The diagram above (which looks a little like Homer Simpson) shows you how to build a 2-dimensional universe from 1-dimensional universes. There is a lot in the diagram so we will go over it a piece at a time then you can use your imagination to build another universe.

Start with the original 1-dimensional line, the one that passes through the points labeled O, and O+ on the diagram. Remember that we need only 2 points on the line, any 2 of them will generate the whole line. We then need to find another point, labeled P, that is not on the original line. The lines radiating out of the point P represent all the lines connecting P with all the points on the original line “O”.

Each one of the line segments between point P with a point on the line spans a different distance. We find a line of symmetry with respect to P by finding the segment with the shortest distance. This segment goes to the point labeled O. Point O is an important point of reflection, a point of symmetry on the line with respect to P, any point on the positive side of O and its reflection on the negative side of O has the same distance to Point P.

We use the label “r” to refer to the distance between point P and point O. In math language, the symbol “r” reminds us of a “radial” distance from the point P. This r distance is referred to as the “distance” between P and the line. If we put together all the points (on the radial lines) that are a distance r from point P, we form a circle of radius r around P. Notice that since r is the shortest distance between P and the line, this circle only touches the line at point O and cannot reach the line anywhere else. Can you find this circle in the diagram?

The line passing through O and P is a special line of symmetry in this case. It is often called the “normal” of the line that passes through point P. It is also called the perpendicular to the line passing through P. Any set of points on the “positive” side of a line can be reflected onto the “negative” side by finding the perpendiculars passing through the points and then reflecting the points along their perpendicular lines the same distance on the “negative” side of the line. In this diagram, this symmetry is obvious, every point on the right (+) side of the perpendicular (O P) is a reflection of points the left (-) side and vice versa.

In the diagram, we show the point O+ on the positive side of the original line at a distance r from the point O. Its reflection on the negative side is labeled O-. We can easily see that the line (O O+) is also the perpendicular of the line (O P). If we wanted to, we could reflect this whole diagram through the original line and we would get the diagram flipped upside down on the bottom half. For clarity’s sake, I didn’t do this in the diagram.

You might be starting to see the amazing symmetries represented in this diagram. It gets even more obvious when you look at the midpoint between O+ and P. This is labeled M+. Its reflection is labeled M-. If we draw a line segment between M+ and M-, as shown in the diagram, notice that it is also perpendicular to the line of symmetry. To show this better I drew a circle around point M+ and M- that touches the perpendicular line at one point. It also becomes obvious that M+ and M- are the same distance from the original line O O+, by symmetry. All distances are preserved upon reflection.

Alright, now we use our imagination to see what would happen if we drew another perpendicular to the original line through point M+ and another through M-. We can use the small circles drawn to visualize where the circle touches the line to draw these perpendiculars. We can now reflect everything in the middle of these perpendiculars on either side. Like two mirrors facing each other, the reflections go on forever in both directions.
From these symmetries, the line M- M+ can be shown to always be the same distance away from the original line forever in both directions. These lines will never cross.

We have finally found two parallel 1-dimensional spaces that do not intersect each other. Do you think you could find an infinite number of parallel lines in the vertical direction by using reflections?

This is a concept utilized by the mathematician Rene Descartes. He built what is known as the Cartesian plane, composed of infinitesimally close parallel vertical and horizontal lines all normal to each other. These concepts give us many ways to look at 2-dimensional space. One of the most important qualities of this space is that it is assumed to be exactly the same in all directions. Very flat indeed.