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Finite Geometries of

Fano and Young

A. FANO’S GEOMETRY

GINO FANO (1871-1952) Fano was an Italian mathematician whose work w a s m a i n l y o n p r o j e c t i v e a n d a l g e b r a i c geometry. Fano was a pioneer in finite people to try to set geometry on an abstract footing. He is best known as the founder of the finite geometry. UNDEFINED TERMS: point, line, and incident Model for Fano’s Geometry

Axiom 1 : There exists at least one line

Axiom 2: Every line of the geometry has exactly three points on it.

Axiom 3: Not all points of the geometry are on the same line

Axiom 4: For two distinct points there exists exactly one line on

both of them.

Axiom 5: Each two lines have at least one point on both of them.

Theorem 1.8: Fano's geometry consists of exactly seven points and

seven lines.

Proof:

Assume that there is an 8th point. By axiom 4 it must be on a line with

point 1. By axiom 5 this line must meet the line containing points 3,

and 7. But the line cannot meet at one of these points otherwise axiom

4 is violated. So, the point of intersection would have to be a fourth

point on the line 3 4 7 which contradicts axiom 2.

B.YOUNG’S FINITE GEOMETRY

John Wesley Young v Mathematics professor at Dartmouth College v Introduced the axioms of projective geometry v H e w a s a p r o p o n e n t o f E u c l i d e a n geometry and held it to be substantially "more convenient to employ" than non- Euclidean geometry.

Axioms of Young’s Geometry

1. There exists at least one line.

2. Every line of the geometry has exactly three

points on it.

3. Not all points of the geometry are on the

same line.

4. For two distinct points, there exists exactly

one line on both of them.

5. If a point does not lie on a given line, then

there exists exactly one line on that point that

does not intersect the given line.

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

To prove:

Let p be a point and by axiom 2 let L be the line

P (point) is not on L (line) thus we say that it satisfies the theorem 1.

Theorem 2: For every point, there are exactly four lines on that point. To prove:

1. Let E be a point Lines: DEF, AEI, BEH and CEG Thus, there are 4 lines on point E
2. Let I be a point Lines: AFI, GHI, CFI and BDI Thus, there are 4 lines on point E

Theorem 5. In Young's geometry, there are exactly 9 points. To prove: 9 points Let L1 (on A,B,C), L2 (on D,E,F), and L3 (on G,H,I), be three lines and every line has three points. Thus, there are exactly 9 points

Summary

Fano’s Geometry

3 1 4 5 0 6 2

q 7 points

q 7 lines

Thank

you!