Interactive Geometry · physics.connermccarthy.com

The Dimension Tradeoff

Global dimension (illustrative) 3

Volume left in the inner 90% radius
(10%-thick outer shell)

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…at d = 3, for comparison

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shell (fixed by slider) real core volume (shrinks with d)

Dimension (d)3

Shell thickness (% of radius)10%

Fraction of volume remaining in the untouched core

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interior_fraction = (1 − thickness)^d

man / king woman / queen predicted (king − man + woman)

Cosine similarity: predicted vector vs. actual "queen"

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cos(king − man + woman, queen)

Euclidean residual: –

±θ° band around equator equator / poles

Dimension (d)3

Band half-width (degrees from equator)20°

Fraction of surface area within the equatorial band

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computed by numerical integration of cos(φ)^(d−2) over the band, ÷ over the full sphere

Dimensions

Dimension (d)2

Shortest pass (real, d‑dim)

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Longest pass (real, d‑dim)

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Shortest ÷ longest pass — when this → 1, every player is equally far away

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N players on a d-dimensional field; you hold the ball, distances are real d-dim Euclidean

hit (inside target) miss

Dimension (d)3

Ball radius (% of box half-width)70%

Probability a random dart lands inside the target zone

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p = V_d(r) / 2^d (exact d-dimensional ball volume formula)

Expected tries for one hit: –

Live Monte Carlo: –

Property	Low dimension (2–3)	High dimension (100–1000+)
Volume location	Spread throughout the ball	Concentrated in a thin outer shell
Sphere surface	Spread across all latitudes	Collapsed near one equator
Nearest vs. farthest neighbor	Very different distances	Nearly identical distances
Two random balls overlapping	Common	Vanishingly rare
Independent "directions" available	2–3 (exactly orthogonal)	Astronomically many (nearly orthogonal)
Good for…	Geometric intuition, physical space	Encoding many features with low interference
Bad for…	Representing many independent features	Distance-based search, nearest-neighbor lookup

All visualizations are 2D/3D stylized metaphors; the numbers shown are computed using the real mathematics of n‑dimensional geometry.