Bubble Core

The geometric heart of the Quadrian Arena. Two angles from origin — θ_H and θ_v — always sum to 90°. As n increases, they converge toward 45° and the core angle α collapses to zero.

Position n:

Core Angles

Z = n / 32—

θ_H = H + Z—

θ_v = V − Z—

α = θ_v − θ_H—

β = θ_H—

γ = θ_v—

h = |CD|—

θ_H + θ_v90°

Points on Unit Circle

C = (cos θ_v, sin θ_v)—

D = (cos θ_H, sin θ_H)—

Pythagorean Table — a² + b² = c² (a = b)

a = b	c = √(2a²)	Value

Key insight: For any value of n, θ_H + θ_v ≡ 90°. At n = 1440, both angles reach exactly 45° — the core angle α collapses to zero and points C and D converge to (√2/2, √2/2). The distance h between C and D measures the angular asymmetry.