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The mathematics of memory

The Neural Map winds your memories onto mathematical objects. Every topology below is a real parametric structure: the equations are the exact ones the engine computes (and the map displays) when your memory takes that shape.

Why mathematics? Because a form with structure can be navigated. A spiral has an inside and an outside, a knot has crossings, an attractor has flows. Position becomes meaning, and your memory becomes a space you learn the way you learn a place. Sailors had the stars; you get nineteen skies.

Chaotic attractors​

Lorenz Chaotic Attractor​

Integrating the Lorenz system from a seed point produces non-linear chaotic dynamics: the butterfly.

dxdt=σ(y−x), dydt=x(ρ−z)−y, dzdt=xy−βz(x0,y0,z0)=(0.1,0,0)\begin{gathered} \frac{dx}{dt} = \sigma(y-x),\ \frac{dy}{dt} = x(\rho-z)-y,\ \frac{dz}{dt} = xy-\beta z \\ (x_0, y_0, z_0) = (0.1, 0, 0) \end{gathered}

Rössler Attractor​

The Rössler system's chaotic differential equations, folded into a single-loop ribbon.

dxdt=−y−z, dydt=x+ay, dzdt=b+z(x−c)(x0,y0,z0)=(0.1,0.1,0.1)\begin{gathered} \frac{dx}{dt} = -y - z,\ \frac{dy}{dt} = x + a y,\ \frac{dz}{dt} = b + z(x - c) \\ (x_0, y_0, z_0) = (0.1, 0.1, 0.1) \end{gathered}

A memory wound onto the Rössler attractor

Minimal surfaces​

Enneper Minimal Surface​

A self-intersecting minimal surface with organic symmetry, parameterized by complex analytical coordinates.

x=scale⋅(u−u33+uv2)y=scale⋅(v−v33+vu2)z=scale⋅(u2−v2)\begin{gathered} x = scale \cdot (u - \frac{u^3}{3} + u v^2) \\ y = scale \cdot (v - \frac{v^3}{3} + v u^2) \\ z = scale \cdot (u^2 - v^2) \end{gathered}

A memory wound onto the Enneper minimal surface

Catenoid Surface​

The minimal surface of a rotated catenary, the physical shape of soap film tension.

x=ccosh⁡(u)cos⁡(v)y=u⋅Lz=ccosh⁡(u)sin⁡(v)\begin{gathered} x = c\cosh(u)\cos(v) \\ y = u\cdot L \\ z = c\cosh(u)\sin(v) \end{gathered}

Hyperbolic Paraboloid​

An open saddle, with hyperbolic geometry and infinite expansion tangents.

z=x2−y21.2Rx=rncos⁡(θn)y=rnsin⁡(θn)\begin{gathered} z = \frac{x^2 - y^2}{1.2 R} \\ x = r_n \cos(\theta_n) \\ y = r_n \sin(\theta_n) \end{gathered}

Pseudospheres & shells​

Dini's Surface​

A helicoidal pseudosphere of constant negative curvature, a spiraling horn.

x=acos⁡(u)sin⁡(v)y=asin⁡(u)sin⁡(v)z=a(cos⁡(v)+ln⁡tan⁡(v/2))+bu\begin{gathered} x = a \cos(u)\sin(v) \\ y = a \sin(u)\sin(v) \\ z = a(\cos(v) + \ln\tan(v/2)) + b u \end{gathered}

A memory wound onto Dini's surface

Logarithmic Seashell​

A logarithmic spiral cone, memory mapped to shell morphogenesis.

x=aebu(1+cos⁡(v))cos⁡(u)y=aebu(1+cos⁡(v))sin⁡(u)z=aebusin⁡(v)−H\begin{gathered} x = a e^{bu}(1+\cos(v))\cos(u) \\ y = a e^{bu}(1+\cos(v))\sin(u) \\ z = a e^{bu}\sin(v) - H \end{gathered}

Tori & non-orientable surfaces​

Toroidal Ring​

Prime sequences wrapped on a torus, structural harmonics split into major and minor windings.

x=(R+rtubecos⁡(ϕn))cos⁡(θn)y=rtubesin⁡(ϕn)z=(R+rtubecos⁡(ϕn))sin⁡(θn)\begin{gathered} x = (R + r_{\text{tube}} \cos(\phi_n)) \cos(\theta_n) \\ y = r_{\text{tube}} \sin(\phi_n) \\ z = (R + r_{\text{tube}} \cos(\phi_n)) \sin(\theta_n) \end{gathered}

A memory wound onto the toroidal ring

Clifford Flat Torus​

The 4-dimensional flat Clifford torus, stereographically projected from the 3-sphere into 3-space.

x=Rcos⁡(u)2−cos⁡(v)y=Rsin⁡(u)2−cos⁡(v)z=Rsin⁡(v)2−cos⁡(v)\begin{gathered} x = \frac{R\cos(u)}{2-\cos(v)} \\ y = \frac{R\sin(u)}{2-\cos(v)} \\ z = \frac{R\sin(v)}{2-\cos(v)} \end{gathered}

Möbius Strip​

One side, one edge, non-orientable flows.

x=(R+vncos⁡(θn/2))cos⁡(θn)y=vnsin⁡(θn/2)z=(R+vncos⁡(θn/2))sin⁡(θn)\begin{gathered} x = (R + v_n \cos(\theta_n / 2)) \cos(\theta_n) \\ y = v_n \sin(\theta_n / 2) \\ z = (R + v_n \cos(\theta_n / 2)) \sin(\theta_n) \end{gathered}

Figure-8 Klein Bottle​

A 3D projection of the non-orientable Klein bottle, a manifold that passes through itself.

x=(R+cos⁡(θn2)sin⁡(vn)−sin⁡(θn2)sin⁡(2vn))cos⁡(θn)y=sin⁡(θn2)sin⁡(vn)+cos⁡(θn2)sin⁡(2vn)z=(R+cos⁡(θn2)sin⁡(vn)−sin⁡(θn2)sin⁡(2vn))sin⁡(θn)\begin{gathered} x = (R + \cos(\frac{\theta_n}{2})\sin(v_n) - \sin(\frac{\theta_n}{2})\sin(2v_n))\cos(\theta_n) \\ y = \sin(\frac{\theta_n}{2})\sin(v_n) + \cos(\frac{\theta_n}{2})\sin(2v_n) \\ z = (R + \cos(\frac{\theta_n}{2})\sin(v_n) - \sin(\frac{\theta_n}{2})\sin(2v_n))\sin(\theta_n) \end{gathered}

Knots & links​

Trefoil Knot Manifold​

The Fermat prime spiral wrapped onto the simplest non-trivial knot.

x=R(sin⁡(θn)+2sin⁡(2θn))y=R(cos⁡(θn)−2cos⁡(2θn))z=−Rsin⁡(3θn)\begin{gathered} x = R (\sin(\theta_n) + 2 \sin(2\theta_n)) \\ y = R (\cos(\theta_n) - 2 \cos(2\theta_n)) \\ z = -R \sin(3\theta_n) \end{gathered}

Torus Knot (8,3)​

Eight times along the tube, three times around the axis.

r=R+r0cos⁡(8θ)x=rcos⁡(3θ)y=r0sin⁡(8θ)z=rsin⁡(3θ)\begin{gathered} r = R + r_0\cos(8\theta) \\ x = r\cos(3\theta) \\ y = r_0\sin(8\theta) \\ z = r\sin(3\theta) \end{gathered}

Borromean Rings​

Three orthogonal rings, none linked to any other, yet the trio cannot be separated.

R1:(x,y)∈ellipse, z≈0R2:(y,z)∈ellipse, x≈0R3:(x,z)∈ellipse, y≈0\begin{gathered} R_1: (x,y) \in \text{ellipse},\ z \approx 0 \\ R_2: (y,z) \in \text{ellipse},\ x \approx 0 \\ R_3: (x,z) \in \text{ellipse},\ y \approx 0 \end{gathered}

Helices & spirals​

Cylindrical Helix​

Fermat's prime spiral extruded along the Y axis, heights scrambled to expose nested structures.

x=rncos⁡(θn)y=hn⋅Lz=rnsin⁡(θn)\begin{gathered} x = r_n \cos(\theta_n) \\ y = h_n \cdot L \\ z = r_n \sin(\theta_n) \end{gathered}

Double Helix Structure​

Two intertwined helices, DNA-like double strands.

x1=Rcos⁡(θn), y1=hn⋅L, z1=Rsin⁡(θn)x2=Rcos⁡(θn+π), y2=hn⋅L, z2=Rsin⁡(θn+π)\begin{gathered} x_1 = R \cos(\theta_n),\ y_1 = h_n \cdot L,\ z_1 = R \sin(\theta_n) \\ x_2 = R \cos(\theta_n + \pi),\ y_2 = h_n \cdot L,\ z_2 = R \sin(\theta_n + \pi) \end{gathered}

Fermat Spiral Disk​

A flat Fermat spiral with a whisper of depth for perspective.

x=rncos⁡(θn)y=rnsin⁡(θn)z∈[−25,25]\begin{gathered} x = r_n \cos(\theta_n) \\ y = r_n \sin(\theta_n) \\ z \in [-25, 25] \end{gathered}

Spheres & stars​

Spherical Nebula​

Spherical Fibonacci coordinates, nodes dispersed volumetrically, never compressed into a shell.

cos⁡(ϕn)=1−2iN−1x=rnsin⁡(ϕn)cos⁡(θn)y=rncos⁡(ϕn)z=rnsin⁡(ϕn)sin⁡(θn)\begin{gathered} \cos(\phi_n) = 1 - \frac{2i}{N-1} \\ x = r_n \sin(\phi_n) \cos(\theta_n) \\ y = r_n \cos(\phi_n) \\ z = r_n \sin(\phi_n) \sin(\theta_n) \end{gathered}

Astroidal Star​

An astroidal ellipsoid, six orthogonal cusps.

x=Rcos⁡3(v)cos⁡3(u)y=Rsin⁡3(v)z=Rcos⁡3(v)sin⁡3(u)\begin{gathered} x = R \cos^3(v)\cos^3(u) \\ y = R \sin^3(v) \\ z = R \cos^3(v)\sin^3(u) \end{gathered}

Every one of these is available from the map's Geometry panel. Switch topologies freely, and your memory re-winds in place.

And if memory needs a cosmos to be navigable, maybe that's why the cosmos feels so much like a memory.