iSoul Time has three dimensions

Circular/harmonic motion

The outline below is also in pdf form here.

Angular speed (velocity) and angular pace (legerity)

  • Speed, v = Δst, Pace, u = Δts so u = 1/v and v = 1/u except if u or v are zero
    • Zero speed: no motion but time changes because time is independent
    • Zero pace: no motion but length changes because length is independent

Circular motion in space and time

distance, x; distime, t; radius r (or a or R or A); period radius q; circumference S = 2πr = wavelength λ; period T = 2πq = wavetime μ; angular velocity, v; angular legerity, u; arc length, s; arc time, w

  • Circle in space
    • space angle θ, arc length s, radius r
    • angle in space: θs/r; r = s/θ; s = rθ; 1/θ = r/s; 1/r = θ/s
    • angular time rate: ωθ/t; t = θ/ω; θ = ωt; 1/ω = t/θ; 1/t = ω/θ
  • Circle in time
    • time angle φ, arc time w, period radius q
    • angle in time, φw/q; q = w/φ; w = ; 1/φ = q/w; 1/q = φ/w
    • angular space rate: ψφ/x; x = φ/ψ; φ = ψx; 1/ψ = x/φ; 1/x = ψ/φ

Angular time rates

  • Independent variable is time, dependent variable is length
  • Angular velocity: time rate of rotation, ωθ/t t = θ/ω; θ = ωt
  • Wave speed: wavelength per unit time, v = s/t = S/T = r/q = ωr
    • frequency, f ≡ 1/T = v/S = v/λ = v/s = vh = h/u
    • wavelength, λ = v/f
  • Wave speed normalized
    • revolutions: If S = 1, then v = 1/T = f
    • space radians: If r = 1, then s = θ = φ and v = θ/t = φ/t = s/t = ω = 1/q = 2π/T = 2πf
      • ω = 2πf = 2π/T = θ/t = φ/t; q = T/2π; f = ω/2π

Angular space rates

  • Independent variable is length, dependent variable is time
  • Angular legerity: space rate of rotation, ψφ/x; x = φ/ψ; φ = ψx
  • Wave pace: wavetime per unit length, u = w/s = T/S = q/r = ψq
    • periodicity, h ≡ 1/S = u/T = u/μ = s/v = uf = f/v
    • wavetime, μ = u/h
  • Wave pace normalized
    • revolutions: If T = 1, then u = 1/S = h
    • time radians: If q = 1, then w = φ = θ and u = φ/x = θ/x = w/s = ψ = 1/r = 2π/S = 2πh
      • ψ = 2πh = 2π/S = θ/x = φ/x; r = S/2π; h = ψ/2π

Parametric equations

|x| = x = √(x1² + x2²)                   |t| = t = √(t1² + t2²)

x(θ) = r cos(θ) i + r sin(θ) j
x(t) = r cos(ωt) i + r sin(ωt) j; v(t) = ωr; a(t) = −ω²x(t)
x(s) = r cos(s/r) i  + r sin(s/r) j

t(φ) = q cos(φ) i  + q sin(φ) j
t(x) = q cos(ψx) i  + q sin(ψx) j; u(x) = ψq; b(x) = −ω²t(x)
t
(w) = q cos(w/q) i  + q sin(w/q) j

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