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This is a follow up to question Is there an effective way to generalize this approach of affinely extending the number line?. This question is about possible ways of extending this system as to go beyond the reals, in particular extending the complex numbers $\mathbb C$ as to get $\hat{\mathbb C}$. So we'll work with Noah's definitions but over the complex numbers and so convergence to be understood as applied to those numbers.

Are the following models compatible with this method?

  1. Square model: add all numbers $a + bi$ where $a,b \in \hat{\mathbb R}$, and stipulate that: $$ a + bi = c + di \iff (a=c \land b=d) $$. In this model parallel lines never intersect at a point at infinity.

  2. Rhomb model: we only add to $\mathbb C$ four distinct signed infinite complex numbers that are "$+\infty, -\infty, +\infty i, -\infty i$", with the following equality rules:

$$ \forall a, b \in \hat{\mathbb R} \, \forall r \in \{-\infty, \infty\}: r + bi \leadsto r \land a + ri \leadsto ri $$. In this model, all parallel lines intersect at some point at infinity.

  1. Cylinder model: We add to $\mathbb C$ all complex numbers $a + \infty i$ for $a \in \hat {\mathbb R}$, and set $ a + (-\infty \times i) = a + (+\infty \times i) = a + \infty i$, for all $a \in \hat{\mathbb R}$, to be especially noticed is that the imaginary number $\infty i$ is an unsigned number; also add all numbers $ \pm \infty + b i$ for all $b \in \mathbb R$; and we keep the same equality rule in 1. To be noted is that we don't have numbers $a + -\infty i; a + +\infty i$ for any $a \in \hat{\mathbb R}$. In this model, no parallel lines intersect at a point at infinity.

  2. The fold model: To $\mathbb {C}$ we only add three distinct infinite complex numbers "$+\infty, -\infty, \infty i$". The last is unsigned. Equality rules are the same in 2. So, Geometrically the $Y$-axis will be wrapped upon the infinite imaginary point $\infty i$ representing $\sqrt{-\infty}$. So, this is visualized by an infinite rhombus folded around its horizontal diagonal such that its upper and lower points overlap each other. In this model, all parallel lines intersect at some point at infinity.

Zuhair
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  • You are forgetting the most important: the completion of $\mathbb{C}$ with a single point at infinity (Cech compactification) with identification with the sphere. – Jean Marie Dec 06 '22 at 20:41
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    @JeanMarie, I know that, this posting comes as a follow up to earlier posting (which included the projective extension) and it is in reference to Noah's answer to the affine extension of the reals, so here we are already trying to extend a system that affinely extend the reals, we are trying to extend it to the complex plane, so we cannot have the one point compactification. – Zuhair Dec 06 '22 at 20:57

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