MATH325: Lecture 19


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More Axioms of Taxicab Space

Last class we learned about taxicab geometry and talked about what definitions we might use and what axioms might be true in taxicab geometry.

Given the difficulty associated with defining a line in taxicab geometry, we won't discuss the parallel postulate. However, we can discuss axioms 6 through 8.

How would we define an isometry of taxicab space?

Is it true that given any two points in taxicab space, we can find an isometry that takes one to the other?

Given a point P and two points Q and R so that |PQ| = |QR|, is it true that there's an isometry of taxicab space that fixes P and moves Q to R?

Are there any reflections that are isometries of taxicab space? If so, what are they?

Using linear algebra, we can describe all of the isometries of taxicab space that fix a specified point.

Think of the point that stays fixed as the origin, (0,0). Let your x-axis run east/west from the origin, and your y-axis run north/south. Then you can label the intersections in your street grid with Cartesian coordinates: (0,1) is one block north of (0,0); (1,0) is one block east of (0,0); and so on.

If an isometry has to leave (0,0) where it is, what can it do with the rest of the points in taxicab space? It can move (1,0) to: (1,0), (0,1), (-1,0) or (0,-1). Any other point is too far away from (0,0); isometries have to preserve distance.

It can move (0,1) to (1,0), (0,1), (-1,0) or (0,-1), but whichever of those it moves to it must be the next point either clockwise or counterclockwise from the image of (1,0).

Linear algebra lets us represent all of these possibilites using matrices. To find out where the isometry corresponding to a matrix sends a point, we multiply on the left by a 2 by 2 matrix. The matrix shown below fixes all the points whose x and y coordinates are equal, and swaps the x and y coordinates of all the other points in taxicab space.

[0 1] [a]   [b]
[   ]*[ ] = [ ]
[1 0] [b]   [a]
What are the other isometries that fix (0,0) and preserve taxicab distance? Take some time and make a list of them.

We could use vector addition to describe some isometries that do not fix (0,0). Can we describe all the isometries of taxicab space as combinations of some isometry that fixes (0,0) and a vector addition?


[1 0]   [1  0]   [0 1]   [ 0 1]   [-1 0]   [-1 0]   [0 -1]   [0 -1]
[   ]   [    ]   [   ]   [    ]   [    ]   [    ]   [    ]   [    ]
[0 1]   [0 -1]   [1 0]   [-1 0]   [0  1]   [0 -1]   [1  0]   [-1 0]

Finite Geometry

The plane, the hyperbolic plane, and the sphere all had infinitely many points. In each of these geometries, any pair of points you choose will have a midpoint.

In taxicab geometry, there are many pairs of points that do not have any points between them. Taxicab geometry is discrete while Euclidean, hyperbolic and spherical geometry is continuous.

However, the points in the taxicab plane go on forever. In a finite geometry there are only finitely many points!

A very simple example of a finite geometry consists of the four vertices of a square.  Each pair of vertices determines a line.  In this geometry there are four points and six lines.

What happens to our eight axioms in this situation? It may or may not make sense to talk about isometries; axioms 6 through 8 can often be handled by listing all possible isometries!

With only four points, we're not going to talk much about distance or care much about circles. Again, we could list all the possible circles if we wanted to, and usually we don't want to. Angles are similarly uninteresting, and there's no place to extend lines to. In the end, only the first, second and fifth axioms turn out to be interesting. We also add an axiom to make sure that there are some points in our geometry.

Axiom F1: Any two points determine a line. (Notice that this is more a statement about what lines exist than about what they look like!)

Axiom F3: There exist four points, no three of which belong to the same line. (This forces our geometry to be two dimensional and non-empty.)

Affine Geometry

In the geometry in which all the existing points are the vertices of a square, given a line L and a point P not on L, there exists a unique line through P that doesn't intersect L. We can check the parallel postulate in this geometry!

Axiom F2a: Given a line L and a point P not on L, there exists a unique line through P that doesn't intersect L.

Notice that combining axioms F3 and F2p forces a certain amount of structure on your geometry. If we try to declare that our geometry will have five points A, B, C, D and E and that any three consecutive letters determine a line, we can't find a line through D parallel to the line containing A, B and C. Finding definitions of points and lines that satisfy these axioms is tricky but useful.

Projective Geometry

As far as I'm aware, mathematicians don't study hyperbolic versions of finite geometries. There's no version of axiom F2 that allows multiple parallel lines. However, you can rewrite the finite geometry parallel postulate to be similar to the what we found in spherical geometry:

Axiom F2b: Any two distinct lines intersect in exactly one point.

Finite geometries that obey axioms F1 and F3 and this version of axiom F2 are also tricky to find and useful to have. The smallest and most famous of these geometries is the Fano plane, which consists of seven points and seven lines.

Connections to Abstract Algebra

I said it was "tricky" to find examples of finite geometries. However, many examples of finite geometries are easy to find.

You may remember modular arithmetic from a previous course or from abstract algebra. When calculating modulo n you add, subtract and multiply as usual, but instead of the usual answer your result is the remainder when the result is divided by n. So, when working modulo 5, 3 + 3 = 1.

If n is prime or a power of a prime you can also define division modulo n, and the numbers modulo n form a field. If we put these numbers on the x- and y- axes of a coordinate system, we'll get n2 points which form a discrete geometry that obeys the parallel postulate (Axiom F2a). A similar process can be used to describe a discrete geometry which obeys Axiom F2b.

With so many examples of finite geometries to study, mathematicians still do not know if there are any finite geometries that behave significantly differently from these examples! (See this Wikipedia article for more information.)

Conclusion

Euclid's five postulates form the foundations of modern geometry. From axioms (or postulates), definitions and common notions we prove lemmas, theorems and corollaries. Our ability to prove absolute truths about geometry is limited chiefly by our ability to recognize our own assumptions and errors.