What if mathematics began with the turn?
Circular Mathematics measures in turns, counts rotations, and treats the straight line as a circle of infinite radius.
Everything it claims is checked against established mathematics: it rewrites, organizes and reveals, and never contradicts.
- From above: a circle
- From the side: a sine wave
- Along the axis: a straight line
- about 10 minutesTake the journeyShort, hands-on lessons that build the idea from a single turn.
- the core toolOpen the calculatorType numbers and angles, and read every answer as a turn.
- curves, waves, natureExplore the labsInteractive experiments with spirals, waves, clocks and closing curves.
Sacred Geometry Workshop βThe thesis, its seals and its frontier β
What it is
A mathematical language in which the turn is the primary object: notation, units, types and ways of seeing. Everything it states as mathematics must be true in established mathematics. It rewrites, organizes and reveals, without contradicting.
What it is not
Not a refutation of established mathematics. It does not change the value of any constant. It is not, by itself, a new physics.
Four guiding theses
All four are demonstrable. Each one can be seen at work.
- T1The hidden integersMeasured in turns, the angular invariants of geometry become integers or simple fractions: the tangent of a simple closed curve, traversed counterclockwise, turns exactly 1 turn, a triangle's angles sum to half a turn, a polyhedron's angular defects sum to 2 turns.C-6.1C-6.5See area by the curve β
- T2Ο lives on the borderIn the domain of the turn, the half turn is exact: Β½. The number Ο only appears when a turn is translated into length or area.C-2.2C-7.1See Ο as half a turn β
- T3Rationality is closureTurning repeatedly by p/q of a turn, in lowest terms, closes the cycle in exactly q steps. Turning by an irrational fraction never closes.C-5.1C-5.2See closing curves β
- T4Every number is a turn with a radiusThe spin number rβ³t makes multiplication a scaling followed by a turn, and reveals β1, i, roots and logarithms as turns.C-4.1C-4.4See multiplication as turning β