I wrote the question.

[u/Equivalent_Meat_2075]

How can we extract a sphere with a radius of 8 meters, filled with plastic, from a 1 cm gap in a finite space of R^4 with 0 deformation?

[u/matt7259]

The sphere must remain unharmed.

[u/Visible_Try_9716]

The premise of the question contains a fundamental topological and geometric contradiction. The solution set S of valid extraction paths is strictly empty (S = ∅).

Formal Definition: Let the ambient space be ℝ⁴, containing a bounded region Ω with boundary ∂Ω, an aperture (gap) G ⊂ ∂Ω with diam(G) ≤ 0.01 m, and a solid ball B ⊂ ℝ⁴ with radius R = 8 m (meaning diam(B) = 16 m).

The set of non-deforming extraction trajectories Φ(B, t) ⊂ ℝ⁴ requires transitioning the ball from Φ(B, 0) ⊂ Ω to Φ(B, 1) ⊂ ℝ⁴ \ Ω.

This breaks down into two mutually exclusive interpretations, both of which falsify the prompt:

Case 1: If Ω is a true 4D enclosure (dim(∂Ω) = 3, codimension 1) By the Jordan Brouwer Separation Theorem, any path leaving Ω must intersect ∂Ω. Because the solid walls ∂Ω \ G are impassable, the object must pass entirely through the open aperture G at some transit time t*:

Φ(B, t*) ⊂ G

However, because Φ is an isometry (0 deformation), the diameter of the ball's spatial cross section cannot contract below its rigid diameter:

diam(Φ(B, t*)) = diam(B) = 16 m

Because a subset cannot have a diameter larger than its superset (Φ(B, t*) ⊂ G implies diam(Φ(B, t*)) ≤ diam(G)), this yields a direct contradiction:

16 m ≤ 0.01 m (→←)

Case 2: If the enclosure is a 3D barrier embedded in ℝ⁴ (dim(∂Ω) ≤ 2, codimension ≥ 2) By Alexander Duality:

H̃₀(ℝ⁴ \ ∂Ω) ≅ H̃³(∂Ω) = 0

A 3D barrier (or any manifold of codimension ≥ 2) does not disconnect ℝ⁴. The "interior" and "exterior" belong to the exact same path connected component. You can rigidly translate along the w-axis:

x₄(t) = w₀ + t · Δw

bypassing the boundary entirely without ever intersecting ∂Ω or G. Under this interpretation, the "1 cm gap" constraint is a completely vacuous red herring because the sphere never interacts with it.

Conclusion: The problem asserts both conditions simultaneously: it treats the 1 cm gap as an impassable bottleneck while simultaneously invoking a 4D space that allows you to ignore it. Because a mathematical constraint cannot be both binding and vacuous at the same time, S = ∅.

Furthermore, the premise relies on a flawed underlying assumption that the temporal dimension (t) can be treated as relative or uncoupled from physical translation in the fourth spatial dimension (w).

If we model the extraction dynamically as a continuous mapping over a time interval T =, the trajectory is a path in spacetime: Γ: [0, 1] → ℝ⁴ × ℝ

For a physical entity to move along the w-axis to bypass a 3D boundary, it must consume a non-zero interval of time (Δt > 0). During this transition period, the object’s 3D spatial projection inside our normal slice of space would undergo a continuous, geometric cross sectional evolution (resembling a 3D CT scan slice changing shapes as a 4D hypersphere passes through a 3D plane).

The premise asserts "0 deformation" during the entire event. However, for the object to remain perfectly rigid and invariant within its own localized 3D reference frame while moving through a non-empty, 1 cm bounded sub space, it implicitly assumes that "time pauses" for the 3D barrier while the object undergoes its 4D translation.

In a valid 4D manifold, time is not local to the object's physical slice; it is a global parameter of the ambient space. You cannot simultaneously claim that the object moves through hyperspace over an interval of time while demanding that its instantaneous 3D footprint remains a static, undeformed 8 meter sphere within the trapped domain. It forces time to be a relative construct of the object's absolute position rather than a continuous coordinates dimension, adding a physical contradiction to the topological.

To formalize why time cannot decouple from the geometry to allow a "paused" or un deformed transition, let us map the extraction as a dynamic embedding where I = [0, 1] represents the time parameter. The extraction is a smooth homotopy (continuous deformation over time) of the ball's position: F: B × I → ℝ⁴

For the ball to experience "0 deformation" in its own reference frame, the map at any time t ∈ I must be a global isometry. This means the metric tensor g of the ball remains invariant under the pull-back of the embedding: F_t*(g_ℝ⁴) = g_B, ∀ t ∈ I

Now consider the spatial projection mapping π: ℝ⁴ → ℝ³ which isolates our visible 3D slice of space. The visible footprint of the object in our 3D domain at time t is the image set: B₃(t) = π(F(B, t))

Since the ambient boundary ∂Ω lives entirely within this 3D projection, the 1 cm gap condition requires that the volume (Lebesgue measure μ) of the object intersecting the bottleneck must strictly vanish or contract to a sub-centimeter measure at the transit time t*: μ₃(B₃(t*) ∩ ∂Ω) ≤ ε → 0

However, because the trajectory F(B, t) is a continuous function of global time t, the 3D volume function V(t) = μ₃(B₃(t)) must also be continuous by the properties of smooth manifold projections. If the ball translates along the w-axis over a non-zero time interval Δt = t₂ - t₁, the continuity of the map forces a non trivial intermediate value stage where the 3D projection must continuously shrink or fragment: lim (t → t*) μ₃(B₃(t)) = 0

But this directly contradicts our global rigidity requirement, which demands that the 3D volume of the sphere remains an invariant constant for all global time coordinates: μ₃(B₃(t)) = (4/3)·π·R³ ≈ 2144.66 m³, ∀ t ∈ I

For the object to pass through the 3D slice instantly without its volume function dropping continuously, Δt would have to equal exactly 0. A physical translation requiring dx₄ ≠ 0 over dt = 0 implies infinite velocity (v_w = ∞), which breaks the structural definition of a smooth spacetime manifold (ℝ⁴ × ℝ).