Ditorus (EntityTopic, 11)

From Hi.gher. Space

Revision as of 20:38, 2 February 2014

The ditorus is a four-dimensional torus formed by taking an uncapped torinder and connecting its ends either in a loop or through its inside. Its toratopic dual is itself.

Equations

• Variables:

R ⇒ major-major radius of the ditorus
r ⇒ major-minor radius of the ditorus
ρ ⇒ minor-minor radius of the ditorus

• All points (x, y, z, w) that lie on the surcell of a ditorus will satisfy the following equation:

(√((√(x² + y²) − ρ)² + z²) − r)² + w² = R²

• The parametric equations are:

x = (R + (r + ρ cos θ₃) cos θ₂) cos θ₁
y = (R + (r + ρ cos θ₃) cos θ₂) sin θ₁
z = (r + ρ cos θ₃) sin θ₂
w = a sin θ₃

• The hypervolumes of a ditorus are given by:

total surface area = 0
surcell volume = 8π³Rrρ
bulk = 4π³ρ²rR

• The realmic cross-sections (n) of a ditorus are:

Unknown

Cross-sections

Jonathan Bowers aka Polyhedron Dude created these three excellent cross-section renderings:
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