I’m surprised to discover that people enjoyed Geometry a lot more than I thought they did. Even more surprisingly, it appears that math people hate Geometry, while everybody else either likes it, or has a mild dislike for it, nothing like the deep dislike that some of my mathematics students seem to have!
Because of the great variety of attitudes towards the subject, people in different places seem to have very different degrees of geometry experience, and what I’m posting today might be old hat to some, and quite refreshingly new to others.
Triangles are very interesting things, mostly because they provide a framework for lots of other structures. One very easy exercise is to make a circle that goes through all three points (vertices, or corners) of the triangle.
Constructing a Circle through the Three Vertices of a Triangle
The principle is simple, if you take two points at a time. Suppose our triangle is ABC. Considering just A and B to start with, there is a line of points an equal distance from A and B, as shown at right. Each of these points might be a different distance from A, but they’re the same distance from B.
We can do the same with the points B and C; again we get a whole new line of points the same distance from B and C. Now, it is clear that these two lines of points have one point in common! Let’s call it O. This point O is the same distance from A and B, and the same distance from B and C. (As we say in mathematics, by transitivity, it is the same distance from C as from A, but we won’t fuss about that.) It follows that we can draw a circle centered at this point O which will pass through all three of A, B, and C.How does one make the first line, which is all the points equal distances from A and B? Easy: join AB, find the midpoint. So this is one of the points. Now make a perpendicular; that gives us the whole line.
When anyone sees the triangle with the circle around it, it merely looks as if someone had taken a circle, and picked three random points on it, and joined them to make a triangle. It is a little more convincing to have the three points move on circles of their own, and observe the circle around the triangle follow along. Even here, the eye is deceived into believing that the circles are the fundamental thing, but a little thought would make you see that this is unlikely, since it is difficult to make a circle dance. Here is a video showing the dancing triangle, with its circumcircle. The music is by J. S. Bach, and it is the (organ) Fugue in A minor, BWV 543, played using a Marimba sample.
[Without appearing to obsess over whether and why people have a strong aversion to mathematics, I’d like to pursue what the root causes might be, perhaps in a later post.
My suspicion is that it is partly because people give up too quickly on mathematics, which leads to the second problem, namely that Mathematics is a vast area of knowledge, and a lot of it is at the intermediate level, about the level of high-school study, so that most laymen have already abandoned relating to the subject by that time. Of course, an enormous amount of mathematics is at the advanced level, and this part of mathematics is very diverse, because it has all sorts of made-up mathematics, created almost exclusively for the purpose of getting someone a doctorate or a publication, and of no use whatsoever, but the rest of it is quite successful attempts, I have to concede, to organize and illuminate what is already known, by underscoring the commonality of it.
But this middle level has been around for ages. It just so happens that only a small part of it has to actually do with numbers and arithmetic; the rest is all about logical relationships. This is not exactly your everyday logic, but more complex reasoning. Here is an example: for every pair of fractions you can give me, I can give you a number that lies between them that is not a fraction. That sentence is quantified, which means that the phrases “for every” and “I can find you” are used, which are definite, but not the normal logical equipment people use.]
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