Math in Plain English: What is a circle?

This is a continuation of yesterday’s post, but this is not part of the main story of getting towards my area of research.

I’m sure everyone knows what a circle is roughly, but let’s consult a dictionary for the word “circle,” here’s Merriam-Webster’s definition (with my own corrections): “a closed plane curve every point of which is equidistant from a fixed point (called the center) within the curve.” So there’s a center and a circle is all the points that have the same distance from that center. Everything else is superfluous, even the bit about the “curve”. So ignoring the word “curve” for now, what’s striking is that nothing about Euclidean geometry is necessary here. You have a point (which we give the name center) and a distance (which is called the radius), and you look at all the points that have that distance from the center. So all we need is a notion of “point” and “distance.”

What a coincidence! That’s exactly what a metric space is! It’s, I don’t know, as though we tailor-made that definition so we can talk about circles or something. So we see that our usual picture of a circle, as a nicely round figure is what it is, because of our sense of distance. Just imagine spinning around with your arms stretched out, the figure traced out by your hands is, of course, a circle.

Before we go into examples, we have to clarify some language. In everyday English, we use the same word circle for both the outline of the shape and the shape including all the points inside. In addition, there’s a problem of whether we just include those insides or have both insides and outline. These things have widely different purposes in math. So to clear up the ambiguity we give them separate names: a circle is the outline, so the points with the same distance; an open disk is the inside, so the points with distance within the radius from the center (but not equal to it); and the closed disk is both the inside and outline, the points with distance within or the same as the radius. “Inside” as we will see is not very useful, so it’s better to think in terms of distances. It should be noted that the words “open ball” and “closed ball” are often used instead of “open disk” and “closed disk.”

So looking back at our examples from last time:

  1. If we consider the major cities of the world. A circle centered in New York City with radius 300 miles are all the major cities exactly 300 miles away from New York, which is actually probably nothing. But the disks are a bit more interesting: the disk centered in New York City with radius 300 miles are all the major cities less than 300 miles away, for example, Philadelphia would be in the disk.
  2. In the second example, where we look at every point of the globe, circles are exactly the same as the image we have in mind. Disks are slightly different, since the globe isn’t flat, the disks look like domes rather than our flat disks.
  3. When we consider the distance being least time it takes to get between places, we get interesting results due to politics. A disk centered at a city near a boundary, it may take more time to get across the boundary despite being physically closer. So the disk might be shaped by the political boundaries. A more physical constraint occurs if one lives on a large island, it may take more time to get on a boat and travel to a nearby city off the island than it is to travel to the other side of the island, despite being closer physically. In this case it is the water that is shaping the disk. So we see that we won’t get nice uniform shapes all the time. The center plays a much larger role in shaping the circles and disks than they did in the ordinary geometry case.
  4. If you live in Paris, then it’s straightforward to take a train to your favorite point. But if you don’t live in Paris and your radius is smaller than your distance to Paris, then you can’t make any transfers. So you can only travel along the one route that goes through Paris. Your circle is exactly two points: the two places along that one route exactly that distance away from the center. The disk is also strange, it’s the points along that one route that have distance less than the radius. But if your radius is bigger than the distance to Paris, you’re in luck. You can make any transfer you like and so you can take any train provided that you have enough in your radius to have gotten to Paris and then go to the point of your choice. Here even in the generalized mathematical, make everything symmetric and easier to deal with case, the circles and disks not only depend on whether you are centered in Paris or not, but also the different radii don’t give you scaled versions of the circles and disks. The shape fundamentally changes from a puny part of a train line to part of a train line and a little bit of every train line centered at Paris depending on whether radius is large enough to let you get to Paris or not!
  5. So if you look at English words with that distance we described last time, then the open disk of radius 1/3 centered at the word “center” would be all the words that begin with “cir-“, the closed disk of radius 1/3 with center “circle” would be the words that begin with “ci-” and the circle of radius 1/3 centered at “circle” would be the words that begin with “ci-” but whose third letter is not “r”.

The main lesson, if blog posts need lessons, is that the concept of circle depends on the notion of distance. So if allow yourself to expand to a variety of notions of distance, you can expand to a variety of types of circles.

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About minimalrho
Unemployed guy with a PhD in math.

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