Incidence Geometry
Essay Preview: Incidence Geometry
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When one thinks about geometry one usually thinks about a two column proof. However geometry goes way beyond that. Incidence geometry deals with axioms, theorems, proofs and the undefined terms: point, line, and lie on. “Incidence geometry is essentially geometry based on the first postulate in Euclids The Elements” (Chi). In 300 BC, Euclids, The Elements made Euclid one of the most important mathematicians in the world. He created a base for other mathematicians to build from and it resulted in a “Geometry explosion” in the 19th century (Hollyer). Incidence geometry deals with the incidence between objects like points and lines (Balbiani). This means that you can always prove a theorem with a previous theorem and the axiomatic system.
Incidence geometry contains an axiomatic system that has three very important axioms for incidence geometry. The three axioms are as follows:
Incidence Axiom 1: For every pair of distinct points P and Q there exists exactly one line l such that both P and Q lie on l.
Incidence Axiom 2: For every line l there exist at least two distinct points P and Q such that both P and Q lie on l.
Incidence Axiom 3: There exist three points that do not all lie on any one line.
Models that interpret these three axioms are models of Three-point geometry, Four-point geometry, Five-point geometry, Fanos geometry, the Cartesian plane and the Klein disk. All these geometries show that the axiomatic system is true when drawn out. Lets take a look at a few examples:
Three point Geometry :
Five Point Geometry : Fanos Geometry:
The three models pictured above hold true for all the axioms in the axiomatic system therefore they are all models for incidence geometry. They are made up of points and lines, like mentioned before, is the biggest part of incidence geometry. They all contain a specific number of points and lines, and must always contain this same number for each model. If you notice, the Fanos geometry has a circle in the middle and one would think it doesnt hold up to the axiomatic system because it is a circle and not a line. However, in incidence geometry a line is made up of points and never specifies on direction therefore a circle is just a rounded line.
Two examples of theorems in Incidence geometry that can be proven by the Axiomatic system are as follows:
Theorem 2.6.5: If l is any line, then there exist lines m and n such that l, m, and n are distinct and both m and n intersect l
Proof: Let P and Q be any two distinct points on line l (Axiom 2). Let R be any point not on line l (2.6.3).