Notice: This material will be included in a forthcoming (summer 2000) book with the tentative title Experiencing Geometry in Two- and Three-Dimensional Spaces. This new book will be an expanded and updated version of Experiencing Geometry on Plane and Sphere. This material is in draft form and may not be duplicated or quoted without the author's written permission, except for purposes of review or trying out the material with students. As always comments are welcome and will affect the final draft. Send comments to dwh2@cornell.edu.

Chapter 1

Straightness and Symmetry

Straight is that of which the middle is in front of both extremities.

— Plato, Parmenides, 137 E [A: Plato]

A straight line is a line which lies evenly with the points on itself.

— Euclid, Elements, Definition 4 [A: Euclid]

Wisdom will save you from the ways of wicked men, from men whose words are perverse, who leave the straight paths to walk in the dark ways,...whose paths are crooked and who are devious in their ways.

— The Holy Bible, Proverbs 2:12-15 [A: Bible]

Verily, this is My Way,

Leading straight, follow it:

Follow not other paths:

They will scatter you about

From His great path:

— Holy Qur-An, Sura VI, verse 153 [A: Koran]

In keeping with the spirit of the approach to geometry discussed in the Preface and Message to the Reader, we begin with a question that encourages you to explore deeply a concept that is fundamental to all that will follow. We ask you to build a notion of straightness for yourself at the beginning of the course rather than accept a certain number of assumptions about straightness. Though it is difficult to formalize, straightness is a natural human concept.

Problem 1. When Do You Call a Line Straight?

Look to your experiences. It might help to think about how you would explain straightness to a 5-year-old (or how the 5-year-old might explain it to you!). If you use a "ruler," how do you know if the ruler is straight? How can you check it? What properties do straight lines have that distinguish them from non-straight lines?

Think about the question in four related ways:

1. How can you check in a practical way if something is straight — without assuming that you have a ruler, for then we will ask, "How can you check that the ruler is straight?"

2. How do you construct something straight — lay out fence posts in a straight line, or draw a straight line?

3. What symmetries does a straight line have?

4. Can you write a definition of "straight line"?

Suggestions

Look at your experience. At first, you will look for examples of physical world or natural straightness. Go out and actually try walking along a straight line and then along a curved path; try drawing a straight line and checking that a line already drawn is straight.

Look for things that you call "straight." Where do you see straight lines? Why do you say they are straight? Look for both physical lines and nonphysical uses of the word "straight."

You are likely to bring up many ideas of straightness. It is necessary then to think about what is common among all of these straight phenomena.

As you look for properties of straight lines that distinguish them from non-straight lines, you will probably remember the following statement (which is often taken as a definition in high school geometry): A line is the shortest distance between two points. But can you ever measure the lengths of all the paths between two points? How do you find the shortest path? If the shortest path between two points is in fact a straight line, then is the converse true? Is a straight line between two points always the shortest path? We will return to these questions in later chapters.

A powerful approach to this problem is to think about lines in terms of symmetry. This will become increasingly important as we go on to other surfaces (spheres, cones, cylinders, etc.) Two symmetries of lines are:

• reflection symmetry in the line, also called bilateral symmetry — reflecting (or mirroring) an object over the line.

Figure 1.1. Reflection symmetry of a straight line.

• Half-turn symmetry — rotating 180° about any point on the line.

Figure 1.2. Half-turn symmetry of a straight line.

Although we are focusing on a symmetry of the line in each of these examples, notice that the symmetry is not a property of the line by itself but includes the line and the space around the line. The symmetries preserve the local environment of the line. Notice how in reflection and half-turn symmetry the line and its local environment are both part of the symmetry action and how the relationship between them is integral to the action. In fact, reflection in the line does not move the line at all but exhibits a way in which the space on the two sides of the line are the same.

Definitions. An isometry is a transformation of a region of space that preserves distances and angle measures. A symmetry of a figure is an isometry of a region of space that takes part of the figure in the region onto (possibly another part of) itself. You will show in Problem 18.3 that every isometry is either a translation, a rotation, a reflection, or a composition of them.

Try to think of other symmetries as well (there are quite a few). Some symmetries hold only for straight lines, while some work with other curves too. Try to determine which ones are specific to straight lines and why. Also think of practical applications of these symmetries for constructing a straight line or for determining if a line is straight.

How Do You Construct a Straight Line?

As for how to construct a straight line, one method is simply to fold a piece of paper; the edges of the paper needn't even be straight. This utilizes symmetry (can you see which one?) to produce the straight line. Carpenters also use symmetry to determine straightness — they put two boards face to face, plane the edges until they look straight, and then turn one board over so the planed edges are touching. See Figure 1.3. They then hold the boards up to the light. If the edges are not straight, there will be gaps between the boards which light will shine through.

Figure 1.3. Carpenter's method for checking straightness.

When grinding an extremely accurate flat mirror, the following technique is sometimes used: Take three approximately flat pieces of glass and put pumice between the first and second pieces and grind them together. Then do the same for the second and the third pieces and then for the third and first pieces. Repeat many times and all three pieces of glass will become very accurately flat. See Figure 1.4. Do you see why this works? What does this have to do with straightness?

Figure 1.4. Grinding flat mirrors.

Imagine (or actually do it!) walking while pulling a long silk thread with a small stone attached. When will the stone follow along your path? Why? This property is used to pick up a fallen water skier. The boat travels by the skier along a straight line and thus the tow rope follows the path of the boat. Then the boat turns in an arc in front of the skier. Since the boat is no longer following a straight path, the tow rope will move in toward the fallen skier.

Another idea to keep in mind is that straightness must be thought of as a local property. Part of a line can be straight even though the whole line may not be. For example, if we agree that this line is straight,

and then we add a squiggly part on the end, like this:

then would we now say that the original part of the line is not straight, even though it hasn't changed, only been added to? Also note that we are not making any distinction here between "line" and "line segment." The more generic term "line" generally works well to refer to any and all lines and line segments, both straight and non-straight.

Now look at symmetries. What symmetries does a straight line have? How do they fit with the examples that you have found and those mentioned above? Can we use any of the symmetries of a line to define straightness?

Think about and formulate some answers for these questions before you read any further. You are the one laying down the definitions. Do not take anything for granted unless you see why it is true. No answers are predetermined. You may come up with something that we have never imagined. Consequently, it is important that you persist in following your own ideas. Reread the section "How to Use This Book" on pages xviii and xix.

This icon will be used throughout the book to indicate places where you should pause and not read further until you have expressed your thinking and ideas through writing or through talking to someone else.

The Symmetries of a Line

Reflection-in-the-line symmetry: It is most useful to think of reflection as a "mirror" action with the line as an axis rather than as a "flip-over" action which involves an action in 3-space. In this way one can extend the notion of reflection symmetry to a sphere (the flip-over action is not possible on a sphere). Notice that this symmetry cannot be used as a definition for straightness because we use straightness to define reflection symmetry — the definition would be circular.

Figure 1.5. Reflection-in-the-line symmetry.

• Practical application: One can produce a straight line by folding a piece of paper because this action forces reflection-in-itself symmetry along the crease. Above we showed a carpenter's example.
  

Figure 1.6. Reflection-perpendicular-to-the-line symmetry.

• Practical applications: You call tell if a straight segment is perpendicular to a mirror by seeing if it looks straight with its reflection. Also, a straight line can be folded onto itself.
  

Figure 1.7. Half-turn symmetry.

• Practical applications: This is the principle behind hinges. If this is applied in 3-space, then you get a universal joint (a hinge that can fold in any direction).
  

Figure 1.8. Rigid-motion-along-itself symmetry.

• Practical applications: Slide joints such as in trombones, drawers, nuts and bolts, pulleys, etc. all utilize this symmetry.
  

Figure 1.9. 3-dimensional rotation symmetry.

• Practical applications: This symmetry can be used to check the straightness of any long thin object such as a stick by twirling the stick with itself as the axis. If the stick does not appear to wobble, then it is straight. This is used for pool cues, axles, etc.
  

Figure 1.10. Central symmetry.

• In 3-space, central symmetry produces a result different than any single rotation or reflection (though one can check that it does give the same result as the composition of three reflections through mutually perpendicular planes). To experience central symmetry in 3-space, hold your hands in front of you with the palms facing each other and your left thumb up and your right thumb down. Your two hands now have approximate central symmetry about a point midway between the center of the palms.
  

Figure 1.11. Similarity symmetry.

• Logarithmic spirals like the chambered nautilus have self-similarity as do many fractals. (See example in Figure 1.12.)

Figure 1.12. Logarithmic spiral.

Clearly, other objects besides lines have some of the symmetries mentioned here. It is important for you to construct your own such examples and for you to attempt to find an object that has all of the symmetries but is not a line. This will help you to experience that straightness and the seven symmetries discussed here are intimately related. You should come to the conclusion that while other curves and figures have some of these symmetries, only the straight line has all of them.

Returning to one of the original questions, how would we construct a straight line? One way would be to use a "straight edge" — something that we accept as straight. Notice that this is different from the way that we would draw a circle. When using a compass to draw a circle, we are not starting with a figure that we accept as circular; instead, we are using a fundamental property of circles that the points on a circle are a fixed distance from the center. Can we use the symmetry properties of a straight line to construct a straight line? Remember the examples earlier in this chapter. Is there a tool (serving the role of a compass) which will draw a straight line? For an interesting discussion of this question see How to Draw a Straight Line: A Lecture on Linkages by A.B. Kempe [Z: Kempe] which shows the following apparatus:

Figure 1.13. Apparatus for drawing a straight line.

The links labeled with the same letter must have the same length. The fact that this apparatus draws a straight line is the subject of Problem 3.3b. See [SE: Hilbert, pp. 272-3] for another discussion of this topic. The discovery of this linkage about 1870 is variously attributed to the French army officer, Charles-Nicolas Peaucellier (1832-1913), and to Lippman Lipkin, who lived in Lithuania and studied in Saint Petersburg. (See Kempe and Hilbert and Phillip Davis' delightful little book The Thread [Z: Davis], Chapter IV.)

Local (and Infinitesimal) Straightness

Previously, you saw how a straight line has reflection-in-the-line symmetry and half-turn symmetry: One side of the line is the same as the other. But, as pointed out above, straightness is a local property in that whether a segment of a line is straight depends only on what is near the segment and does not depend on anything happening away from the line. Thus each of the symmetries must be able to be thought of (and experienced) as applying only locally. This will become particularly important later when we investigate straightness on the cone and cylinder. (See the discussions in Chapter 4.) For now, it can be experienced in the following way:

When a piece of paper is folded not in the center, the crease is still straight even though the two sides of the crease on the paper are not the same. So what is the role of the sides when we are checking for straightness using reflection symmetry? Think about what is important near the crease in order to have reflection symmetry.

Figure 1.14. Reflection symmetry is local.

When we talk about straightness as a local property, you may bring out some notions of scale. For example, if one sees only a small portion of a very large circle, it will be indistinguishable from a straight line. This can be experienced easily on many of the modern graphing programs for computers. Also a microscope with a zoom lens will provide an experience of zooming. If a curve is smooth (or differentiable), then if one "zooms in" on any point of the curve, eventually the curve will be indistinguishable from a straight line segment. This property is called infinitesimally straight, or in more standard terminology, differentiable. We say that a curve is infinitesimally straight at a point p if there is a straight line l such that if we zoom in enough on p, the line and the curve become indistinguishable.¹ See Figure 1.15. When the curve is parametrized by arc length it is equivalent to the curve having a well-defined velocity vector at each point.

Figure 1.15. Infinitesimally straight.

In contrast, we can say that a curve is locally straight at a point if that point has a neighborhood that is straight. In the physical world the usual use of both smooth and locally straight are dependent on the scale at which they are viewed. For example, we may look at an arch made out of wood that at a distance appears as a smooth curve; then as we move in closer we see that the curve is made by many short straight pieces of finished (planed) boards, but when we are close enough to touch it, we see that its surface is made up of smooth waves or ripples, and under a microscope we see the non-smoothness of numerous twisting fibers. See Figure 1.16.

Figure 1.16. Straightness and smoothness depend on the scale.

f(x) - t(x) = f(x) - f(p) - f ¢(p)(x-p) = { [f(x) - f(p)]/(x-p) - f ¢(p) }(x-p) < e.

In general, the value of d might depend on p as well as on e. Often the term smooth is used to mean continuously differentiable which the interested reader can check is equivalent (on closed finite intervals) to, for each e > 0, there being one d > 0 that works for all p.