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Cosmos and Infinity

AndreyGlasberg

Is the space really unbounded?

Move 1
Andrey
One-point compactification:Rn∪{∞}≅Sn⊂Rn+1Inverse stereographic map:π−1 ⁣:R2→S2∖{(0,0,1)}π−1(u,v)=(2uu2+v2+1, 2vu2+v2+1, u2+v2−1u2+v2+1)Asymptotic boundary:lim⁡∥(u,v)∥→∞π−1(u,v)=(0,0,1)=North Pole\begin{aligned} \text{One-point compactification:}& \quad \mathbb{R}^n \cup \{\infty\} \cong S^n \subset \mathbb{R}^{n+1} \\[6pt] \text{Inverse stereographic map:}& \quad \pi^{-1}\colon \mathbb{R}^2 \to S^2 \setminus \{(0,0,1)\} \\[4pt] \pi^{-1}(u, v) &= \left( \frac{2u}{u^2 + v^2 + 1}, \, \frac{2v}{u^2 + v^2 + 1}, \, \frac{u^2 + v^2 - 1}{u^2 + v^2 + 1} \right) \\[6pt] \text{Asymptotic boundary:}& \quad \lim_{\|(u,v)\| \to \infty} \pi^{-1}(u, v) = (0, 0, 1) = \text{North Pole} \end{aligned}
Move 2
Glasberg

To understand how stereographic projection is related to the topic we need first to define terms. Could you clarify what you mean by infinity in context of cosmos: From geometrical point of view I can think of two concepts: having a boundary? being unbounded? These are distinct concepts in mathematics, so it might be helpful to make sure we’re using the terminology in the same way. A closed disk is bounded and has a boundary — its circumference. An infinite half-plane is unbounded, but it also has a boundary — the straight line that bounds the half-plane. A sphere is bounded but, as a manifold, it has no boundary.

Move 3
Andrey

Illustration of Move 2 — Let us illustrate how these are formally defined - which one applies here?

Closed Disk: D‾(x0,r)={x∈R2∣∥x−x0∥≤r}(r>0,x0∈R2)Half-Plane: H={x∈R2∣a⊤x≤b}(a∈R2∖{0},b∈R)Sphere: S2(x0,r)={x∈R3∣∥x−x0∥=r}(r>0,x0∈R3)\begin{aligned} \text{Closed Disk: } & \overline{D}(x_0, r) = \{ x \in \mathbb{R}^2 \mid \|x - x_0\| \le r \} \quad (r > 0, x_0 \in \mathbb{R}^2) \\[6pt] \text{Half-Plane: } & H = \{ x \in \mathbb{R}^2 \mid a^\top x \le b \} \quad (a \in \mathbb{R}^2 \setminus \{0\}, b \in \mathbb{R}) \\[6pt] \text{Sphere: } & S^2(x_0, r) = \{ x \in \mathbb{R}^3 \mid \|x - x_0\| = r \} \quad (r > 0, x_0 \in \mathbb{R}^3) \end{aligned}
Move 4
Glasberg

Analogy of Move 3

The idea that the universe has literal boundaries feels implausible, because it immediately raises the question of what lies beyond them. Still, as far as I understand, the notion of a bounded universe—specifically, a space whose diameter is finite—doesn’t contradict any established physical principles. It also isn’t supported by observational evidence. This makes it interesting to explore the implications of assuming a compact universe (closed and finite in diameter). And do you think we can assume the universe is simply connected, meaning it has no “holes” or nontrivial topological loops, which would significantly constrain the possible global geometries? It is also worth considering what additional structural assumptions, beyond compactness and simple connectedness, might be reasonable.

Move 5
Andrey

Inversion of Move 4 — Perhaps the most human question here is not about the formal topology as it stands (even if it can be determined), but about our perception. The question of bounds is also a question of our capacity to experience them.

Everyone in Flatland has width and length, but no height whatever.

Carl Sagan, Cosmos
Move 6
Glasberg

If we rely primarily on perception, similar to the limitations of the two-dimensional Flatlander, we can easily drift into subjective interpretations that are difficult to verify or falsify. Bertrand Russell touched upon this distinction in his essay “On the Value of Scepticism”, whose doctrine is “that it is undesirable to believe a proposition when there is no ground whatever for supposing it true.” In other words, while certain perspectives are conceivable, they are most helpful when supported by evidence or consequences that can be tested.

A verifiable alternative is to discuss models of the universe that are at least mathematically consistent and therefore exist, at least in the sense of Plato’s heaven. Such models can be described, analyzed, and compared. The focus then shifts from whether a model can simply be imagined to whether it yields clear, testable implications.

This is where Poincaré’s perspective on topology becomes particularly relevant. Topology asks questions about the global structure of a space, such as whether every closed loop can be continuously contracted to a point. His own formulation in Dernières pensées (1913), that in analysis situs “two figures are equivalent whenever one can pass from one to the other by a continuous deformation,” captures this idea nicely.

So rather than focusing on what the universe looks like from a particular vantage point, we might explore questions such as: What would the universe look like if it were bounded and simply connected, so that every loop could be continuously contracted to a point? And how would its structure differ if that condition did not hold?

Move 7
Andrey

Development of Move 6 — Riemann saw it in 1854: a world can be endless to walk through and still finite to hold. Poincaré tells us what shape that world has to be.

Poincareˊ conjecture (Perelman, 2002–03):M3 compact, ∂M=∅, π1(M)=1  ⟹  M≅S3Bounded means compact (Hopf–Rinow):(M,g) complete, diam⁡(M)<∞  ⟹  M compactMove 1, one dimension up:R3∪{∞}≅S3⊂R4Homogeneous and isotropic, so constant K:K≤0  ⟹  M~∈{R3,H3}  ⟹  ∣π1(M)∣=∞so π1(M)=1  ⟹  K=1/R2>0Testable, if closed:ΩK=−c2H02R2<0Planck 2018 + BAO:ΩK=0.0007±0.0019  ⟹  R≳18 c/H0≈260 Gly (95%)A space that is bounded, has no edge, and lets every loop shrink to a pointcan only be the 3-sphere. That is move 1’s picture one dimension up: ordinary spacewith a single point added at infinity. If you travel far enough in a straight line and youcome back to where you started, from behind.If space is also the same everywhere and in every direction, the sphere must be round,with a curvature we can measure. Planck finds space flat within its errors, so the spherewould have to be enormous: a radius of at least about 260 billion light-years, farbeyond everything we can observe.\begin{array}{l} \begin{aligned} \text{Poincaré conjecture (Perelman, 2002–03):}& \quad M^3 \text{ compact},\ \partial M = \varnothing,\ \pi_1(M) = 1 \;\Longrightarrow\; M \cong S^3 \\[6pt] \text{Bounded means compact (Hopf–Rinow):}& \quad (M, g) \text{ complete},\ \operatorname{diam}(M) < \infty \;\Longrightarrow\; M \text{ compact} \\[6pt] \text{Move 1, one dimension up:}& \quad \mathbb{R}^3 \cup \{\infty\} \cong S^3 \subset \mathbb{R}^4 \\[6pt] \text{Homogeneous and isotropic, so constant } K\text{:}& \quad K \le 0 \;\Longrightarrow\; \widetilde{M} \in \{\mathbb{R}^3, \mathbb{H}^3\} \;\Longrightarrow\; |\pi_1(M)| = \infty \\[4pt] & \quad \text{so } \pi_1(M) = 1 \;\Longrightarrow\; K = 1/R^2 > 0 \\[6pt] \text{Testable, if closed:}& \quad \Omega_K = -\frac{c^2}{H_0^2 R^2} < 0 \\[4pt] \text{Planck 2018 + BAO:}& \quad \Omega_K = 0.0007 \pm 0.0019 \;\Longrightarrow\; R \gtrsim 18\, c/H_0 \approx 260\ \text{Gly}\ (95\%) \end{aligned} \\[14pt] \\ \\ \text{A space that is bounded, has no edge, and lets every loop shrink to a point} \\ \text{can only be the 3-sphere. That is move 1's picture one dimension up: ordinary space} \\ \text{with a single point added at infinity. If you travel far enough in a straight line and you} \\ \text{come back to where you started, from behind.} \\[6pt] \text{If space is also the same everywhere and in every direction, the sphere must be round,} \\ \text{with a curvature we can measure. Planck finds space flat within its errors, so the sphere} \\ \text{would have to be enormous: a radius of at least about 260 billion light-years, far} \\ \text{beyond everything we can observe.} \end{array}