## Wheels

The miraculous paradox of smooth round objects conquering space by simply tumbling over and over, instead of laboriously lifting heavy limbs in order to progress, must have given young mankind a most salutary shock.

—Vladimir Nabokov, Speak, Memory

Things would be very different without the wheel. Transportation aside, if we consider wheels as simple machines—pulleys, gears, gyroscopes and so on—it is hard to imagine any advanced society without them. H. G. Wells, in The War of the Worlds, describes a Martian civilization far ahead of ours but using no wheels in its intricate machinery. Wells may have intended this to be a put-on; one can easily understand how the American Indian could have missed discovering the wheel, but a society capable of sending spaceships from Mars to the earth?

Until recently the wheel was believed to have originated in Mesopotamia. Pictures of wheeled Mesopotamian carts date back to 3000 b.c. and actual remains of massive disk wheels have been unearthed that date back to 2700 b.c. Since World War II, however, Russian archaeologists have found pottery models of wheeled carts in the Caucasus that suggest the wheel may have originated in southern Russia even earlier than it did in Mesopotamia. There could have been two or more independent inventions. Or it may have spread by cultural diffusion as John Updike describes it in a stanza of his poem, Wheel:

The Eskimos had never heard

Of centripetal force when Byrd

Bicycled up onto a floe

And told them, "This how white man go."

It seems surprising that evolution never hit on the wheef-ás a means for making animals go, but on second thoughtprte realizes how difficult it would be for biological mectónisms to make wheeled feet rotate. Perhaps the tumbleWeed is the closes* nature ever came to wheeled transport. (On the other hand, it be Dutch artist Maurits C. Escher designed a creature capable of curling itself into a wheel and rolling along at high speeds. Who can be sure such creatures have not evolved on other planets?) There may also be submicroscopic swivel devices inside the cells of living bodies on the earth, designed to unwind and rewind double-helix strands of DNA, but their existence is still conjectural.

A rolling wheel has many paradoxical properties. It is easy to see that points near its top have a much faster ground speed than points near its bottom. Maximum speed is reached by a point on the rim when it is exactly at the top, minimum speed (zero) when the point touches the ground. On flanged train wheels whose rims extend slightly below a track, there is even a short segment in which a point on the rim moves backward. G. K. Chesterton, in an essay on wheels in his book Alarms and Discursions, likens the wheel to a healthy society in having "a part that perpetually leaps helplessly at the sky; and a part that perpetually bows down its head into the dust." He reminds his readers, in a characteristically Chestertonian remark, that "one cannot have a Revolution without revolving."

The most subtle of all wheel paradoxes is surprisingly little known, considering that it was first mentioned in the Mechanics, a Greek work attributed to Aristotle but more likely written by a later disciple. "Aristotle's wheel," as the paradox is called, is the subject of a large literature to which such eminent mathematicians as Galileo, Descartes, Fermat and many others contributed. As the large wheel in Figure 1 rolls from A to B, the rim of the small wheel rolls along a parallel line from C to D. (If the two lines are actual tracks, the double wheel obviously cannot roll smoothly along both. It either rolls on the upper track while the large wheel continuously slides backward on the lower track, or it rolls on the lower track while the small wheel slips forward on the upper track. This is not, however, the heart of the paradox.) Assume that the bottom wheel rolls without slipping from A to B. At every instant that a unique

Aristotle's wheel paradox point on the rim of the large wheel touches line AB, a unique point on the small wheel is in contact with line CD. In other words, all points on the small circle can be put into one-to-one correspondence with all points on the large circle. No points on either circle are left out. This seems to prove that the two circumferences have equal lengths.

Aristotle's wheel is closely related to Zeno's well-known paradoxes of motion, and it is no less deep. Modern mathematicians are not puzzled by it because they know that the number of points on any segment of a curve is what Georg Cantor called aleph-one, the second of his transfinite numbers. It represents the "power of the continuum." All points on a one-inch segment can be put in one-to-one correspondence with all points on a line a million miles long as well as on a line of infinite length. Moreover, it is not hard to prove that there are aleph-one points within a square or cube of any size, or within an infinite Euclidian space having any finite number of dimensions. Of course, mathematicians before Cantor were not familiar with the peculiar properties of transfinite numbers, and it is amusing to read their fumbling attempts to resolve the wheel paradox.

Galileo's approach was to consider what happens when the two wheels are replaced by regular polygons such as squares [.see Figure 2]. After the large square has made a complete turn along AB, the sides of the small square have coincided with CD

Figure 2

Figure 2

in four segments separated by three jumped spaces. If the wheels are pentagons, the small pentagon will jump four spaces on each rotation, and so on for higher-order polygons. As the number of sides increases, the gaps also/ increase in number but decrease in length. When the limit is ifeached—the circle with an infinite number of sides—the gaps will be infinite in number but each will be infinitely short. These Galilean gaps are none other than the mystifying "infinitesimals" that later so muddied the early development of calculus.

And now we are in a quagmire. If the gaps made by the small wheel are infinitely short, why should their sum cause the wheel to slide a finite distance as the large wheel rolls smoothly along its track? Readers interested in how later mathematicians replied to Galileo, and argued with one another, will find the details in the articles listed in this chapter's bibliography.

As a wheel travels a straight line, any point on its circumference generates the familiar cycloid curve. When a wheel rolls on the inside of a circle, points on its circumference generate curves called hypocycloids. When it rolls on the outside of a circle, points on the circumference generate epicycloids. Let R/r be the ratio of the radii, R for the large circle, r for the small. If R/r is irrational, a point a on the rolling circle, once in contact with point b on the fixed circle, will never touch b again even though the wheel rolls forever. The curve generated by a will have an aleph-null infinity of cusps. If R/r is rational, a and b will touch again after a finite number of revolutions. If R/r is integral, a returns to b after exactly one revolution.

Consider hypocycloids traced by a circle of radius r as it rolls inside a larger circle of radius R. When R/r is 2, 3, 4, ... , points a and b touch again after one revolution and the curve will have R/r cusps. For example, a three-cusped deltoid results when R/r equals 3 [see Figure 3, left]. The same deltoid is produced when R/r is 3/2; that is, when the rolling circle's radius

Figure 3

Figure 3

is two-thirds that of the fixed circle. All line segments tangent to the deltoid, with ends on the curve, have the same length. A four-cusped astroid is generated when R/r equals 4 or 4/3 [see Figure 3, middle]. The two ratios apply to all higher-order hy-pocycloids of this type: when R/r is either n or ra/(n — 1), the rolling circle produces an n-cusped curve.

There is a surprising result when R/r equals 2 [see Figure 3, ng/tf]. The hypocycloid degenerates into a straight line coinciding with a diameter of the larger circle. Its two ends may be regarded as degenerate cusps. Can you guess the shape of the region swept over by a given diameter of the smaller circle? It is a region bounded by an astroid. This is the same as saying that the astroid is the envelope of a line segment that rotates while it keeps its ends on two perpendicular axes, as shown in Figure 4.

Figure 4

Figure 4

The simplest case of an epicycloid traced by a point on the rim of a wheel rolling outside another circle is seen when the two circles are equal. The result is a heart-shaped curve called

Figure 5

Figure 5

the cardioid [see Figure 5]. All chords drawn through its cusp have the same length. The cardioid in the illustration was drawn by dividing the fixed circle into 32 equal arcs and then drawing a set of circles whose centers are on this fixed circle and that pass through other points on the same circle. The figure can be shaded to produce a dazzling Op-art pattern [see Figure 6]. (Both pictures are from Hermann von Baravalle, Geometrie als Sprache der Formen, Stuttgart, 1963.)

Figure 6

Figure 6

The cardioid is also generated by a point on the circumference of a circle that rolls twice around a fixed circle inside it chat is half as large in diameter. This fact underlies a problem chat was incorrectly answered in The American Mathematical Wfomthly for December, 1959 (Problem E 1362) but correctly answered in the March 1960 issue of the same journal. Imagine a giil whose bare waist is a perfect circle. Rolling around her waist, while she remains motionless, is a hula hoop with a diameter twice that of her waist. When a point on the hoop, couching the girl's navel, first returns to her navel, how far has th-at point traveled? Since the point traces a cardioid, this is equivalent to asking for the cardioid's length. It is not hard to show that it is four times the diameter of the hoop or eight times the diameter of the girl's waist.

"When a rolling circle is half the diameter of a fixed circle that it touches externally, the epicycloid is the two-cusped nephroid (meaning kidney-shaped) that is shown in Figure 7. The drawing both shows the rolling circle and demonstrates a method of constructing the nephroid as the envelope of circles whose centers are on the fixed circle and that are tangent to the vertical central axis. As before, the curve can also be generated by rolling a circle around a smaller circle inside it; in this case, when R/r is 3/2. This is the same ratio as that which produces a deltoid, but now it is the larger circle that does the rolling.

Figure 7

Figure 7

The cardioid and the nephroid are both caustics, curves enveloped by reflected light rays. The cardioid appears when the rays originate at a point on the circumference and are reflected by the circumference. The nephroid is produced by parallel rays crossing the circle, or from rays originating ait the cusp of a cardioid and reflected by the cardioid. The cusped curve that one often sees on the surface of tea or coffee in a cup, when slanting light falls across the liquid from a wimdiow or other light source far to one side, is a good approximation of a nephroid cusp. Pleasant approximations are also fr equently seen on photographs that appear in girlie magazines.

There are varied and perplexing problems that involve non-circular "wheels." For example, suppose a square wheel rolls without slipping on a track that is a series of equal arcs, convex sides up. What kind of curve must each arc be to prevent the center of the wheel from moving up and down? (In other words, the wheel's center must travel a straight horizontal path.) The curve is a familiar one and, amazingly, the same curve applies to similar tracks for wheels that are regular polygons with any number of sides. The answer will be disclosed in the answer section at the end of this chapter.

And can any reader solve this new riddle From Stephen Barr: What type of conveyance has eight wheels, carries only one person and never pollutes the atmosphere?

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