What question did this study set out to answer?

To explain why physical space has exactly three large dimensions using three independent proofs.

March 23, 2026Open Access

BCT Letter 89: Why Space Has Three Dimensions — D4 Triality, the Hopf Fibration, and BCT Compactification

Key Points

To explain why physical space has exactly three large dimensions using three independent proofs.
Derives proofs based on the Hopf fibration, D4 triality, and crystallographic constraints.
Utilizes mathematical representations and symmetries related to SO(8).
Investigates the role of the vacuum structure in determining dimensionality.
Demonstrates that the Hopf fibration leads to compactification at the Planck scale.
Shows that D4 triality enforces the existence of exactly three large spatial dimensions.
Establishes that crystallographic constraints support the three-dimensional model.

Abstract

Letter 89 of the BCT Superfluid Lattice Model programme. Derives from first principles why physical space has exactly three large dimensions, via three independent convergent proofs. (1) Hopf fibration: the OHC vacuum manifold is S³, whose Hopf fibration S³→S² compactifies one dimension at the Planck scale, leaving three observable dimensions. (2) D4 triality: the triality symmetry of SO (8), unique to D4, permutes three representations — 8ᵥ (spatial dimensions), 8ₛ+ (quark generations), 8ₛ- (lepton generations) — forcing exactly three large dimensions by the same theorem that forces three quark generations (Letter 61). (3) Crystallographic: the BCT constraint c/a=√2 sets the 110 direction to Planck-scale length, compactifying it and leaving exactly three large directions. All three arguments converge. Space is three-dimensional because the vacuum is a D4-symmetric superfluid on an S³ manifold with c/a=√2. Companion to Letter 88 (10. 5281/zenodo. 19140621). Prediction #126.

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