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What Happens When 4D Light Folds Into 3D?
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What Happens When 4D Light Folds Into 3D?

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What happens when 4D light folds into 3D space? In 1931, Heinz Hopf discovered an astonishing geometric miracle: the 3-sphere S³ can be completely decomposed into a family of non-intersecting circular fibers. Stereographically projected into 3D space, these circles form nested tori that weave through space with zero collisions. Every circle is linked with every other circle exactly once.

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

#MathArt #HopfFibration #4DGeometry #Topology #HigherDimensions #GeometricArt #Blender3D

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Base fiber before bridges and framing\text{Base fiber before bridges and framing}x1=cos⁡θ2cos⁡(ψ+ϕ),x2=cos⁡θ2sin⁡(ψ+ϕ)x_1=\cos\frac{\theta}{2}\cos(\psi+\phi),\quad x_2=\cos\frac{\theta}{2}\sin(\psi+\phi)x3=sin⁡θ2cos⁡ψ,x4=sin⁡θ2sin⁡ψx_3=\sin\frac{\theta}{2}\cos\psi,\quad x_4=\sin\frac{\theta}{2}\sin\psi(X,Y,Z)=(x1,x2,x3)1−x4,0≤ψ≤2π(X,Y,Z)=\frac{(x_1,x_2,x_3)}{1-x_4},\quad 0\leq\psi\leq2\pi
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