A different Kind of Geometry

Introduction

So in class we’ve been talking a lot about non-Euclidean geometries recently. Now while both spherical and hyperbolic geometries are pretty different from Euclidean geometry, they all shared some things in common. One such thing was that they were continuous.  So instead of dealing with continuous geometries let’s talk about a finite geometry.

Definitions

The following definitions and theorems, for the most part, were taken from the ARML (American Regions Mathematics League) Novermber 1999 Power Contest titled Twenty-five Point Affine Geometry. There are some modifications.

Definition 1: A point is any letter from A to I. There are nine points in this initial geometry. The points are arranged in blocks as follow:

A B C
D E F
G H I

 

A E I
H C D
F G B

 

(Note that there is more than one arrangement for which all of the definitions would be satisfied)

Definition 2: A line is any row or column in one of the blocks. Therefore, a line contains 3 distinct points. There are only 12 lines and every point is on 4 different lines. For instance, DEF is a line and point E is on the lines DEF, BEH, AEI, and ECG. Note that lines do NOT have endpoints, but rather are cyclical in nature. So in the line ADG, after G we come back to A again.

Theorem 1: Given any two distinct points, there is one and only one line containing both of them. For example, given points E and A, only the line AEI contains them both.

Definition 3: two lines are parallel if they have no points in common. The points are arranged in the blocks so that parallel lines must lie in the same block and both be rows or both be columns.

Theorem 2: Given any two distinct lines, either they are parallel or they have only one point in common.

Theorem 3: Given a line and a point not on the line, there is one and only one line containing the point and parallel to the given line. For instance, given the line DEF and the point H, the line GHI is the only line that is both parallel to DEF and contains H.

Definition 4: Two lines are perpendicular if one of them is a row and the other a column in the same block. For instance, ABC and CFI are perpendicular and intersect, while line ABC and AEI intersect at point A but are not perpendicular.

Theorem 4: Through any point, there is one and only one line perpendicular to a given line. For example, given ABC and C, only the line CFI is perpendicular to ABC and contains C.

Definition 5: The distance between any two points is the least number of steps separating the points. Furthermore, since the lines do not have end points, it is important to note that, for instance, on the line ABC, the distance between A and B is 1, the distance between B and C is 1, and the distance between A and C is also 1 and not 2.  In other words, these lines can be thought of as cyclic. Furthermore row and column distances are not equivalent. In other words, the distance between A and B is not the same as the distance between A and D. The first is equal to 1, while the second is equal to 1’.

Definition 6: Point x is the midpoints of two points a and b if x is on the line containing points a and b and the distance between a and x is the same as the distance between x and b.

Note: two figures are considered different if they have a different set of points.

Definition 7: A triangle is a set of 3 non-collinear points.

Definition 8: A quadrilateral is a set of 4 points. Not 3 of which are collinear

Definition 9: A circle is the set of four points a given distance (called the radius) from a given point (called the center)

In relation to Euclidean geometry

Postulates

The first postulate, namely that a straight line segment can be drawn joining any two points, still holds up in this geometry. This is pretty evident from theorem 1. If you don’t believe that theorem, feel free to actually go through each and every case. It’s finite so you can actually do this.

The second postulate does not hold, namely that you can extend line segments indefinitely into straight lines, unless you apply some weird definitions. We did define our lines as not having end points (see definition 5), but as extending the line further would simply be looping the line around itself over and over it can’t really be extended indefinitely.

The third postulate, namely that given a straight line segment, a circle can be drawn having the segment as radius and one endpoint as center, does hold, provided that we create an intuitive definition of line segment, and that we modify our selection of distances to only include 1 and 1′. Since these are the only two distances within this geometry, we are still choosing from “all possible distances”, there just aren’t very many of them. The line segment will give us the distance which we call the radius and one of the endpoints which we will use at the center, from which we can construct the remainder of the circle.

The fourth postulate, that all right angles are congruent, doesn’t really hold. The issue here is that the term “angle” isn’t defined. We have 3 types of lines: those that intersect but are not perpendicular, those that are perpendicular, and those that are parallel. From this one could attempt to construct 3 classes of angles defined by the intersection types of these lines, but there wouldn’t necessarily be a good way of comparing the angles.

The parallel postulate does hold, but not quite directly. It hold in the sense that Playfair’s axiom, which is equivalent to the parallel postulate, holds, and is actually stated in theorem 3. However while the Playfair’s axiom does hold, Euclid’s 5th postulate, that is two lines are drawn which intersect a third in such a way that the sum of inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough, is once again problematic due to a lack of a definition for the term “angle”.

Other dimensions

So far we’ve stayed within the constraints of these 9 points and two blocks. This should be suspicious to you as the title of the ARML Power Contest was Twenty-five Point Affine Geometry. The original contest took place using 25 points, A through Y, as arranged below.

A B C D E
F G H I J
K L M N O
P Q R S T
U V W X Y

 

A I L T W
S V E H K
G O R U D
Y C F N Q
M P X B J

 

A X Q O H
R K I B Y
J C U S L
V T M F D
N G E W P

Ok so what changed from our earlier dimension to this one?

Well first off, the number of points on a line has been increased from 3 to 5. Furthermore instead of each point being either 1 or 1’ away from each other point, each point is now either 1, 2, 1’, or 2’ away from each other point. The total number of lines goes up to 30 and each point is on 6 lines. Finally, circles have a center and 6 other points instead of 4.

But what entirely new things did we gain by adding this additional dimension?

Well for one thing we now have pentagons! If we define pentagons as a set of 5 points no 3 of which are collinear, then we can in fact construct pentagons on the 25 point geometry, something which could not be done on the 9 point geometry. One instance of a pentagon in this new geometry is ABGFN.

Furthermore, you can also actually make hexagons, provided that we define a hexagon as a set of 6 points no 3 of which are collinear. One such example would be ABFGNX.

We can also introduce the following definitions:

Definition 10: A parabola is a the set of points equidistant from a given point (called the focus) and a given line (called the directrix)

Definition 11: A line tangent to a parabola intersects the parabola at only one point and is not perpendicular to the directrix.  Of the five points of a parabola, one is the vertex and the other four form two pairs of endpoints of a focal chord, which is a cord going through the focus.

Note that these parabolas did technically exist in the 9 point geometry; they just don’t mesh very well with the lines defined in definition 11. For instance, the parabola with a focus at E and with the directrix ADG is defined as the set of {F, I}, as both F and I are the same distance from ADG as they are from E. However, this means that this parabola has no focal chords and we can’t tell which point is the vertex. In the 25 point geometry, by contrast, one example of a parabola is the one with a focus at M and with the directrix of AFKPU. This parabola is defined as the set of {L, J, T} and {D, X}. All 5 of these points are the same distance from the line and from the focus, but what that distance actually is varies, unlike the parabolas in the 9 point geometry. Furthermore, it has a vertex at L and JMPX and TMFD are the focal chords. So while parabolas, from just the definition of a parabola, do exist in the 9 point geometry, they don’t interact very well with the lines associated with parabolas.

Generalized dimensions

We can actually create a geometry of this style for any (2n+1)2 number of points where n is an integer greater than or equal to 1. Since each point needs to be connected by a line to every other point, we can easily figure out how many square blocks we need to make this work. Let s be the number of square blocks. Then s*2*((2n+1)-1) =(2n+1)2-1, where (2n+1)2 is the total number of points.  This is because the total number of connections required per point is given by the right side while the left side is in the form of number of connections per block times the number of blocks. Thus s, the number of square blocks, is equal to n+1. It is also clear from this that you cannot create a geometry similar to these ones with (2n)2 points where n is any integer greater than or equal to 1 because the number of required square blocks would be given by

s*2*(2n-1)=(2n)2-1

where s and n represent the same things as before. Solving for s yields

s=(4n2-1)/(4n-2)

which is clearly not an integer as the top will always be an odd number while the bottom will always be an even number. This means that s will not take on an integer value which means that we need a non-integer number of square blocks, which doesn’t actually make sense.

Symmetry

Look back at the original 9 point geometry we defined. The point E is somewhat unique, in that it occupies the center of the first square. This is not true for any other points…or is it? Recall that lines don’t have end points. This means that if I go up from the point B in the first square, I end up at the point H. Going up from A will place me at G, and up from C will place me at I. You could, in fact, move that bottom row to above the current top row without actually changing anything about the geometry. In this new layout, B is in the center. This process also holds for left to right and so on, so anything that holds for one of the points, holds for all of the points. It may take some rearranging for this to become clear, but it’s definitely possible to do so.

Generalized Dimension properties

So obviously a geometry with (2n+1)2 points will have that many points, and will have the number of squares given by the formulas in the previous paragraph. Lines will contain 2n+1 points and each point will be in 2*s number of lines where s is the number of square blocks. The number of possible distance is given by 2n. A circle will be defined by the radius and 2*s points where s is the number of square blocks. The number of possible circles is given by the number of points multiplies by the total number of possible distances, which results in         (2n)*(2n+1)2 total circles.

To find the largest possible polygon, we use the following process. As a polygon must contain at least one point, we pick a point. The remaining number of points is now given by (2n+1)2 -1. As the polygon must be connected by lines, we will now choose a point that lies on the same line as the first point. Furthermore, as no three points can be collinear, by our definition of a polygon, the other points that were on this line can no longer be considered. Thus choosing this second point removes another 2n points from the remaining points. This means the total number of remaining points is now given by (2n+1)2-1-2n. Each time we choose another point we remove another 2n points from the remaining points. Thus, in general, remaining points    p=(2n+1)2-1-2nk where k is the number of points we have already chosen. By setting p=0, the least possible value p can take on while still making any sense, we can solve for k in any given geometry. So for instance, in our 9 point geometry, n=1, we end up with 0=9-1-2k,  from which we can clearly get that k=4, so the largest possible polygon in the 9 point geometry is a quadrilateral. In the 25 point geometry, n=2, we get 0=25-1-4k, from which we clearly get that k=6, so the largest possibly polygon in the 25 point geometry is a hexagon.

In general, we can solve 0=(2n+1)2-1-2nk to get k=2n+2. Thus the largest possible polygon in a given (2n+1)2 point geometry will have 2n+2 vertices.

Conclusion

In this finite affine (meaning that given two points there is only 1 line between them, Playfair’s axiom holds, and that it is possible to chose 4 points so that no 3 of them are colinear) geometry, there are many interesting properties. Some have been discussed here, like the largest possible polygon in given dimension, while others, like some properties of parabolas, have been left out. This is by no means a full exploration into the properties of this specific geometry, but rather a basic exploration into how some things within this geometry work. You should prove more things about it.

References:

Weisstein, Eric W. “Euclid’s Postulates.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/EuclidsPostulates.html

ARML (American Regions Mathematics League) Novermber 1999 Power Contest titled Twenty-five Point Affine

http://en.wikipedia.org/wiki/Finite_geometry

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