Real projective plane

In mathematics, the real projective plane is a special kind of space that is similar to the flat plane we are used to, but it works a little differently. Unlike the regular Euclidean plane, the real projective plane does not have ideas like distance, circles, angles, or parallel lines. This makes it very useful for studying how shapes and lines relate to each other when viewed from different angles.

One of the biggest differences is that in the real projective plane, any two lines will always meet at a point, even if they seem to be running side by side forever in our normal world. This idea comes from how we see things in pictures and photographs, where lines and objects appear to meet when we look at them from certain points.

We can imagine the real projective plane using lines that all pass through a single point in three-dimensional space. These lines can be thought of as 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 looks at shapes in new ways. The real projective plane can be shaped and placed in different ways in regular space. One important idea is that the projective plane cannot fit perfectly into three-dimensional space without overlapping itself.

We can think of the projective plane as half of a sphere where opposite points on the edge are considered the same. Another way to show the projective plane is through 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], where multiplying all three numbers by the same non-zero value gives the same point. Points where the last number is 1 are the regular points we’re used to, while 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 understand how points and lines relate in new and interesting ways.

When we look at lines using these coordinates, we find that what seems like a point can sometimes act like a line, and vice versa. This idea is called duality. It means that the rules for points and lines are deeply connected, and studying one helps us understand the other.

Embedding into 4-dimensional space

The real projective plane can be placed inside a space with four dimensions. It is created by taking a special shape called a two-sphere and following a particular rule that connects each point to its opposite point. There is a way to turn this shape into points in four-dimensional space, and this turning keeps the shape intact without overlapping. This special placement can also be shown in a three-dimensional space known as the Roman surface.

Higher non-orientable surfaces

When we connect projective planes together, we create special surfaces that are not oriented in a consistent direction. This is done by cutting a small circle from each surface and then joining the edges together. If we join two projective planes in this way, we get a shape called the Klein bottle.

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