Real projective plane

In mathematics, the real projective plane is a special kind of space. It is like the flat plane we know, but it works a little differently.

Unlike the regular Euclidean plane, the real projective plane does not use ideas like distance, circles, angles, or parallel lines. This helps us study how shapes and lines relate to each other from different angles.

One big difference is that in the real projective plane, any two lines will always meet at a point. This is different from our normal world, where some lines may never meet. This idea comes from how we see things in pictures and photographs, where lines and objects can appear to meet from certain views.

We can imagine the real projective plane using lines that all pass through a single point in three-dimensional space. These lines are like the "points" of the projective plane. There are many ways to build this shape, such as connecting the edge of a Möbius strip to itself or joining opposite sides of a square in a special way. However, this shape cannot be placed in our normal three-dimensional space without overlapping itself.

The fundamental polygon of the projective plane – A is identified with A and B is identified with B, each with a twist

The Möbius strip – because of the twist between the identified red A sides of the square, the dotted line is a single edge

Examples

Projective geometry studies shapes in new ways. The real projective plane can be shaped and placed differently in regular space. One key idea is that the projective plane cannot fit perfectly into three-dimensional space without overlapping itself.

We can imagine the projective plane as half of a sphere where points on the edge that are opposite each other are the same. Another way to show the projective plane is with a special surface called Boy's surface, which fits into three-dimensional space but touches itself in some places. There are also flat drawings and mappings of the projective plane that help us understand it better. One way is to glue a flat circle to a special shape called a cross-cap, which creates a surface that matches the real projective plane.

Figure 1. Two views of a cross-capped disk.

Figure 2. Two views of a cross-capped disk which has been sliced open.

Figure 3. Two alternative views of a self-intersecting disk.

Homogeneous coordinates

Main article: Homogeneous coordinates

In math, we can describe points on a special flat space called the real projective plane using something called homogeneous coordinates. These coordinates look like [x : y : z]. If we multiply all three numbers by the same non-zero value, we still get the same point.

Points where the last number is 1 are the regular points we know. Points where the last number is 0 are special points called “points at infinity.” These points at infinity form a line called the “line at infinity.”

Lines in this space can also be described using similar coordinates. A line is given by three numbers [a : b : c], and a point [x : y : z] lies on this line if ax + by + cz = 0. This helps us see how points and lines are related in new ways.

When we study lines with these coordinates, we find that sometimes a point can act like a line, and a line can act like a point. This idea is called duality. It shows that the rules for points and lines are closely connected. Studying one helps us understand the other.

Embedding into 4-dimensional space

The real projective plane can be placed in a space with four dimensions. It is made from a special shape called a two-sphere by connecting each point to its opposite point. This shape can be turned into points in four-dimensional space without overlapping. This special placement can also be shown in a three-dimensional space called the Roman-surface.

Higher non-orientable surfaces

When we connect projective planes together, we make special surfaces that do not have a consistent direction. We do this by cutting a small circle from each surface and joining the edges together. If we join two projective planes like this, we get a shape called the Klein bottle.

The article on the fundamental polygon talks more about these special surfaces.