In mathematics, we spend a lot of time on definitions-and for good reason. Definitions are the foundation on which formal mathematics rests. When we need to know whether something is really true, we go back to the definition.
But most of the time when we think about mathematical objects, we don't have the definition in mind. Most of the time, we have an image in our heads and we refer to that. Or we have an example or a set of associations.
So there is an important difference between how formal mathematics works and how our minds work. That's OK as long as we are aware of it. This packet uses the example of circles to examine this difference.
(If you are interested in a vivid illustration of this difference, see a blog post I wrote relating a conversation I had with my 4-year old daughter.)
Here are some common responses to the question What is a circle? Notice that each of them is focused on what circles look like, or on properties of circles, but none is really good enough to precisely sort circles from non-circles.
A circle is round. Lots of things are round, but are not circles. Eggs are round. Coffee cans and soccer balls are round. But eggs are not circles. Coffee cans have bases that are circles, but they are (approximately) cylinder, not circles. Likewise soccer balls are three-dimensional objects that very, very round. But they are not circles. There is a difference between a circle and a sphere.
The figure below is round, but is not a circle.
A circle is something that has no end Neither does a line (in the mathematical sense, lines are infinitely long). Neither does the figure below.
A circle is a wheel This is a useful image, but it doesn't really sort circles from non-circles in a formal mathematical sense. For example, a wheel has thickness (the width of the tire print on the road), but a circle does not. And wheels have other properties that circles do not (axles, spokes, etc.)
These sorts of descriptions of circles are extremely useful. But there comes a time in mathematics that we need to have a definition of a circle, and none of these is good enough.
An example of a question that the above images are not good enough to answer is the following:
Why do compasses draw circles (and not ellipses or some other mathematical object)?
So in mathematics, a circle is defined as:
a set of points in a plane such that each point is a common distance from a given point, called the center.
This is a more abstract way of talking about circles than the characteristics listed above. But it definitively sorts circles from non-circles.
Eggs: Some points on an eggshell are further from the center than others are. For example, the pointy end is further from the middle of the egg than the blunt end is. And an egg isn't a set of points in the plane; it's a set of points in space.
Wheels: Again, not a figure in the plane.
Ellipses: The figure below is round, but its center is much closer to the top and bottom edges than to the left and right-hand edges.
Odd figures: Where would the center of this figure be? Wherever we put it, there will be a whole bunch of different distances to points on the figure so it's not a circle.
If a circle is the set of points a common distance from a center, then the circle does not include the points further than that (shaded below):
And the circle does not include the points closer than that:
It only includes the points that are that exact distance away:
What mathematicians mean by circle is what most of us think of as the outside edge of the circle (Fig. C). The stuff inside-what we might think of as the circle-mathematicians call the disk (Fig. B).
So technically, a circle has area equal to 0. When we ask about the area of a circle, we really mean the area of the disk. The area of the circle is the amount of space taken up by that set of black points around the edge; it's infinitely small.
But these are technicalities we don't need to worry about most of the time.
So back to the question from earlier on; Why does a compass draw a circle?
We are thinking here of the geometric tool (not the North-South navigational object), as below.
Let's go back to the definition. The pencil draws traces out a set of points in the plane (i.e. on the paper). The arms are designed so that the tip of the pencil stays the same distance from the tip of the other arm-either through friction or a screw-tightening adjustment. And the pointy end ensures that this common distance is from a single point. We press the compass down and the sharp tip makes that end stay put.
So a compass draws a circle because it keeps the pencil a constant distance from a single point.
A compass is not designed to draw round things, it is designed to preserve distances. Distance is the important thing about circles.
Imagine you live in the city whose map is below:
You start at point A and you go for a walk of length exactly five blocks, and during this walk you stay on the streets-no shortcuts across lawns, vacant lots or alleys. Where might you end up?
That is, what is the set of points that is exactly 5 blocks from point A? According to the definition, this set of points is a circle. But it certainly has a funny shape! Why?
The answer is because we are measuring distance in a different way. By only allowing you to follow the streets, we are changing the meaning of distance, and this results in new geometric results. A square is not a circle in our usual geometry, but this circle sure looks like a square in this system.
And what about the Earth's equator? Is it a line (as it appears on a map) or a circle (as it appears on a globe)? It is the set of points a common distance from the center of the Earth, so it's a circle, right? But if you walked the equator, you wouldn't have to turn even the tiniest amount to your right or to your left, so it's a straight line right?
(Answer: In Euclidean, 3-D geometry, the equator is a circle. In 2-D spherical geometry, it's a straight line-these are distinctions that mathematicians only figured out how to deal with in recent history-the last 150 years or so).