Introduction

[ ♫ intro music ♫ ] >> KEVIN: Hi, I'm Kevin, and I'm a geek. I really like working on mathematical problems and explaining them to people. I've retired from a job at a big industrial research lab where a lot of my work consisted of just that. Now that I'm on my own for a while, I'm looking for whether I can get the same sort of fun out of explaining things in videos. A lot of what I've done involves various aspects of geometry, but I'm likely to stray far and wide because the field of mathematics is all connected. For this video, I'll examine one of my favorite geometric figures, the cardioid.

Cardioid on a spirograph

One of the easiest ways to think anbout a cardioid is to imagine drawing one on a spirograph. For anyone who doesn't remember, the spirograph was a popular toy in the 1970's that consisted of a set of gears. You'd pin one gear to a drawing board, and then hold a pen in a hole of another gear and turn it around, meshed with the first one. You could use it to draw a whole variety of swirly patterns. The cardioid is probably the most boring pattern that you could possibly get! To get it, imagine pinning one wheel to the drawing board. Let a second wheel of exactly equal size roll against it. Put the pen on the circumference of the second wheel. Some spirographs actually allowed this, but having a perforated gear tooth. Let the second wheel roll completely about the first, and you have a bean-shaped figure. Giovanni Salvemini di Castiglone, the mathematician who named it, thought it looked like a heart shape, and named it a cardioid from the Greek word for heart.

Definitions

We'll want to have a few words for the pieces of a cardioid. The cusp is the place where the curve touches the fixed wheel. The vertex is the point opposite the cusp. The distance from cusp to vertex is called the diameter. It has a lot in common with the diameter of the circle, which we'll see when we get to the analytic geometry of the cardioid. Let's look at how big the diameter is. Each wheel has radius 'a'... ...and when we place the fixed wheel... ...and the moving wheel, they lie side-by-side exactly within the cardioid.

Rotations and revolutions

So the diameter of the cardioid is twice the diameter of the wheels. Now, let's look at the motion of the wheels. How many times does the moving wheel revolve about the fixed one, and how many times does it rotate about its own axis? I'll give you a few moments to think about it. Feel free to pause the video until you think you have the answer. [ ♫ music plays ♫ ] OK, are you ready? Watch just the center of the moving wheel as it goes around. It goes around exactly once. Now watch the arrow in the moving wheel as it turns about its own axis.

Alternative spirograph setup

It turns about its own axis twice. There's another way to set up a cardioid on a spirograph. Instead of a rolling gear, use a ring gear that's twice the diameter of the fixed gear, and roll it. So, how many times does the center of the ring go around the fixed wheel, and how many times does the ring turn around its own axis? This one is a little tougher that the last one, so feel free to pause and think if you need to, [ ♫ music plays ♫ ] Ready? OK, let's see if you're right. Watch the red arrow, which points to the center of the ring. Can you see that it goes around twice? Now watch the yellow arrow, which shows the ring's rotation around its own axis. The yellow arrow goes around only once. This is just the reverse of using the gear. The gear revolved once and rotated twice.

Ruler and compass construction

The ring revolves twice but rotates once. Are both these cardioids the same? Can we prove it? I'm going to stick here to the ancient Greek methods, as you'd encounter in a high-school geometry class. Sometimes, in high-school geometry, you'll see discussions of curves in those terms. You might see an ellipse defined with two foci and a length, or a parabola defined with focus and directrix. But usually they immediately skip over this stuff and use it as an opportunity to introduce analytic geometry. But the Greeks, without analytic geometry, or even algebra, had a nearly complete theory of conic sections. Since their methods are connected to some relatively modern mathematics like the theory of quadratic forms, it's good to understand their approach. Sharpen your pencil, break out your ruler and compass, and let's try to set up the problem as they would! Let's start by examining what it means for the wheel to roll. We can describe it with some basic geometric facts. The wheels have to stay in contact, so their centers are connected by a straight line and the radii are, of course, equal. The wheels also must have rolled the same distance, which means that the lengths of the arcs from the starting point of contact to the current point of contact must be equal. Remember that if we measure angles in radians, the arc length is the central angle times the radius. Since the arc lengths and radii are both equal, the central angles have to be equal, too. We see the same thing with the rolling ring. The ring has to be touching the wheel, which sets up one line with equal segments. And the ring and the wheel have to have rolled the same distance, which gives us another pair of equal arcs. Again using the arc length formula, we find that the central angle on the ring has to be half the central angle on the stationary wheel. And Thales' Theorem tells us that an inscribed angle is half the central angle, so the points P, C, and R in the diagram have to lie on the same line. This gives us two straightedge-and-compass constructions that will start with an arbitrary angle, and construct a point on the cardioid. Let's do the one that starts with a point on the wheel. How do we set up the problem? We need to give the location of the center, the location of the cusp, and an angle of rotation, or equivalently, a point of contact between the wheels. If you like, we will build a formal statement of the problem on the right. So choose the center... ...and choose the cusp, Draw the fixed wheel with the compass. Draw the radius from center to cusp. Choose the point of contact. Use the straightedge to draw the radius through the point of contact, and extend the line out one more radius with the compass. Use the compass to draw the moving wheel. Use straightedge and compass to duplicate the central angle and intersect the new ray with the moving wheel. Since radii and central angles are equal, arc lengths are equal. Every possible choice of angle gives another point on the cardioid. The cardioid is the locus of all those points. [ ♫ music plays ♫ ] We can use nearly the same construction for points on the cardioid defined by the ring gear.

Both cardioids are the same

But now, what does it mean in Greek geometry for two curves defined in this way to be the same? We have to prove that every point on the first curve is a point on the second, and vice versa. That means that every point anywhere on the plane is either on both curves, or neither. I'll just show the proof that every point on the cardioid defined by the ring is a point on the one defined by the wheel, but once you have one, the other is obvious. Of course, we have to specify what it is we're trying to find! By what we already showed about doubling the central angle, we can find the center of the ring, and the point of contact, immediately. And we draw the radius from the center of the ring through the cusp. I claim this will always hit the ring at the point P. The diagram we just made has equal angles all over the place - we constructed them that way. ∡COA, ∡AOB and ∡AWP are all equal. ∡CRB, the one angle that we didn't copy. is given to us by Thales. OWPR is a parallelogram, and there are equal line segments everywhere. RP, OW, and RB are all equal to the diameter of the fixed wheel, and OA is equal to its radius. The arc length formula shows that the arcs CP and PB are of equal length, and we're done. We can prove the converse, that every point on the cardioid defined by the ring is a point on the one defined by the wheel, by essentially running the same steps in reverse So now that we have two good straightedge-and-compass constructions for the cardioid, I'll give in to your hurried geometry teacher and shift from Euclid to Descartes.

Coming attractions

Next time, we'll use analytic geomtry to discover some facts about the cardioid, so stay tuned for that! Until then, stay safe, stay healthy, and keep calculating! [ ♫ music plays ♫ ]'