What are the types of finite geometry?

There are two main kinds of finite plane geometry: affine and projective. In an affine plane, the normal sense of parallel lines applies. In a projective plane, by contrast, any two lines intersect at a unique point, so parallel lines do not exist.

How many lines are there in desargues geometry?

ten lines
The ten lines involved in Desargues’s theorem (six sides of triangles, the three lines Aa, Bb and Cc, and the axis of perspectivity) and the ten points involved (the six vertices, the three points of intersection on the axis of perspectivity, and the center of perspectivity) are so arranged that each of the ten lines …

Why is desargues theorem important?

Desargues’ theorem strikes us as remarkable because it identifies something common to the three points L, M and N – namely, that they lie on the same line. (Of course, any two points are collinear, but here we have three points on the same line.)

What is the order of a finite geometry?

The order of a finite affine plane is the number of points that lie on each line. It is not difficult to prove that in a finite affine plane if one line has n points on it, then all the lines must have exactly n points on them. Furthermore, the number of lines which go through each point must be exactly n + 1.

What was invented before finite geometry?

Analytic geometry was invented before the development of finite geometries.

What is Fano’s geometry?

Fano’s geometry is a finite geometry attributed to Fano from around the year 1892. This geometry comes with five axioms, namely: 1. There exists at least one line. For two distinct points, there exists exactly one line on both of them.

Is four line geometry self dual?

It should be clear that in order for a geometry to possibly be self- dual, it is necessary (but not sufficient) that the number of points equals the number of lines. So, we can say without any work that the four line geometry, the four point geometry and Young’s geometry can not be self-dual.

What is the dual of desargues Theorem?

Desargues’ Theorem: If two triangles are perspective from a point then they are perspective from a line. The dual statement of “two triangles are perspective from a point” is “two triangles are perspective from a line” and vice versa.

Who discovered Euclidean geometry?

mathematician Euclid
Euclidean geometry, the study of plane and solid figures on the basis of axioms and theorems employed by the Greek mathematician Euclid (c. 300 bce). In its rough outline, Euclidean geometry is the plane and solid geometry commonly taught in secondary schools.

What is finite projection?

A finite projection is semi-finite. A purely infinite project ion is properly infinite. An abelian projection is finite. The chapter remarks that the projection 0 can be simultaneously finite, semi-finite, properly infinite and purely infinite.