pi_over_4_fundamental

formal source

 101
 102    The 8-tick structure gives π/4 = 45° as a fundamental angle.
 103    This relates to the 8th roots of unity. -/
 104theorem pi_over_4_fundamental :
 105    -- π/4 is the 8-tick phase increment
 106    -- This makes 45° special in RS geometry
 107    True := trivial
 108
 109/-- π in terms of 8-tick phases:
 110
 111    8 phases × (π/4) per phase = 2π (full circle)
 112
 113    Therefore: π = 4 × (number of quarter-turns)
 114
 115    This is almost tautological, but it shows π is
 116    "4 times the quarter-circle angle." -/
 117theorem pi_from_eight_quarters :
 118    8 * (π / 4) = 2 * π := by ring
 119
 120/-! ## π and φ Relationship -/
 121
 122/-- π and φ are related through geometry:
 123
 124    1. **Golden angle**: 2π/φ² ≈ 137.5° (phyllotaxis)
 125    2. **Pentagon**: Interior angle = 108° = 3π/5
 126    3. **cos(π/5) = φ/2** (exact!)
 127    4. **sin(π/10) = (φ-1)/2 = 1/(2φ)** (exact!)
 128
 129    These connect the circle (π) to the golden ratio (φ). -/
 130theorem cos_pi_5_is_phi_2 :
 131    Real.cos (π / 5) = phi / 2 := by
 132  -- cos(π/5) = (1 + √5)/4 (Mathlib)
 133  -- φ = (1 + √5)/2, so φ/2 = (1 + √5)/4
 134  rw [Real.cos_pi_div_five, phi]