Axiomatic Euclidean geometry (Affine Geometry)

Geometry Level 5

Let $(A,L)$ be a pair of sets with $A \neq \emptyset$ , where the elemets of $A$ will be called points , and $L$ is a set of subsets of $A$ ( $L \subseteq \mathbb{P}(A)$ ) where elements of $L$ will be called lines .

Then $(A,L)$ is said to be an affine (euclidean) plane if it fulfills the 4 next axioms:

(Axiom I) Each element of $L$ has at least 2 points,i.e, each line has at least 2 points.(Actually, this is not an axiom, it can be deducted of the rest of following axioms.I have just written it as an axiom to expedite the solution. Without this assumption may be a long way until a complete solution)

(Axiom II) There exists at least 3 points in $A$ not aligned, i.es, there exists at least 3 points in $A$ which are not contained in any line.

(Axiom III) For all two distinct points given $a, b \in A$ there exists one and only one element $l\ \in L$ such that $a,b \in l$ ,i.e, for all two distinct points given there is one and only one line that contains both,

(Axiom IV. Euclides's Postulate) Given a line $l \in L$ and a point $a \in A$ , there exists one and only one line $l'$ parallel to $l$ which contains to $a$ . (Two lines are parallel if they are equals or their intersection is the empty set).

Then, if $(A,L)$ is an affine (euclidean) plane, which is the minimum number possible of points in $A$ ?

4 8

\infty

10 9 3 6

1 solution

Guillermo Templado
Sep 8, 2016

How you can see in the above picture, the set of points $A,B,C,D$ with the segments (lines) joing them, $AB , AC, AD, BC, BD, CD$ is an affine plane. Note that $AD$ and $BC$ don't cut each other ("it is a hole"). You can check that they form an affine plane, they fulfills all the axioms. For instance, the parallel line to $AB$ passing through $C$ is $CD$ , the line parrallel to $AD$ passing through $C$ is $BC$ ... (equal for the other lines and points).For each two points there is a line joining them. There are 3 points not aligned, for instance, $A,B,C$ . Furthemore, all the lines have exactly two points. There are 6 lines.... and, it's an affine plane with minimum number of points, given $A, B$ there exists $C$ which isn't in the line $AB$ , and for $C$ there exists one and only line with at least two points ( $CD$ ) containing $C$ which is parallel to $AB$ .

Note.- other example is $\mathbb{R}^2$ with euclidean geometry. This classic affine plane has $\infty$ points and lines... When the axiom IV is replaced,for instance, for each two distinct lines in a plane they cut each other at asingle point, we create the projective geometry... etc

Axiomatic Euclidean geometry (Affine Geometry)

1 solution

0 pending reports