Kaluza-Klein Compactified Dimension

Compactification

Extra-dimension radius R 1.00

Observer distance 18

KK mode n 1

Winding speed 1.0x

Perceived shape

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R / distance

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KK mass m_n = n·ħc/R

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Mode gap Δm

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KK mass tower

How it works

The long tube is ordinary 3D space, drawn here as a single spatial line for clarity. Wrapped around every point of that line is a tiny circle of circumference 2πR — a compactified extra dimension, exactly like the Kaluza–Klein proposal that unified electromagnetism with gravity by adding a curled-up fifth dimension. A garden hose viewed from an airplane looks like a 1D line; viewed up close, it is clearly a 2D surface with a circular cross-section. Shrink R, or back the camera away, and the tube collapses visually into a line — the same reason a compact dimension small enough (near the Planck length in the most basic Kaluza–Klein picture) has never been directly observed.

m_n = n · ħc / R      (n = 0, 1, 2, 3, …)
p_extra = n·ħ/R  →  quantized momentum around the circle
E² = p_3D²c² + m_n²c⁴   (each n behaves as a heavier ordinary particle)

Extra-dimension radius R — the size of the curled-up circle. Large R: the tube visibly looks like a tube. Small R: it visually degenerates into a line, mirroring why a small enough compact dimension is undetectable at any given experimental resolution.
Observer distance — moving the "camera" back has the same visual effect as shrinking R: resolution is always about the ratio of R to how closely you can look, never R alone.
KK mode n — the winding number of the highlighted particle's quantized momentum around the compact circle. Each n is a distinct momentum state, which 3D physics sees as a distinct particle of mass m_n = n·ħc/R — the "Kaluza–Klein tower".
Winding speed — animation rate of the highlighted particle's helical path, purely visual.

As R shrinks, the tower's mass spacing Δm = ħc/R grows, pushing every excited KK mode to energies past what current colliders can reach — the standard explanation for why no extra-dimensional partner particles have shown up yet, even though the idea itself is decades old.

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3D "brane" line Compact circle (extra dim) KK-mode particle (n)

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