Geometry in Our Three-Dimensional World

The following sections are included:

• Preface
• About the Authors
• Contents

When we speak of geometry, we typically think of the geometric figures that we can draw on a piece of paper, like triangles, squares, and circles. This is two-dimensional geometry and has for a long-time dominated school mathematics, and indeed many were fascinated by the intriguing wonders revealed in the relationships of two-dimensional figures. However, we live in a world with three spatial dimensions — length, width, depth — that is even richer in intricate geometric facts and relations. Sometimes, in everyday life, we encounter phenomena that are hard to explain unless one has a good understanding of the three-dimensional nature of the world in which we live. One example, where one is easily fooled by a two-dimensional world view, is the following phenomenon, which we call the flight-path reality.

The talent of spatial understanding is a prerequisite for developing and understanding spatial geometry. It is, therefore, worthwhile to analyze in more detail this ability, which is partly innate and partly acquired. Extensive psychological research in recent years has shown that visual perception and spatial awareness consist of several components, each of which represents a very specialized skill. This chapter describes the components of spatial thinking in detail and illustrates them with a collection of sample problems that readers can use to test their own skills.

Now that we have seen how important spatial perception is for our lives and how the brain processes spatial sensations, we will ask how these spatial impressions can be conveyed from person to person. One then can ask how a spatial scene can be represented on a piece of paper so that the viewer can correctly interpret the scene. This question has played an important role in art history, since it has long been regarded as the task of a draftsman or painter to present on paper a real or imagined scene in a realistic way. This is by no means an easy task. After all, one has to convince the viewer to imagine a three-dimensional scene by looking at a two-dimensional flat piece of paper or canvas. An understanding of how this can be achieved has developed slowly and gradually over a period of several centuries.

The geometry that can be done on the surface of the sphere is often referred to as spherical geometry as opposed to the geometry in a plane, which is called plane geometry, or Euclidean geometry. Although there are some similarities between spherical and plane geometry, there are also important differences as we will see as we navigate this very interesting topic. Spherical geometry is a special case of Riemannian geometry, named after the German mathematician Bernhard Riemann (1826–1866) who departed from Euclidean geometry to develop a geometry of various surfaces in higher dimensions.

The following sections are included:

• Why the Earth is not Flat
• Day and Night
• A Formula for the Length of a Day
• Seasons
• Phases of the Moon
• Solar Eclipse
• Size of the Moon’s Shadow
• Annular Solar Eclipse
• Frequency of Eclipses
• Lunar Eclipses
• Summary

In our everyday world, we often encounter solid shapes which exhibit regularity or symmetry, a sample of which we show in Figure 6.1. The symmetry of shapes has typically generated some fascination. The most basic symmetric shapes are usually classified as cubes, prisms, pyramids, cylinders, cones, or spheres. Industrial design often uses these solid shapes as basic elements. Many buildings have the form of a rectangular solid, but there are also buildings today in the shape of cylinders, pyramids and other unusual solids…

The following sections are included:

• Ball Games
• Platonic Solids and Archimedean Solids
• The Four Common Balls and Their Associated Solids
• The Soccer Ball
• The Beach Ball
• The Basketball
• The Volleyball

What is mathematics? One common answer has been: Mathematics is the science of patterns. This appropriate description stems from the English mathematician G.H. Hardy (1877–1947), one of the last century’s finest thinkers. However, the world is full of patterns, though, only very few observers recognize the mathematics involved while strolling around. Throughout this chapter, we will take a journey to discover beautiful patterns and reveal the underlying mathematics.

Modern container ships are giants of the sea, approximately 1,300 ft (400m) long and 200ft (60m) wide and equipped with diesel engines producing 100,000 kW of power. Container shipping is the basis for world-wide trade and a lot of mathematics is involved in the transport logistics as well as in the basic problem of finding the best way of packing objects into a container in order to maximize its full capacity. After a brief historical account of the origin of the standardized intermodal shipping container, this chapter presents various packing problems in three-dimensional geometry, leading to interesting mathematical questions such as Kepler’s conjecture for the optimal packing of equal spheres and the related “kissing problem” of finding the maximum number of equal spheres that can be arranged in such a way that they touch (“kiss”) a common center sphere of the same size. We will also discuss three-dimensional packing puzzles as well as certain aspects of real-world packing problems as they appear in the loading of shipping containers.

Mathematical topology deals with those properties of shapes and surfaces that are not changed under continuous deformations. One of these properties is one-sidedness, and it is shared by a class of surfaces in three dimensions that paradoxically appear to have only one side. The best-known example is the Möbius strip, which can easily be made from a simple strip of paper.

In our intrinsically three-dimensional world, there is nothing that better approximates a two-dimensional sub-space than an ordinary sheet of paper. For all practical intents and purposes, paper has width and height, but not depth; the thickness of a sheet of paper is a measure we generally ignore…

Knot theory is the study of knots from a mathematical perspective. Obviously, tying a knot is only possible in three dimensions, a knot cannot be formed in the plane. Thus, it is not surprising that the various knots that can be created and their geometric properties are intimately related to properties of three-dimensional space itself. In this chapter, we will first consider practical knots, some of which you may be familiar with. We will then explain the notion of a mathematical knot and briefly review the history of knot theory. As we go along, all mathematical knots with up to six crossings will be presented. The question of how to determine whether two knots are equivalent will lead us to the so-called Reidemeister moves, which is a method to simplify knot diagrams, and to knot invariants such as tricolorability.

We are so accustomed to living in a three-dimensional world that we do not generally give any thought to the more abstract aspects of moving around in our surroundings. We simply have the feeling of being able to enjoy unencumbered movement in every imaginable direction (ignoring the effects of walls and gravity for the moment), and this level of awareness is quite sufficient as far as it concerns our own mobility…

One of the basic problems of computer graphics is the representation of three-dimensional shapes. These virtual shapes are then assembled into complex photo-realistic scenes that are usually referred to as Computer Generated Imagery (CGI). CGI has become an important industry through the interest of large film studios in creating artificial scenes, characters, buildings, and landscapes. Realistic spaceships, futuristic cities and scary monsters have become a familiar sight in the movies.

Four-dimensional “hyperspace” is a well-known motif in science-fiction literature. The general idea is that one could take a shortcut through hyperspace to quickly cover large distances in normal space, and thus, be able to move faster than light. As exotic and esoteric as the idea of hyperspace may seem, higher-dimensional spaces are nothing special in mathematics, where they even represent a very common topic. Multi-dimensional or even infinite-dimensional spaces are taught at an early stage in advanced mathematics courses at universities. The reason is that even in the description of very simple, everyday systems, more than three dimensions are needed to capture the variety of possible behaviors.

The following section is included:

• Index