Easy, early proofs in Young's geometry

The axioms for Young's geometry (finite affine plane of order 3) appear as follows in the text:
  1. There exists at least one line.
  2. There are exactly three points on every line.
  3. Not all points are on the same line.
  4. There is exactly one line on any two distinct points.
  5. For each line l and each point P not on l, there exists exactly one line on P which is not on any point of l.

Proposition 1. For every point, there is a line not on that point.

Proof. Let p be a point. We will find a line l which is not on p. By Y1, a line l' exists. If p is not on l', we are done: take l=l'. If p is on l', then by Y2 there are two other points, p1 and p2, on l'. By Y3, there is a point q not on l'. By Y4, there is a line l'' on q and p1. The point p is not on l'' because if p and p1 were both on l'', then by Y4, l'' would be l', but, by assumption, q is not on l', while q is on l''. Take l=l''. QED

Proposition 2. For every point, there are at least four lines on that point.

Proof. Let p be a point. Using Prop. 1, let l be a line which is not on p. By Y2, there are 3 points, p1, p2, and p3, on l . By Y4, for each pair {p,p1}, {p,p2}, and {p,p3}, there is a line li , i=1,2,3, on {p,pi }. No two of these lines are the same, since, by Y4, if pi and pj are on a line, that line must be l , but l is not on p. There is a fourth line, l4, on p, by Y5. QED

Proposition 3. For every point, there are exactly four lines on that point.

Proof. Let p be a point. By the proof of Prop. 2, there is a line l not on p, lines l1,l2, and l3 each on p and each intersecting l , and a line l4 on p parallel to l . There can be no fifth line on p which is parallel to l , since, by Y5, l4 is the only line with these properties. There can be no fifth line on p which intersects l , since, for instance, a line on p and p1 must be l1, by Y4, and similarly for p2 and p3. QED

Proposition 4. In Young's geometry, each line is parallel to exactly two lines.

Proof. Let l be a line. By Y3 it's on exactly 3 points, p1, p2, and p3. By Prop. 2 there are 3 other lines on p1: call them l1, l2, and l3. By Y3, the line l1 is on two other (by Y4) points, q1 and q2. By Y5, for i=1,2, there is exactly one line mi on qi such that mi is parallel to l . It's easy (but necessary) to show that m1 and m2 are distinct. So there are at least two parallels to l . If there were another line m3 parallel to l , then by Y5, l would be the unique parallel to m3 on the point p1. Then since l1 is on p1, l1 must intersect m3. The point of intersection is q1 or q2. But then, by Y4, m3 would be either m1 or m2, a contradiction. QED

Proposition 5. In Young's geometry, there are exactly 12 lines.

Proof. By Y1 there is a line l , which is on three points p1, p2, and p3, by Y3. On each of these points there are exactly three other lines by Prop. 3. No two of these additional lines can be the same, since none may be on two points of l . So far we have 10 lines, and there can be no other line which intersects l . By Prop. 4, there are exactly two lines parallel to l . Any 13th line must either intersect or be parallel to l , but this is impossible as noted. QED

Proposition 6. In Young's geometry, there are three lines, no two of which intersect.

Proof. We know there is a line l , and lines l1 and l2 parallel to l (by Y1 and Prop. 4). If l1 intersected l2 at a point p, there would be two parallels to l through p, contradicting Y5. QED

Proposition 7.In Young's geometry, there are exactly 9 points.

Proof. Let l1 (on p1,p2,p3), l2 (on p4,p5,p6)), and l3 (on p7,p8,p9)), be three lines as in Prop. 6. If there were a 10th point, q, then by Y4 there is a line m on q and p1. Since we already have a line through p1 parallel to l2 , and a line through p1 parallel to l3 (in both cases, it's l1), m must be on points of l2 and l3. This would force m to have at least four points, violating Y2. QED

Proposition 8. Given any three mutually parallel lines in a Young's geometry, every other line intersects all three.

Proof. Exercise.

Len Smiley
Last modified: Tue Jan 23 11:02:08 AKST 2001