https://orcid.org/0000-0003-2023-8529
This section will review those books whose content and level reflect the general editorial policy of Technometrics. Publishers should send books for review to Ejaz Ahmed, Department of Mathematics and Sciences, Brock University, St. Catharines, ON L2S 3A1 ([email protected]).
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Geometry in our Three-Dimensional World
Alfred S. Posamentier,
Bernd Thaller,
Christian Dorner,
Robert Geretschläger,
Guenter Maresch,
Christian Spreitzer, and
David StuhlpfarrerFirdous Ahmad Mala 127
Dark Data: Why What You Don’t Know Matters
David J. HandMohieddine Rahmouni 129
Vedic Mathematics: A Mathematical Tale from the Ancient Veda to Modern Times
Giuseppe Dattoli,
Silvia Licciardi, and
Marcello ArtioliFirdous Ahmad Mala 131
Foundations of Statistics for Data Scientists: With R and Python
Alan Agresti and Maria KateriRoger Sauter 132
Directional Statistics for Innovative Applications: A Bicentennial Tribute to Florence Nightingale
Ashis SenGupta and
Barry C. Arnold, eds.Shuangzhe Liu 133
Chance, Logic and Intuition: An Introduction to the Counter-Intuitive Logic of Chance
Steven TijmsStan Lipovetsky 134
Geometry has been a source of fascination since ancient times. It is hard to overemphasize the importance of geometry for human welfare. Plato recognized this long back as is evident from his famous proclamation concerning his academy: “Let no one ignorant of geometry enter here” (Anglin Citation1994).
Geometry is learned with sincere and strenuous efforts. It is so today. And it was so during the time of Euclid as is evident from Euclid’s famous repartee to Ptolemy: “There is no royal road to geometry” (Ball Citation1960).
This book is an outcome of the efforts of seven mathematicians and mathematics educators who have undertaken to collectively share their experiences and their expertise with a view to help and enlighten the readers about the various beautiful aspects of geometry.
Chapter 1 is about dimensions and spatial coordinates. Since our world is primarily a three-dimensional world, we are pervaded with a plethora of three-dimensional objects. As such, this introductory chapter introduces certain everyday observations that have to do with spatial geometry. The role of optical illusions in befooling our spatial perception is also brought to thefore.
Chapter 2 takes the readers from a basic understanding of spatial coordinates to that of spatial understanding, visual perception, and spatial skills that are gaining unprecedented importance in several professional areas. Making use of recent psychological research, certain exercises are presented that one can employ to train one’s spatial abilities.
Chapter 3 is devoted to the representation of three-dimensional objects in two dimensions. This is particularly helpful for its pedagogical implications. Teachers are often required to make use of drawings of three-dimensional objects on a plane board to enhance students’ understanding. A brief history of the art and science of perspective drawing has been traced. The creation of perceptivity for bringing about the illusion of space and distance has also been discussed. This has naturally been extended to concepts of camera obscura (originally a darkened room containing a tiny hole or lens for projecting an image onto a wall) and photography. Camera obscura has long been used for producing geometrically correct constructions (Gutruf and Stachel Citation2010).
Chapter 4 brings to the fore some other forms of geometry such as spherical geometry. How the sum of the angles of a triangle in spherical geometry could range from 180 degrees to 540 degrees could come to many as a surprise. Mapping a spherical surface onto a plane is a very useful concept and its use by cartographers to produce Earth’s maps typically seen in an atlas is a standout of this chapter. Special attention has been paid to the Mercator projection that is used nowadays in satellite navigation. Mercator is claimed to have worked out his projection by the mere usage of a compass and a ruler (Pápay Citation2022). The chapter also discusses the various natural phenomena that convinced the ancient Greeks of the Earth’s spherical shape.
Optics is mostly regarded as a part of physics, earlier known as natural philosophy. However, given the amount of geometry in optics, it is no exaggeration to declare optics as another branch of geometry. Chapter 5 discusses more of optics than anything else. It discusses the shape of the Earth and attempts to explain the reasons why the Earth is not flat. It also talks about the formation of days and nights with a special emphasis on the celestial sphere model. Very interestingly, the chapter contains an account of a formula, involving trigonometric functions, for the length of a day. It also clarifies why there are seasons, phases of the Moon, and eclipses. A formula for the size of the Moon’s shadow is also an engaging part of the chapter.
Chapter 6 is a joy to read and discusses, among other things, Platonic and Archimedean solids. These solids are polyhedral and are the natural spatial equivalents of plane objects such as triangles, squares, and other polygons. The fact that there are only five Platonic solids can be easily established thanks to Euler’s characteristics formula. Using the concept of duality, it is shown that a dodecahedron (with 20 faces and 12 vertices) is the dual of an icosahedron (with 12 faces and 20 vertices). A hexahedron (aka a cube) is shown as the dual of an octahedron. A tetrahedron, however, is self-dual and is without a Platonic partner. Somewhat less popular but equally interesting Kepler-Poinsot solids have also been discussed in considerable detail.
A large number of sports make use of one or the other kinds of balls. Chapter 7 is about the shapes of these balls. It discusses four of the most popular sports balls—the soccer ball, the beach ball, the basketball, and the volleyball. These balls were derived from certain specific polyhedra. A soccer ball is a truncated icosahedron, a beach ball is a hexagonal prism, a basketball is an analogue of a regular octahedron and a volleyball is a reflection of the properties of a cube.
Chapter 8 is about unity in diversity. It is about Voronoi diagrams. As per its mathematical definition, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects (Aurenhammer and Klein Citation2000). Examples include corn on cobs, foam bubbles, heads of garlic, giraffe’s skin, and honeycombs. It is surprising how Voronoi diagrams have found their applications in economics, public health, art and architecture.
These concepts are further manifested in Chapter 9 that is primarily concerned with packing. Crystallographers and geometers have common concerns. Symmetry is one of them (Mala Citation2022a). Packing is another. Packing is the issue of optimization vis-à-vis space-saving.
Topology is a branch of mathematics in which shapes obtained from each other by compression or stretching, or more technically by continuous deformations, are considered equivalent. For a topologist, a donut is not different from a coffee cup. One-sided surfaces such as the Mobius strip and nonorientable surfaces such as Klein’s bottle are some of the biggest surprises geometry offers us. Such is the subject matter of Chapter 10.
Paper-folding, aka origami, is the folding of paper to give rise to three-dimensional shapes. Tessellation is the covering of a surface or plane by tiles with no overlaps. Origami and tessellation fall into the domains of both mathematics and art. Chapter 11 is about these aspects of geometry.
Knot theory, the study of closed curves in three dimensions and their possible deformations without one part cutting through the other, is a branch of geometry that has remained a hot debate in recent times. It so happened that U.S. congressmen denounced the waste of public money on this theory. They believed that spending on it was simply a waste. Little did they know that this area of research in geometry holds a central position in low-dimensional topology and has applications to DNA and quantum mechanics (Stewart Citation2021). A ring is an example of a knot. Chapter 12 discusses the drawing and the classification of knots. Finding out whether two descriptions represent a single knot is a central concern of this chapter.
What do mechanical machines have to do with geometry? Everything. The fact remains that a profound knowledge of mathematics, in general, and geometry, in particular, reveals some of the most mysterious connections that exist between the abstraction of mathematics and the concreteness of the real world. Chapter 13 brings some of such surprising connections to the front. It discusses, besides other things, the roles that geometry plays in robotic arms (Clothier and Shang Citation2010), and a Stewart platform (Ronga and Vust Citation1992) which is a kind of parallel manipulator with six prismatic actuators commonly applied in flight simulators, crane technology, robotics, and orthopedic surgery.
Modeling is becoming increasingly popular and attempts to assign numbers to everyday happenings (Mala Citation2022b). The ultimate chapter of the book presents the various mathematical principles that underlie computer-aided designs. The chapter discusses the creation and the realization (using three-dimensional printers) of virtual models.
Given the interplay and effectiveness of mathematics in our lives (Mala Citation2022c, Citation2022d), one often feels an urge to revise, reread and restudy all the mathematics one has missed. It is exactly when books such as this one (Mala Citation2022e) can come to one’s rescue.
The book aims at raising awareness of our experiences with three-dimensional geometry.
Government Degree College Sopore, Baramulla,
Jammu and Kashmir, India
[email protected]
References
- Anglin, W. S. (1994), Mathematics: A Concise History and Philosophy, New York: Springer.
- Aurenhammer, F., and Klein, R. (2000), “Voronoi Diagrams,” in Handbook of Computational Geometry (Vol. 5), eds. J.-R. Sack and J. Urrutia, pp. 201–290, Amsterdam: North Holland.
- Ball, W. W. R. (1960), A Short Account of the History of Mathematics, New York: Courier Corporation.
- Clothier, K. E., and Shang, Y. (2010), “A Geometric Approach for Robotic Arm Kinematics with Hardware Design, Electrical Design, and Implementation,” Journal of Robotics, 2010, 984823.
- Gutruf, G., and Stachel, H. (2010), “The Hidden Geometry in Vermeer’s ‘The Art of Painting’,” Journal for Geometry and Graphics, 14, 187–202.
- Mala, F. A. (2022a), “A First Course in Group Theory, by Bijan Davvaz. Springer, 2021. Softcover, pp. xv+ 291. ISBN 978-981-16-6364-2,” Acta Crystallographica Section A: Foundations and Advances, 78, 1–2.
- Mala, F. A. (2022b), “Your Life in Numbers: Modeling Society Through Data by Pablo Jensen,” The Mathematical Intelligencer, 1–2.
- Mala, F. A. (2022c), “Philosophy of Mathematics: Classic and Contemporary Studies,” The Mathematical Intelligencer, 1–3.
- Mala, F. A. (2022d), “The Psychology of Mathematics: A Journey of Personal Mathematical Empowerment for Educators and Curious Minds by Anderson Norton,” World Literature Today, 96, 79–79.
- Mala, F. A. (2022e), “All the Math You’ll Ever Need: A Self-Teaching Guide (3rd Ed.),” Technometrics, 64, 579–580.
- Pápay, G. (2022), “Mercator’s Geometric Method in the Construction of His Projection from 1569,” KN-Journal of Cartography and Geographic Information, 1–7.
- Ronga, F., and Vust, T. (1992), “Stewart Platforms Without Computer,” in Proceedings of the International Conference on Real, Analytic and Algebraic Geometry, pp. 197–212.
- Stewart, I. (2021), What’s the Use? The Unreasonable Effectiveness of Mathematics, London: Profile Books.