Developing new axioms for Elementary Geometry
I have always been bad at geometry. Always.
How does one develop geometric intuition? Let us talk about the fundamental building blocks of Geometry- Congruency. I must confess I never really understood HOW the whole of geometry is motivated by a seemingly useless and, let’s face it, mysterious concept of Congruency.
Two figures which are exactly the same under rigid motion are said to be Congruent. Congruency has always been only about triangles in school. This is an arbitrary restriction, and should be changed (bad teaching at the school level is what drives most people away from Geometry). Congruency conditions can easily be developed for squares, hexagons, dodecagons, you name it.
For example, if two squares have the same side length, they are the same. Simple.
Quadrilaterals- If three sides and three angles of two quadrilaterals are the same, then the two are the same quadrilateral.
Seeing Geometry from this generality sure makes the subject more comprehensible. Even today, whenever I see a geometrical figure, I just get irritated as I start believing that an endless bout of angle chasing and congruency proofs (of triangles) will follow. Now this bout is more motivated than before.
1. Distances and angles- Angles and distances are thought to be different measures in geometry. The concept of distance is fairly intuitive (to me). I have always felt that angles are a non-intuitive and arbitrary construct, and they really are just a way of representing distance. I am going to go off on a tangent and give a distance-centric definition of angle.
Let us take two lines (or any curves), and consider distances between points. There are many reasonable ways of dong this. One way is: take a point on , and drop a perpendicular on . Let the point of intersection be . Now consider . After we consider the distances between all such pairs of points, we can assign an “angle” to the pair of lines. But how does this assignment work? There are multiple reasonable ways, again. We could consider distances only between a reference point on and the point on the perpendicular to , etc. We could also see the general trend of increasing or decreasing distances between points. Whatever. Depending upon these distances, the assignment of the angle between the two lines is done. Hence, angles have a totally distance- motivated definition.
2. Parallel lines- I want to give an intuitive feel for how parallel lines are constructed. Take two intersecting lines. Now move one line (along the base line) in such a way that the distances between points remain the same (find a reasonable way to parse this statement). Then the initial line and the moved line are parallel, and the angle between the moved line and base line is the same as that between the initial line and base line (as the distances remain the same).
3. Tangent- An imaginary construct. A line that touches a curve only at one point. Impossible to construct by hand. But then again, a point would be an imaginary construct too. We’re talking about limits here anyway.
4. Area and volume- The formulae are only indicative of “how much” matter there is (or capacity for matter); both area and volume are proportional to this “how much”. As the formulae are pretty random, a separation of units does the trick- we can only compare two volumes, and not an area and a volume, although technically we SHOULD be able to compare both as both are representative of the amount of matter (or capacity of a vessel, etc). However, the volume and area formulae are certainly getting something right. If I take two arbitrary vessels of different shapes but same volume (as calculated by the formula), they will contain the exact same amount of water!. Hence, the current formula is clearly proportional to the amount of matter (if such a concept can ever be fully fleshed out in the future).