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How Six Triangles Fold Into an Impossible Vortex

BEHIND THE WORK

Discovered by Dániel Erdély, the Spidron is an impossible geometric arm formed by alternating equilateral and isosceles triangles scaling down by 1/√3. When six Spidron arms interlock at a common center, they fold flat Euclidean space into a continuous 3D kinetic accordion vortex. This video traces the complete hexagonal Spidron tessellation in one unbroken trajectory with exact 0.000mm periodic closure.

Related Video: The Architecture of Infinity | The Seven Sacred Portals

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

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THE MATHEMATICAL RULE

\text{Spidron Logarithmic Spiral and Folded Relief}

s_i = 3^{-i\cdot 0.375},\quad \theta_i = \frac{k\pi}{3} + \frac{i\pi}{6},\quad i=0,\ldots,7,\ k=0,\ldots,5

z(r,\theta) = -0.10(1-r)^{1.2} + 0.045\cos(6\theta)\sin(\pi r) + 0.02\sin(12\theta+4r)(1-r^2)

\mathbf r(u) = \left( r_u\cos\theta_u,\, r_u\sin\theta_u,\, z(r_u,\theta_u) \right)

WATCH IT COME TO LIFE

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Can Pure Math Grow a Diamond Snowflake?

\text{Sector replication and linear edge interpolation}

R_k=\begin{pmatrix}\cos(k\pi/3)&-\sin(k\pi/3)&0\\\sin(k\pi/3)&\cos(k\pi/3)&0\\0&0&1\end{pmatrix},\quad k=0,\ldots,5

\mathbf Q_{k,j}=R_k\mathbf P_j,\quad\mathbf P_j\in\text{stored sector vertices}

\mathbf q(\lambda)=(1-\lambda)\mathbf Q_i+\lambda\mathbf Q_{i+1},\quad0\leq\lambda\leq1

Can a single mathematical line grow an intricate hexagonal ice crystal with zero human intervention? Governed by D6 dihedral crystal invariants and self-similar dendritic branching at exact 60-degree intervals, this diam

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Three Spheres Trap an Infinite Void

\text{Sphere geometry before path construction}

\mathbf C_j\in\frac1{\sqrt2}\{(1,1,1),(1,-1,-1),(-1,1,-1),(-1,-1,1)\}

\|\mathbf x-\mathbf C_j\|=1,\quad r_{\mathrm{inner}}=\sqrt{3/2}-1

\|\mathbf x\|=r_{\mathrm{inner}},\quad\mathbf P_{ij}=\frac{\mathbf C_i+\mathbf C_j}{2}

What happens when four spheres kiss in 3-dimensional space? Governed by Descartes' 4-Sphere Theorem, the 3D Apollonian Spherical Foam packs mutually tangent spheres into a cosmic tetrahedral cage. Trapped at the center l

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Can Infinite Circles Kiss Without Ever Crossing?

\text{Nested ring rule used within the circle layout}

\varphi=\frac{1+\sqrt5}{2},\quad r_{k+1}=r_k/\varphi

\mathbf P=\mathbf C+r_0(\cos\alpha,\sin\alpha)

\mathbf C_{k+1}=\mathbf P-r_{k+1}(\cos\alpha,\sin\alpha)

\mathbf q_{k+1}(t)=\mathbf C_{k+1}+r_{k+1}(\cos(\alpha+t),\sin(\alpha+t))

How do infinite circles kiss in perfect harmony without ever crossing each other? Governed by Descartes' Circle Theorem, the Apollonian Gasket packs recursively tangent circles into every curving void. Traversed by a sin

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