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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)?

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    $\begingroup$ $(0,0)$ is a couple as all others elements of $\mathbb{R}\times \mathbb{R}$. Why you say that this couple play a special rule? $\endgroup$ Dec 6, 2018 at 21:17
  • $\begingroup$ What axioms are you willing to start from? From your question, and from your comments to the various answers, it appears that you are not willing to start from the axioms of the real numbers, nor are you willing to start from the axioms of Euclidean geometry, both of which are fully rigorous axiomatic systems from the modern mathematical point of view. All of modern mathematics starts from some axiomatic system. So, what axiomatic system are you willing to start from? $\endgroup$
    – Lee Mosher
    Dec 10, 2018 at 19:57

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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.

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  • $\begingroup$ The type of this defintion that Euclid used is what I am looking for, except I am looking for a defintion that has actually mathematical meaning rather than just intuitive meaning. $\endgroup$
    – LinearGuy
    Dec 6, 2018 at 22:14
  • $\begingroup$ Euclid's axioms and definitions do indeed have mathematical meaning. This was first shown, with full (modern) mathematical rigor, by Hilbert. If you want to see a recent treatment of Euclid to be convinced of this, one good place to start is the book by Hartshorne, "Geometry: Euclid and beyond". $\endgroup$
    – Lee Mosher
    Dec 10, 2018 at 19:54
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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.

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  • $\begingroup$ Yes, I am more or less looking for a defintion that resembles and affine space, but, as far as I know, one still uses $\mathbb R \times \mathbb R$ and then lets the group act on it. $\endgroup$
    – LinearGuy
    Dec 6, 2018 at 22:17
  • $\begingroup$ OK, well if you already know what an affine space is, I don't know what else can satisfy you. In general, I think it's difficult to define specific "non-special things" since the very act of defining it privileges some choice. The way to get at the "generic" is by looking at all the possibilities at once. For instance, how do you get rid of the special "north pole" point on a sphere? By imagining the set of all rotations you can perform on it, i.e. all the symmetries. $\endgroup$
    – zoidberg
    Dec 6, 2018 at 22:22
  • $\begingroup$ @LinearGuy As far as I can see, the group $G$ is isomorphic to $\mathbb R\times\mathbb R$ but there is no need for $X$ to have intrinsic coordinates. Of course, once you declare that some point in $X$ has coordinates $(x_1,x_2)$ then all the points have coordinates and exactly one has coordinates $(0,0)$--so don't do that. $\endgroup$
    – David K
    Dec 6, 2018 at 22:24
  • $\begingroup$ Furthermore, by viewing $\mathbb{R} \times \mathbb{R}$ as a vector space (and often as an oriented inner product space) you get rid of the "specialness" of vectors pointing "along the $x$-axis or along the $y$-axis" since then no one (orthonormal) basis of $\mathbb{R}^2$ is any better than another. This would reflect the isotropy (direction independence / rotation invariance) of Euclidean-style geometry, just as passing to the affine space reflects the homogeneity (location independence / translation invariance). $\endgroup$ Dec 6, 2018 at 22:43
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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

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  • $\begingroup$ Yes, but he never defined the plane and only uses the intuition of it. $\endgroup$
    – LinearGuy
    Dec 6, 2018 at 22:38
  • $\begingroup$ The lines and points are the plane. With these axioms you can derive all the things Euclid proved, with no appeal to intuition, except to guess what you will be able to prove. $\endgroup$ Dec 7, 2018 at 0:43

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