Problem 34: Synthetic Geometry

Problem 34: Synthetic Geometry.

For this problem let's make a brief detour and recall our definition of a geometry from Problem 1:

We defined a geometry as a non empty set of points and a non empty set of lines, where a line is a given subset of the set of points that contains at least two elements. Geometries can be either finite or infinite depending on whether they have a finite or infinite number of points, but for now, we will be looking mostly at finite geometries.

A given geometry is defined by a set of axioms. Any constructed example or model of the geometry must satisfy each and every axiom.

Following Beutelspacher and Rosenbaum we may give a broader (and more precise) defintion of geometry. Although this new defintion may appear rather technical at first, it opens the door to many interesting geometric models, as you will find out for yourself in this problem.

Definition. A synthetic geometry (or just geometry ) is a pair , where is a set and is a relation on that is symmetric and reflexive:

then

for all

This new defintion is broader than the older one, as we can let denote the collection of points p and lines l of one of our previous geometries, and then let represent the incidence relation. That is, precisely when p and l are incident. It would also make sense that we would also then have , , and .

Another example would be to let represent the 8 vertices, 12 edges, and 6 faces of a given cube, with denoting incidence with closure (here an edge is relatively open and so does not contain its endpoints, and a face is relatively open and does not include its boundary edges or vertices): that is, if element x is contained in the closure of element y, then we will write either or . Since any element x lies in its own closure we can write for all .

In general, will consist of those objects of interest to us in our synthetic geometry, and the relation will describe the realtion between these objects that interest us: we will often refer to this relationship as one of incidence. One such possible example is given by tilings of the real affine plane. Please go now to the wonderful site Escher Web Sketch.

Officially then,

Definition. A tiling of the real affine plane is a set of polygonal regions along with the boundary edges and vertices of these regions such that any planar point belongs to exactly one vertex, edge, or region.

Let us now go on to define two new objects:

Definition. Let be a geometry. A flag of is a set of elements of that are mutually incident. A flag is called maximal if there is no element such that is also a flag.

For example, for our cube geometry above any maximal flag consits of exactly three elements: a face, an edge of that face, and a vertex of that edge.

Definition. A geometry will be said to have rank r if there is a partition of such that any maximal flag intersects each set in exactly one element.

For example, for our cube geometry if we let denote the set of 8 vertices, the set of 12 edges, and the set of 6 faces, then has rank 3.

PROBLEM 34:

(a) Let be the set of vertices, edges, and polygonal regions of a tiling of the real affine plane, and let denote incidence with closure. [Note: any edge is relatively open and does not contain its boundary vertices, and any polygonal region is open and does not contain its boundary edges or vertices.] If element x is contained by the closure of element y, then we will write either or . So is a geometry? Solution.

(b) If the answer to (a) was 'yes', then is a geometry of rank r? If yes, what is the value of r? Solution.

READING PROJECT:

Semi-Regular Tilings of the Plane:

What was the most important thing that I learned?
What is the most important questions that remains unanswered?

You may also enjoy perusing Geometry and Imagination in Minneapolis: (Here is the printable postcript version).

References: Beutelspacher and Rosenbaum.