Soft Question - Definition of the Real Plane

Question

I have only seen the real plane to be defined as $\mathbb R \times \mathbb R$.

In Euclidean geometry, the origin does not play any central role, but in this set theoretic definition, it does. Is there a way to define the real plane in a way that does not give any special importance to the origin, that is, is there a way to defined the real plane without using coordinates (or products)?

Answer 1

Euclid never defined the plane in his elements, and indeed never used numbers to measure length, angles, or area. In this sense, the Euclidean plane is not the same as the Cartesian plane. Euclid used a plane to mean a flat, two-dimensional surface that extends infinitely far, independent of numerical values. This definition gives no importance to the origin, as indeed there is no origin.

Answer 2

I think what you are interested in is exactly the difference between an affine space and a vector space. Read the first few paragraphs in that wikipedia article. Basically, you get rid of the "specialness of the origin" by thinking of two copies of the plane. One represents the group, $G$, of translations and the second is the geometric manifold, $X$, that represents your space. For any two points $x$ and $y$ in your space $X$, there is a unique translation $g$ in $G$ that takes $x$ to $y$. Thus, seen in this way, there are no "special" points in $X$. Notice that the origin is special in $G$ since it represents the identity translation, i.e. the one that keeps $X$ fixed. An even more general concept is that of a principal homogenous space.

Answer 3

Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie

Hilbert's axiom system is constructed with six primitive notions: three primitive terms:[5]

point;
line;
plane;

and three primitive relations:[6]

Betweenness, a ternary relation linking points;
Lies on (Containment), three binary relations, one linking points and straight lines, one linking points and planes, and one linking

straight lines and planes; Congruence, two binary relations, one linking line segments and one linking angles, each denoted by an infix ≅.

Line segments, angles, and triangles may each be defined in terms of points and straight lines, using the relations of betweenness and containment.[how?] All points, straight lines, and planes in the following axioms are distinct unless otherwise stated.

https://en.wikipedia.org/wiki/Hilbert%27s_axioms