Going around on circles: mathematics and computer art; part 2. Sheldon P. Gordon; Florence S. Gordon.
Last month we pointed out the strange fascination that mathematicians have for weird situations and how computer graphics provides a spur to these interests. As an illustration of this, we examined a class of curves known in mathematics as hypocycloids, the path traced by a point on a circle rolling around inside a larger circle (Figure 1). This resulted in some fascinating patterns of both a geomatrical and a numerical nature.
Having achieved such nice results, however, no self-repecting mathematician would let the idea die without pursuing it further. All kinds of other questions and speculations come to mind and, again with the power of sophisticated computer graphics, we have a tool to look at some of these other possibilities.
The most likely follow-up question that arises is probably the following: if a hypocycloid is the type of curve that is formed when a circle rolls around inside a larger circle, what happens when the small circle rolls around the outside of the larger one? (See Figure 2). The result is know as an epicycloid, and we shall explore some of its properties in this article.
the easiest way to picture an epicycloid is to think of a piece of radioactive bubble gum attached to a wheel which is rolling arounds the outside of a larger whell. The path of the gum, as recorded by the radioactive track it leaves behind, is the epicycloid curve. Clearly, the shape of the path depends on the relative sizes of the two wheels as measured by their radii.
Therefore, let's begin by looking at the shape for different values for the radii. Suppose the rolling circle has radius 1 and the fixed inner circle has radius 2. The resulting shape is shown in Figure 3. On the other hand, if we change things slightly so that the fixed circlehas radius 3, then we obtain the shape in Figure 4. Similarly, if the radius of the fixed circle is 4, then we obtain the shape in Figure 5. Cusps and Revolutions
If we compare these three shapes in conjunction with the values for the radii, then some patterns will become evident. To make things easier, let's write the pair of radii as: (A,B,) = (inner, outer). Thus, we have drawn the pairs: (2,1), (3,1) and (4,1). Notice that each epicycloid shape has the same number of arches as the radius of the inner cirlce. Similarly, it has the same number of sharp points (called cusps) where the point on the rolling circle touches the inner circle. With this in mind, we would expect that the shape with radii (25,1) would have 25 arches and 25 points. This is indeed the case, as is seen in Figure 6.
However, what happens if the outer circle does not have radius 1? SAy it has radius 2 while the inner cirlce has radius 5 (5,2). The result is shown in Figure 7. From this, we see that there are still five arches and five cusps (as expected), but they occur in a more complicated pattern. If you trace out the curve with a pencil starting at the right-most cusp, you will find that the five arches are formed while making two full revolutions around the curve. Equivalently, the bubble gum comes back to its starting point, having rolled around the inner circle precisely twice while forming the five arches. The two revolutions, then, probably suggest some connection with the smaller radius 2 of the rolling circle.
With this conjecture in mind, we might expect that the pair of radii (8,3) would produce an epicycloid having eight arches and cusps, but taking three full revolutions to complete the full curve. This is shown in Figure 8. In a similar way, the pair (40,11) takes 11 full revolutions to complete its 40 arches, as seen in Figure 9.
However, if you look at Figure 10, which shows the result for the pair (40,12), the apparent pattern breaks down. Here, there are only 10 arches and it took only three full revolutions to complete the curve. What happened? A mathematical bent of mind provides the answer.
Let's consider the two numbers 40 and 12. It turns out that both are divisible by 4 and, if this common factor is removed from each of them, we are left with the numbers 10 and 3. In other words, the epicycloid shape appears to depend on the factored values for the radii.
Therefore, the pair (20,6) will lead to the same shape, as will (100,30). (We should point out that it is the shape that is the same; the size of these epicycloids will be vasty different because of the different radii. The automatic scaling done by the computer allows us to ignore size while concentrating on the more interesting question of shape). On the other hand, a very slight change in the radii, say to (101,30) will result in a dramatically different shape, as shown in Figure 11. Shapes
By this stage, there really is no point in continuing to count the number of cusps or revolutions. Instead, it is far more appealing o look on the shapes that are produced as examples of computer generated art. To share the fun of constructing thse geometric patterns, we have attached a simple program that will graph any epicycloid with integer values for the radii. It is written for the TRS-80 Color Computer with Extended Basic, but can be modified easily to function on most other microcomputer models. Incidentally, the program will also operate with non-integer values for the radii, but will usually not trace out the complete curves without introducing several modifications.
For those who are mathematically unclined, the equations for the epicycloid are given by the pair of parametric equations: where B is the radius of the outer rolling circle and A is the radius of the fixed larger circle. For any values of A and B (subject to the property of dividing out common factors), the program will produce different shapes.
To have the program operate quickly, the number of points plotted depends on the values of A and B up to a preset maximum of 400 points. This introduces an interesting feature.
When A and B are large, the 400 point maximum is not large enough for an exact portrayal of the graph. Instead, the program produces a tremendous number of rounding errors that produce even more intricate and dazzling shapes whose general outline is the correct epicycloid shape. For example, the pair (4000,1000) should reduce down to the same shape as (4,1) but the actual result is that shown in Figure 12. Similarly, Figures 13 and 14 show the results for the pairs (6000,1500) and (8888,2222) respectively.
Actually, the designs shows in these figures convey only some of the artistic effects produced by the program. They are just the completed curves and as such are static in nature. In the process of the program running, however, equally striking patterns are constantly being produced and in turn being incorporated within even more complicated shapes.
This is truly an example of dynamic art which can best be appreciated as a process rather than as a final fixed shape. Further, as with no other art form, the computer lets you have an "instant replay" of the dynamic development of the picture whenever you want to see the entire progression of shapes; just run the program again.
With these ideas and suggestions, you can now use the program to generate your own shapes for different values of the radii to enjoy the excitement of producing new shapes and, with a little curiosity and luck, possibly some new mathematical relationships and theories regarding the epicycloids. Historical Note
In case you suspect that curves such as epicycloids are just figments of a mathematician's imagination, the following historical note is in order. Prior to the discoveries of Copernicus and Kepler in the period 1580-1600 AD on the structure of the solar system, the univesally accepted view was geocentric: the earth was at the center of the universe and everything else--the planets and the stars--revolved around it.
Unfortunately, the apparent motion of the known planets conflicted with the idea that their orbits were circular or even elliptic. Picture, if you will, a model of the solar system as displayed in any planetarium. In particular, picture yourself on the moving earth and watch the motion of one of the other planets; it approaches closest to you at one point, then recedes in a wide are and then comesback to the nearest distance somewhere else in the sky and so on.
It may not come as much of a surprise, at this point, that the ancient astronomers believed that the paths taken by the planets in their motion about the earth were nothing more than epicycloids.