Division Algebra Visualizer
Mathematics proves exactly four division algebras exist. This page shows what they are, why they matter, and how every key number in the framework traces back to the dimensions 8.
Click any number to see where it appears. Tap the algebra cards for details.
The Four Division Algebras
A division algebra is a number system where you can always divide — no two nonzero elements multiply to zero. Mathematics proves exactly four exist. These aren't chosen; they're the only ones possible.
How the Numbers Combine
The dimensions {1, 2, 4, 8} and their imaginary parts {0, 1, 3, 7} combine to produce every key integer in the framework. No numbers are chosen — they all trace back to the algebras.
From Algebra to Physics
Each division algebra's symmetry group corresponds to a fundamental force. The connection between division algebras and gauge groups is explored by several professional physicists (Furey, Dixon, and others). Here, we push further to derive numerical constants.
The complex numbers have a single imaginary direction. Rotating in that direction gives U(1) — the gauge symmetry of electromagnetism. This is why electric charge is described by a single number.
The quaternions have three imaginary directions (i, j, k). Rotations among these give SU(2) — the gauge symmetry of the weak force. This is why the weak force has three carriers (W+, W-, Z).
The octonions have seven imaginary directions. Their automorphism group contains SU(3) — the gauge symmetry of the strong force. This is why quarks come in three "colors."
Why 4 dimensions? Time evolution must be associative — grouping sequential events differently can't change the outcome. The quaternions are the largest associative division algebra, so spacetime gets 4 dimensions. The non-associative octonions describe internal symmetries instead.
Number Coherence
The same small set of integers appears everywhere. Click any number below to see all the places it shows up. This coherence — the same numbers reappearing in different contexts — is the framework's central structural claim.
Click any number above to see where it appears across the framework.
The Core Claim
The four division algebras {R, C, H, O} with dimensions {1, 2, 4, 8}generate all the structure of the Standard Model: spacetime dimension (4), gauge group (U(1) x SU(2) x SU(3)), fermion content, and numerical constants like the fine structure constant (1/137...) and the Weinberg angle (sin^2 = 28/121). No free parameters — every number traces back to these four algebras.
This is a speculative framework, not established physics. The mathematical connections are genuine; the physical interpretation is what needs external scrutiny.