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The Geometry That Swallows Itself
Differential geometry & surfaces

The Geometry That Swallows Itself

BEHIND THE WORK

What happens when you invert a 3D doughnut through a sphere? The symmetric torus warps into a Dupin cyclide - an asymmetric geometric horn that appears to swallow its own tail. Every single line on this luminous manifold is an exact circle in three-dimensional space, woven together in one continuous trajectory with 0.000mm periodic closure.

Related Video: How Infinity Fits Inside a Circle (Longform Documentary LF_011) Playlist: Differential Geometry & Minimal Surfaces

Artismathic turns equations into visual art - one function at a time.

#MathArt #DupinCyclide #DifferentialGeometry #MinimalSurfaces #Geometry #3DArt #Blender3D #GenerativeArt #Mathematics #Shorts

THE MATHEMATICAL RULE

Base trajectory before normalization\text{Base trajectory before normalization}a=2, c=1.15, μ=1.45, b=a2−c2a=2,\ c=1.15,\ \mu=1.45,\ b=\sqrt{a^2-c^2}u=31t,v=5t,d=a−ccos⁡ucos⁡v,0≤t≤2πu=31t,\quad v=5t,\quad d=a-c\cos u\cos v,\quad0\leq t\leq2\pix=μ(c−acos⁡ucos⁡v)+b2cos⁡udx=\frac{\mu(c-a\cos u\cos v)+b^2\cos u}{d}y=bsin⁡u(a−μcos⁡v)d,z=bsin⁡v(ccos⁡u−μ)dy=\frac{b\sin u(a-\mu\cos v)}d,\quad z=\frac{b\sin v(c\cos u-\mu)}d
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