Here's some things that might be helpful to remember when you're trying to explain Geometry to someone. First, draw pictures. That seems pretty obvious, but even in a lot of books there aren't enough of pictures to really get the idea across. You need to illustrate the definitions and the theorems. For instance, here's a drawing of a cone. Now in three-dimensions a cone looks like this funnel more or less. To the first time the first time that a student sees the picture they may not realize that for instance the dotted line represents the back side of the cone's circular base, which is invisible to the viewer from the front. So visual aids are quite helpful as well. The second thing which is a little bit harder to accomplish all of the time is to understand even more than you teach. This can be really helpful when a student doesn't quite understand the first presentation. And then you're able to come up with an alternate point of view, another way of explaining it. It's also really helpful when the student asks questions. For instance here's the formula for the area of a circle. Area is equal to pi times the radius squared. And then of course we need to know what pi is. Well pi is this number expanded out to nine places. Three point one four one five nine two six five three. Of course you don't need to tell all the places of pi, but it's good to know a few more than you actually tell. How do we get those digits? Well one way is this expansion invented by Leibnitz and Gregory. Start with four, subtract four-thirds, add four-fifths, subtract four-sevenths, add four-ninths and continue. The top number is always four. The bottom number is going to be the next odd number, and the subtraction and addition alternates. That gives you pi. But how do we know that gives you pi? Now you need some calculus. You first have to be able to show that this sequence converges, and then you have to know that this sequence is actually the expansion of arc tangent times four evaluated at one, which gives you pi. Last but not least, curiosity is one of the real good motivators to learning. So half of mathematics is what we don't know, probably much more than half actually. So it's good to mention as much as you have time for what we don't know. For instance this question which no one knows the answer to. Do the digits zero, one, two, three, four, five, six, seven, eight, nine each occur infinitely often in the expansion of pi, in the decimal portion of pi? No one knows the answer to this.
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