How would Cartesian Plane look at infinity?

Question

I’m relatively new to mathematics, and I admit I’m still trying to grasp the concept of infinity, so there might be some errors in my thinking.

I understand that $\infty$ is not a real number, which means standard mathematical operations don’t apply to it. I’ve come across the idea that there can be different magnitudes of infinity (please correct me if I’m mistaken).

$$ \infty_a > \infty_b $$

Let’s imagine 2 functions, $f_x$ & $f_y$, of time.

At a constant time, $c$,

$$f_x(c) = f_y(c)$$

From this, we can define two perpendicular line segments, $X$ and $Y$, whose midpoint is a fixed point. Their lengths are determined by $f_x$ and $f_y$, starting at the same time.

Hence, at any time $c$,

$$|X| = |Y| \text{, where } |l| = \frac{\text{length of line segment}}{2}$$

Now, let’s introduce another line, $L$, with the same properties as $X$ and $Y$, but with variable angles between $X$ or $Y$.

So, at a constant time,

$$|X| = |Y| = |L| \text{, where } |l| = \frac{\text{length of line segment}}{2}$$

According to the definition of a Cartesian plane, any line drawn from the midpoint is in the Cartesian plane till infinity. In other words it can’t be bigger than axises in magnitude that are in plane.

If we fix the maximum length of the lines, we can conclude that the plane will have the boundary shape of a circle, as the distance from the origin to the end is equal to $|X| = |Y| = |L|$.

Additionally, if time increases infinitely, we can infer that it will always form a circle.

My question is, based on the above reasoning, can we assert that the coordinate system is circular in 2D and spherical in 3D at infinity? Or is this logic only applicable to real numbers, or can we not be certain?

Answer 1

You're right that in some sense, the plane has the same infinite length in every direction. You could think of it as circular for this reason, but this is dubious because "infinite length" is not precise in a geometric way.

Imagine stretching the plane in the $x$-direction, taking every point $(x,y)$ to the point $(2x,y)$. The resulting set of points is the same as the original set, because every original point $(x,y)$ is obtained from a point $(\frac x2,y)$. So "stretching" does nothing to the plane as a whole. In contrast, a stretched circle is no longer a circle. There's no bounded geometric shape that remains the same when stretched, so it's misleading to think of the plane as having such a shape.

Related topics:

• Measure theory is the standard tool for defining sizes like lengths and areas. There's only one infinite size in measure theory, denoted $+\infty$.
• Cardinality is a different notion of size that can be used to compare infinite sets (giving us "different magnitudes of infinity"), but it's not useful in geometry: a small line segment has the same cardinality as the entire plane.
• Topology is a tool for studying how a space "fits together" without using geometric measurements. The infinite plane, the interior of a circle, and the interior of a rectangle all have the same topology.

Answer 2

About "we can conclude that the plane will have the boundary shape of a circle, as the distance from the origin to the end is equal to $|X| = |Y| = |L|$".

This is true for the usual norm, i.e. the $L_2$ norm. But however important is this norm for geometry, in practice other norms are also used: $L_1$, $L_{\infty}$, the pseudo-norm $L_0$, and in general the $L_p$ norms with $p \in \mathbb R_{\gt 0}$. Cf. as usual the relevant Wikipedia page. These norms are much in use in applied maths: operational research, probability theory, data science, A.I., and in some functional analysis.

With these norms, the shape of the unit ball is: a square with its edges parallel to the axes for $L_{\infty}$, a "diamond" i.e. a square with its edges parallel to the diagonals for $L_1$, various shapes intermediate between circle and square for $L_p$ with $p \gt 2$, between diamond and circle for $1 \lt p \lt 2$, between cross and a diamond for $0 \lt p \lt 1$. Cf. Unit Ball with p-norm.

This being said, when considering infinity, whether it is in $\mathbb R$ or $\mathbb R ^2$ or $\mathbb R ^n$, one has to introduce a structure through which infinity is looked at, because infinity in itself is not defined.
For example it can be functions: we look at the behavior of $f(x)$ when $\lVert x \rVert \to +\infty$; and in the case of $\mathbb R ^n, n > 1$, we have to specify the trajectory of $x$ when $\lVert x \rVert \to +\infty$. This enables to say that all conics, in projective geometry, are just ellipses, that happen to cross the line at infinity (for hyperbola) or be tangent to it (for parabola). Note that in this case the "edge at infinity" is not a circle, but a circle with its opposite points glued together: the real projective plane.

Instead of gluing (= making same, using equivalence classes) opposite points of each set of parallel lines, you may want to keep two points for each set of parallel lines, one for each direction. In which case the edge at infinity looks like a circle; although without any metrics defined upon it so far, so that we could as well name it a square, or whatever other Jordan curve, and especially the $L_p$ unit balls shape.
Or you may want to glue together all points at infinity, which means, for example when studying functions $\mathbb R \to \mathbb R ^2$, making a single equivalence class of all functions such that $\lVert f(x) \rVert \to +\infty$ when $x \to +\infty$. In which case the overall shape of this extended cartesian place is that of a sphere.
Or you may want to glue sets of parallel lines so as to generate the two other usual topologies (after sphere and projective plane): torus and Klein bottle.

Or you may want to use other criteria, to define infinity in the cartesian plane, than parallel lines, e.g. algebraic curves with whatever equivalence classes may fit.

In summary: the view that the cartesian plane looks like a circle at infinity requires a succession of specific choices:

• We are interested by lines, i.e. we will use lines to analyze whatever we want to analyze at infinity (e.g. it may be conics). We might have chosen a broader selection of curves, or other techniques.
• We consider as equivalent any parallel lines, and we consider their two directions as two separate points at infinity. Note that the most usual plane completion at infinity, the real projective plane, which has plenty of applications (e.g. optics), considers the two directions of parallel lines as the same point at infinity.
• We consider these points at infinity to form a circle. There should of course be a precise definition of what we mean by that, but it would probably include some kind of norm, and the circle is a shape generated by the $L_2$ norm; other norms generate other shapes.