Circular Mathematics
A laboratory for a mathematics built on rotation and circular motion.π rad = half a turn · 2π rad = a full turn · positive ↺ counter-clockwise · negative ↻ clockwise
definitionL1 · Circular Core
Circular ↔ Traditional
One angle, three languages. Keep the accumulated journey separate from the position on the circle — and nothing gets lost in translation.
tr
circular measure
0.375 tr
radians
2.356194 rad
degrees
135°
lifted angle
0.375=0+0.375
winding 0phase [0, 1) 0.375
Keeping only phase locates the point, but discards winding. Preserve both whenever the journey matters.
position on S¹
0.375 tr
orientation
↺ counter-clockwise
π rad
0,5 tr
scalar π
≈ 3.141593
π remains the traditional scalar C/d. Only the angular quantity π rad translates to 0.5 tr.
computed resultL3 · S¹
Phase Algebra Observatory
Linear distance can look large at the seam. Circular distance sees the short path across zero.
Δ(a,b) ∈ (−0,5; 0,5]
a ⊕ b
0
oriented Δ
0.1
geodesic d
0.1
circular mean
0
Linear gap = 0.9 · circular distance = 0.1. Same data, topology made explicit.
Circular definition
phase(t) = ((t mod 1) + 1) mod 1t = winding + phaseExact bridge
rad = 2π · trgraus = 360 · trWhat this does not know
A winding–phase pair reconstructs an angle, not a complete time history. Different paths may reach the same lifted value.