For over a decade I’ve been struggling with certain math puzzle, first with the help of James Dolan and later also with John Huerta. It’s about something amazing that happens when you roll a ball on another ball that’s *exactly 3 times as big*. John Huerta and I just finished a paper about it, and I’d like to explain that here.

But I’d like to ease into it slowly, so I’ll start by talking about what happens when you roll a circle on another circle that’s the *exact same size*:

Can you see how the rolling circle rotates *twice* as it rolls around the fixed circle *once?* Do you understand why?

The heart-shaped curve traced out by any point on the rolling circle is called a **cardioid**. In her latest video Vi Hart pretends to complain about parabolas while actually telling us quite a lot about them, and much else too:

Naturally, with her last name, she prefers the cardioid. She describes various ways to draw this curve: for example, by turning the hated parabola inside out. Here are my 6 favorite ways:

1) The one we’ve seen already: roll a circle on another circle the same size, and track the motion of a point on the rolling circle:

2) Take a parabola and ‘turn it inside out’, replacing each point with polar coordinates (r, θ) by a point with coordinates (1/r, θ). As long as your parabola doesn’t contain the origin, you get a cardioid:

This ‘turning inside out’ trick is called conformal inversion.

3) Draw all circles whose centers are points on a fixed circle, and which contain a specified point on that circle:

Here’s a nice animation of this process, made available by the Math Images Project under a GNU Free Documentation License:

4) Let light rays emanate from one point on a circle and reflect off all other points on that circle. Draw all these reflected rays, and you’ll see a cardioid:

If you draw all light rays that reflect off some curve, the curve they snuggle up against (their so-called envelope) is called a catacaustic.

5) Draw 36 equally spaced points on a circle numbered 0 to 35, and draw a line between each point n and the point 2n modulo 36. You’ll see a cardioid, approximately:

But there’s nothing special about the number 36. If you take more evenly spaced points, you get a better approximation to a cardioid. You get a perfect cardioid if you connect each point (1, θ) to the point (1, 2θ) with a line, and take the envelope of these lines.

6) Finally, here’s how to draw a cardioid starting with a cardioid! Draw all the **osculating circles** of the cardioid—that is, circles match the cardioid’s curvature as well as its slope at the points they touch. The centers of these circles give another cardioid:

This picture has some distracting lines on it; just look at the big and the little cardioid, and the circles. This trick is an example of an ‘evolute’. The **evolute** of a curve is the set of centers of the osculating circles of that curve.

All the pictures above are from Xah Lee’s wonderful website or the Wikipedia article on cardioids. Click on the picture to see where it came from and get more information.

I ❤ cardioids!

Next time we’ll see what happens when we roll a circle inside a circle that’s *exactly twice as big*.

### Constructions on curves

We’ve seen a few constructions on curves:

• A **roulette** is the curve traced out by a point attached to a given curve as it rolls without slipping along a second curve.

We rolled a circle on a circle and got a cardioid, but you could roll a parabola on another parabola:

This gives a curve called the **cissoid of Diocles**, which in some coordinate system (not the one shown) is given by this cubic equation:

• A **catacaustic** is the envelope of rays emanating from a specified point (perhaps a point at infinite distance, which produces parallel rays) and reflecting off a given curve.

We’ve obtained the cardioid as a catacaustic of the circle. Supposedly if you take the cissoid of Diocles and form its catacaustic using rays emanating from its ‘focus’, you get a cardioid! This would be a seventh way to get a cardioid, but I don’t understand it, even though it’s described on Wolfram Mathworld. I don’t even know what the ‘focus’ of a cissoid of Diocles is. Can you help?

• The **evolute** of a curve is the curve formed by the centers of its tangent circles.

We’ve seen that the cardioid is its own evolute. The evolute of an ellipse looks like this:

It’s called an **astroid**, and it’s given by an equation of this form:

If you take the tangent circles of the black points on the ellipse above, their centers are the sharp pointy ‘cusps’ of the astroid.

### Order from chaos?

Some other famous ways to construct new curves from old ones include the involute, the isoptic, and the pedal. I could describe them… but I won’t. You get the picture: there’s a zoo of curves and constructions on curves, and lots of relations between these constructions. It’s all very beautiful, but also a bit of a mess.

It seems that all these constructions, and their relations, should be studied more systematically in algebraic geometry. It may seem like a somewhat musty and old-fashioned branch of algebraic geometry, but surely there’s a way to make it new and fresh using modern math. Has someone done this?

There’s got to be a better name than tangent circles. I was thinking bitangent circles, but that sounds like they’re tangent at two different points, rather than at the same fat point, which is what you want.

The Mathematical English word for having equal curvature at a point of tangency is “osculating”. So, one has the odd sequence “secant, tangent, osculant”, from latin words for cutting, touching, and kissing; for obvious reasons one reverts to ordinals to describe higher-degree contact conditions.

It took me a while to realize that the ‘obvious reasons’ involved keeping mathematics, and this blog, suitable for children.

I like ‘osculant’, but it turns out Wikipedia prefers the term osculating circle.

So I’ve edited the blog entry to change ‘tangent circle’ to ‘osculating circle’.

Worth it for the Hart link alone! Have shared that with

my step-daughter in law, a college senior thinking of teaching math

There’s a pretty famous cardioid, the set of points that have a single limiting point under the Mandelbrot iteration.

http://en.wikipedia.org/wiki/Mandelbrot_set#Main_cardioid_and_period_bulbs

Nice! You’re talking about region 1 here:

I would describe this region a bit differently than you (in part because I’ve just learned this stuff right this second, and I want to say it precisely so I remember it): it consists of points such that the map

has an attracting fixed point The region labelled 2 is a disc, and it consists of points such that this map has an attracting orbit of period 2.

Great to know that I am not totally on the wrong track -thanks for this post and for Vi Hart’s video. I have just posted some of my visual thoughts here:

• Meditations on the Mandelbrot set.

Wow, that’s great! Everyone should take a look! I find most art based on math a little disappointing, but not this. Really good drawings done by hand have a richness and humanity to them that computer-generated drawings have trouble matching, and when that’s mixed that with the elegance of math I get very happy.

You might like my series on The Beauty of Roots: Part 1, Part 2 and Part 3. Lots of pretty pictures, all drawn by my friends, not me.

Thanks for the “Beauty of Roots” links below – I have seen and enjoyed them, and in fact used some of the concepts as inspiration.

Thanks! I’m reading your blog. It’s a lot of fun!

I just added a new section at the end of this blog article, for devotees of this blog… this foreshadows some ideas we’ll see later.

“Has someone done this?” – Yes, several books. The first I’m personally aware of is Robert C. Yates, “A Handbook on Curves and Their Properties,” J. W. Edwards, 1952. I used this for the basis of my book, J Dennis Lawrence, “A Catalog of Special Plane Curves,” Dover, 1972. A more recent book is Eugene V. Shikin, “Handbook and Atlas of Curves,” CRC Press, 1995. There are others.

Hi! I’ve recently heard about your book, as I’m dipping my toe into the world of plane curves. I’ll try to get ahold of it and read it, and all these others too.

When I asked “has someone done this?”, here’s the kind of thing I was wondering about.

Algebraic geometers tend to like projective geometry, which studies constructions that are invariant under all projective transformations. However, constructions like the evolute and isoptic involve distances and angles, which are not invariant under all projective transformations. For any kind of construction on curves, I’d like to know the minimal kind of geometry needed to carry out these constructions: conformal geometry if only angles are involved, Euclidean geometry if both angles and distances are involved, etcetera.

(The isoptic prominently mentions angles, but it also involves drawing straight lines, so I believe it requires full-fledged Euclidean geometry: I don’t think this construction is invariant under conformal inversion, for example. Anyway, this is the kind of question I’m talking about.)

Then it might be nice to see which constructions send algebraic curves to algebraic curves, since transcendental curves are trickier (though still very much worth studying). If we have some nice constructions that send algebraic curves to algebraic curves, we can ask questions about how these constructions affect various famous invariants of curves, like the degree and genus. We could also look for smaller classes of curves that are preserved by these constructions.

And then there are relations between constructions like “the catacaustic of a curve is the evolute of its orthotomic”, most of which have probably been discovered already.

I can add to the list of books Vasilyev and Gutenmacher “Lines and Curves: A Practical Geometry Handbook”, Birkhauser; 1 edition (July 23, 2004) and C. Zwikker’s “The Advanced Geometry of Plane Curves and Their Applications”, Dover, 2005. At my site http://www.cut-the-knot.org there are several interactive Java gadgets that explore exactly those properties of cardioids that you mentioned.

Last time we rolled a circle on another circle the same size, and looked at the curve traced out by a point on the rolling circle. It’s called a

cardioid. But suppose we roll a circle on another circle that’s twice as big. Then we get anephroid![…]In Part 1 and Part 2 we looked at the delightful curves you get by rolling one circle on another. Now let’s see what happens when you roll one circle

insideanother!“Can you see how the rolling circle rotates twice as it rolls around the fixed circle once? Do you understand why?”

OK, sorry for the basic question, but for the life of me I can’t see how the rolling circle rotates twice – the point on the rolling circle that’s at (1,0) or 0 deg ends up at the same spot after the cardioid has been completed, and it only reaches that spot once. ? This must be a dumb question but I would really appreciate some insight.

The point on the rolling circle that’s initially at (1,0) on the rolling circle faces right

twiceas that circle rollsoncearound big one. If the rolling circle were the Earth and the center of the big circle were the Sun, you’d see the stars go around the sky twice each year. So, we can say the rolling circlerotatestwice as itrevolvesonce.(Of course, the Sun would be actually touching the Earth, but never mind!)

You can probably get what I’m asking in Puzzle 1 and Puzzle 2 now. It’s also good to figure out how many times you see the stars go around the sky during a single year.

It’s a strange sort of optical illusion (perhaps well known by magicians?). I too found it impossible to mentally see from the animation. Even with mathematica demonstrations “frame by frame” I could not see it. I was about to use 2 table coasters to help me out when I remembered that in V. Hart’s video she used paper with lines. Only by looking closely at her video I got it.

Really readily understandable, fun and fascinating this entire series. From geometry to the center of the universe astronomy to engine design and also with some nice user contributed proofs and facts, its a “little fractal” of science – I think this would be quite a good talk to high-school students.

One way that helps me to picture it is to first imagine the moving circle moving around the stationary one but with the same point of the moving circle touching the stationary one. In this case, it is apparent that the moving circle will have rotated only once.

Second, picture the moving circle rotating while its centre remains fixed.

Lastly, the actual example is a combination of these two motions, as the moving circle rolls without slipping around the stationary circle. The combination of the two types of rotation produces two rotations in all.

In Part 1 we rolled a circle on a circle that’s the same size […]

Order and chaos are two sides of the same coin. Indeed, the words “cosmos” and “chaos” share the same greek origin and mean curiously such a different “notions”, likely complementary “a la Böhr”, since order is regularity and chaos is mainly associated to nonlinear and “wild” phenomena. In fact, you should know I have always preferred my own variation of those words “Chaosmos” (Chaotic Cosmos, as portmanteau) to call the main features of this Universe (or Multiverse/Polyverse if you believe there are other bubble Universes beyond the Lemaitre limit, aka the observable radius of our known Universe).

Thank you. Fascinating.