff!

The last chapter provides an appealing introduction to Fractal Geometry together with discussion of many relevant ideas such as contraction mappings, algorithmic geometry and space-filling curves. And Geometry Explorer is again brought into action via ‘Projects’ on Snowflake Curves, Complex Branching Systems and IFS Ferns.

Intended for use with mathematics students at ‘junior or senior collegiate level’, the book requires a background in geometry provided by elementary high school and some expertise with matrix algebra and groups is also recommended. Generally, I very much like this book but, for the following reasons, I have considerable reservations regarding its use with such a target group.

  • Treatment of much of the material, although soundly formulated, could be very demanding for those with weak geometric backgrounds (the majority of undergraduate students, of course).
  • There is a rather sudden transition from the synthetic/axiomatic approach of Euclid to the transformational approach of Klein. And, although isometries are applied to analysis of symmetry and tiling patterns etc, this algebraic machinery is not employed to reveal a range of geometric results akin to the theorems of Euclid. At a very basic level, for example, if two triangles have equal sides, one can find an isometry that maps one onto the other, thus linking the ideas of Euclidean and Kleinian congruence.
  • The exercises are predominantly of an investigative nature and many of them require reports to be written by students. This is all very well, except that students need considerable preparation to meet the demands of such a research-based approach to learning.

Of course, no review is complete without mention of one or two minor quibbles. Firstly, it was not the Arabs who introduced symbols such as x2, and I still shudder on seeing expressions like 1/0 = ∞. Finally, no history of vector analysis is complete without mention of Willard Gibbs!

However, if only for the chapters on axiomatics, non-Euclidean and fractal geometry, this book should be regarded as a very valuable addition to the existing literature. Moreover, it is suitable for use with a much wider readership than specified by the author, but it wouldn’t be my first choice as a self-tuition manual for those with weak geometric backgrounds.

Peter Ruane is retired from university teaching and now dabbles in as many creative diversions as possible.


1 Geometry and the Axiomatic Method

1.1 Early Origins of Geometry

1.2 Thales and Pythagoras

1.2.1 Thales
1.2.2 Pythagoras

1.3 Project 1 - The Ratio Made of Gold

1.3.1 Golden Section
1.3.2 Golden Rectangles

1.4 The Rise of the Axiomatic Method

1.5 Properties of Axiomatic Systems

1.5.1 Consistency
1.5.2 Independence
1.5.3 Completeness
1.5.4 Gödel's Incompleteness Theorem

1.6 Euclid's Axiomatic Geometry

1.6.1 Euclid's Postulates

1.7 Project 2 - A Concrete Axiomatic System

2 Euclidean Geometry

2.1 Angles, Lines, and Parallels

2.2 Congruent Triangles and Pasch's Axiom

2.3 Project 3 - Special Points of a Triangle

2.3.1 Circumcenter
2.3.2 Orthocenter
2.3.3 Incenter

2.4 Measurement and Area in Eucliedean Geometry

2.4.1 Mini-Project - Area in Euclidean Geometry
2.4.2 Cevians and Areas

2.5 Similar Triangles

2.5.1 Mini-Project - Finding Heights

2.6 Circle Geometry

2.7 Project 4 - Circle Inversion and Orthogonality

2.7.1 Orthogonal Circles Redux

3 Analytic Geometry

3.1 The Cartesian Coordinate System

3.2 Vector Geometry

3.3 Project 5 - Bézier Curves

3.4 Angles in Coordinate Geometry

3.5 The Complex Plane

3.5.1 Polar Form
3.5.2 Complex Functions
3.5.3 Analytic Functions and Conformal Maps (Optional)

3.6 Birkhoff's Axiomatic System for Analytic Geometry

4 Constructions

4.1 Euclidean Constructions

4.2 Project 6 - Euclidean Eggs

4.3 Constructibility

4.4 Mini-Project - Origami Construction

5 Transformational Geometry

5.1 Euclidean Isometries

5.2 Reflections

5.2.1 Mini-Project - Isometries through Reflection
5.2.2 Reflection and Symmetry

5.3 Translations

5.3.1 Translational Symmetry

5.4 Rotations

5.4.1 Rotational Symmetry

5.5 Project 7 - Quilts and Transformations

5.6 Glide Reflections

5.6.1 Glide Reflection Symmetry

5.7 Structure and Representation of Isometries

5.7.1 Matrix Form of Isometries
5.7.2 Compositions of Rotations and Translations
5.7.3 Compositions of Reflections and Glide Reflections
5.7.4 Isometries in Computer Graphics
5.7.5 Summary of Isometry Compositions

5.8 Project 8 - Constructing Compositions

6 Symmetry

6.1 Finite Plane Symmetry Groups

6.2 Frieze Groups

6.3 Wallpaper Groups

6.4 Tiling the Plane

6.4.1 Escher
6.4.2 Regular Tessellations of the Plane

6.5 Project 9 - Constructing Tessellations

7 Non-Euclidean Geometry

7.1 Background and History

7.2 Models of Hyperbolic Geometry

7.2.1 Poincaré Model
7.2.2 Mini-Project - The Klein Model

7.3 Basic Results in Hyperbolic Geometry

7.3.1 Parallels in Hyperbolic Geometry
7.3.2 Omega Points and Triangles

7.4 Project 10 - The Saccheri Quadrilateral

7.5 Lambert Quadrilaterals and Triangles

7.5.1 Lambert Quadrilaterals
7.5.2 Triangles in Hyperbolic Geometry

7.6 Area in Hyperbolic Geometry

7.7 Project 11 - Tiling the Hyperbolic Plane

7.8 Models and Isomorphism

8 Non-Euclidean Transformations

8.1 Möbius Transformations

8.1.1 Fixed Points and the Cross Ratio
8.1.2 Geometric Properties of Möbius Transformations

8.2 Isometries in the Poincaré Model

8.3 Isometries in the Klein Model

8.4 Mini-Project - The Upper Half-Plane Model

8.5 Weierstrass Model

8.6 Hyperbolic Calculation

8.6.1 Arclength of Parameterized Curves
8.6.2 Geodesics
8.6.3 The Angle of Parallelism
8.6.4 Right Triangles
8.6.5 Area

8.7 Project 12 - Infinite Real Estate?

9 Fractal Geometry

9.1 The Search for a "Natural" Geometry

9.2 Self-Similarity

9.2.1 Sierpinski's Triangle
9.2.2 Cantor Set

9.3 Similarity Dimension

9.4 Project 13 - An Endlessly Beautiful Snowflake

9.5 Contraction Mappings and the Space of Fractals

9.6 Fractal Dimension

9.7 Project 14 - IFS Ferns

9.8 Algorithmic Geometry

9.8.1 Turtle Geometry

9.9 Grammars and Productions

9.9.1 Space-filling Curves

9.10 Project 15 - Words into Plants: The Geometry of Life

A Book I of Euclid's Elements

A.1 Definitions

A.2 The Postulates (Axioms)

A.3 Common Notions

A.4 Propositions (Theorems)

B Brief Guide to Geometry Explorer

B.1 The Main Geometry Explorer Window

B.2 Selecting Objects

B.3 Active vs. Inactive Tools

B.4 Labels

B.5 Object Coloring

B.6 Online Help

B.7 Undo/Redo of Actions

B.8 Clearing and Resizing the Canvas

B.9 Saving Files as Images

B.10 Main Window Button Panels

B.10.1 Create Panel
B.10.2 Construct Panel
B.10.3 Transform Panel

B.11 Measurement in Geometry Explorer

B.11.1 Neutral Measurements
B.11.2 Euclidean-only Measurements
B.11.3 Hyperbolic-only Measurements
B.11.4 User Input Measurements

B.12 Using Tables

B.13 Using the Calculator

B.14 Hyperbolic Geometry

B.15 Analytic Geometry

B.16 Turtle Geometry

C Birkhoff's Axioms for Euclidean Geometry

D Hilbert's Axioms for Euclidean Geometry

E The 17 Wallpaper Groups