Geometry Chambers

CATENOID MINIMAL SURFACE OBSERVATORY

Catenoid Minimal Surface Observatory

Observe

Two catenoid branches span equal coaxial rings until their connected solutions merge at a limiting separation.

First touch

Change ring separation and radius, select a branch, and compare surface, mean curvature, and area views.

Worth watching

Zero mean curvature does not guarantee the least area. The two-disk area crossing, local stability, and loss of connected solutions are distinct.

FRACTAL OBSERVATORY

Fractal Zoomer

Observe

Recursive boundaries, islands, and fine detail that appear from repeated iteration.

First touch

Zoom first, then switch Mandelbrot and Julia fields to compare fixed and parameter-driven space.

Worth watching

The edge is the interesting part: small movements reveal new rooms inside the boundary.

DIMENSIONAL PROJECTION ROOM

4D Shadow Projector

Observe

Higher-dimensional rotation becoming shadows, slices, and changing silhouettes.

First touch

Adjust projection axes and rotation until the object stops feeling flat.

Worth watching

Look for moments when a simple wireframe suddenly reads as a deeper object.

COMPLEX MAPPING CHAMBER

Complex Function Warper

Observe

The complex-function warping under square, inverse, exponential, sine, and Mobius-style maps.

First touch

Switch maps and follow paired points to see how one plane becomes another.

Worth watching

Straight grids are useful because their bending makes the invisible rule visible.

RIEMANN SHEET NAVIGATOR

Riemann Sheet Navigator

Observe

One complex value continuing along a loop and returning to the same z on another sheet.

First touch

Trace the default loop, drag branch A outside it, and compare swap with identity before trying logarithmic branches.

Worth watching

Unlike pointwise complex mapping, the result depends on the whole path and which branch points it winds around.

POLYNOMIAL ROOT BRAID COMPARATOR

Polynomial Root Braid Comparator

Observe

The same two coefficient loops, followed in opposite orders, becoming two computed root paths, braids, and final arrangements.

First touch

Run TRACE BOTH ORDERS, follow the colored r0/r1/r2 identities through A→B and B→A, then compare their intermediate and final mappings.

Worth watching

The loops are the same; changing only their order can change which tracked root identity reaches each final position.

P ADIC LIFT TREE

P-adic Lift Tree

Observe

Polynomial roots lift from mod p to higher powers of p, forming a computed tree of unique, split, dead, and pending branches.

First touch

Use LIFT ONE LEVEL on x²−2 at p=7, then compare the no-root sieve at x²+1, p=3 with the singular x²−1, p=2 tree.

Worth watching

Every child is a real congruence test: a branch can continue once, split into several digits, end, or never begin.

SPECTRAL DRUM

Spectral Drum

Observe

A drawn membrane becoming six computed eigenmodes, nodal lines, and an ordered vibration spectrum.

First touch

Change the membrane shape, Solve, choose a Strike point, and Play the modal superposition before comparing eigenvalue Delta.

Worth watching

Shape changes the operator and spectrum; the strike point changes modal coefficients without recomputing the modes.

HODGE DECOMPOSITION WORKBENCH

Hodge Decomposition Workbench

Observe

A discrete edge field separating orthogonally into gradient, rotational, and harmonic components as holes change b1 and the harmonic basis count.

First touch

Start with One-hole circulation, press DECOMPOSE, switch the four components, then add or connect holes to watch the harmonic basis count change.

Worth watching

Reconstruction, pairwise orthogonality, kernel residuals, and energy remain numerically certified while the topology changes the harmonic degrees of freedom.

MODULAR FORMS FIELD

Modular Domain Explorer

Observe

The upper half-plane repeating under T-shift, folding under inversion, and settling into a fundamental domain.

First touch

Switch DOMAIN, LATTICE, WAVE, FUNDAMENTAL, PERIOD, and INVERSION while using T-SHIFT, INVERT, and FOLD.

Worth watching

The useful moment is when many copies begin to read as one field represented by a small domain.

RADON TRANSFORM SCANNER

Projection Scanner

Observe

A hidden phantom becoming projection rows, a sinogram, and a visual reconstruction field.

First touch

Collect SCAN rows, switch SINO, RECON, FILTERED, and COMPARE, then raise noise to see uncertainty enter the reconstruction.

Worth watching

The useful moment is when many thin shadows begin to read as an interior shape.

STEREOGRAPHIC PROJECTION FIELD

Sphere-to-Plane Mapper

Observe

A sphere projected onto a plane, with points near infinity stretching outward.

First touch

Switch projection, rotation, sphere, grid, and circle views to compare the same surface in two forms.

Worth watching

Circles and grids are the easiest way to feel how the far edge expands.

HYPERBOLIC TESSELLATION CHAMBER

Infinite Tile Disk

Observe

Tiles compressed toward the rim of a disk, making space feel deeper near the edge.

First touch

Change kaleido, geodesic, and orbit modes, then increase warp to feel the rim pull outward.

Worth watching

The pattern matters less than the pressure of space near the boundary.

HYPERBOLIC ISOMETRY STUDIO

Hyperbolic Isometry Studio

Observe

The same P and Q following computed geodesic paths under B(A(z)) and A(B(z)).

First touch

Drag P, Q, A, and B directly, or select a preset, then run both orders from one shared snapshot.

Worth watching

The endpoints can differ even though each composition preserves d(P,Q).

KNOT INVARIANT CHAMBER

Knot Crossing Inspector

Observe

A knot turning in projection while crossings, over-under order, and writhe stay readable.

First touch

Spin the knot, switch crossing or writhe lenses, and compare trefoil, cinquefoil, and unknot.

Worth watching

The useful moment is when the curve looks different but the knot still feels like itself.

GRAVITY LENS CHAMBER

Gravity Lens Warper

Observe

Background stars, photon-ring cues, and disk particles visibly bending around a central mass.

First touch

Raise mass or lens strength, then compare stars, rays, disk, and distortion views.

Worth watching

It is not the black core alone; the lensing is easiest to read in the stars around it.

SYMMETRY ORBIT CHAMBER

Symmetry Copy Maker

Observe

A seed point copied by rotations, reflections, inversions, and mirror axes into visible symmetry copies.

First touch

Drag the seed point, raise ORDER, then compare ORBIT, AXIS, FIXED, and TRACE lenses.

Worth watching

The useful moment is when the copies move differently while still revealing the same symmetry rule.

ELLIPTIC CURVE GROUP LAW

Elliptic Curve Group Law

Observe

A chord or tangent construction becoming an exact group operation on points of one smooth cubic.

First touch

Place P and Q, trace the chord or tangent, and compare the exact sum with the identity and inverse cases.

Worth watching

The visible curve construction is distinct from a lattice basis change: one adds points, the other changes generators.

LATTICE REDUCTION FORGE

Lattice Reduction Forge

Observe

A long pair of integer generators becoming a shorter basis while the lattice points, determinant, and fundamental area stay fixed.

First touch

Start with Fibonacci bad basis and press REDUCE BASIS once. Then inspect B′ = BU, det(U), and the Before → After length.

Worth watching

The basis description changes; the lattice does not. This is a 2D Gauss reduction model, not high-dimensional LLL or a proof tool.

PRISM DISPERSION OBSERVATORY

Prism Dispersion Observatory

Observe

Wavelength-dependent rays separating into an exit spectrum after two refractions through a prism.

First touch

Change incident angle, apex angle, and material, then APPLY or STEP through the ray calculation.

Worth watching

The spectrum is a Cauchy-model result, not a camera measurement or a physical prism reading.

SIERPINSKI TETRAHEDRON OBSERVATORY

Sierpiński Tetrahedron Observatory

Observe

A recursive tetrahedron keeping its shape while copies, scale, gaps, and volume fraction change.

First touch

Change recursion depth, rotate the view, and spread the copies to inspect each layer.

Worth watching

The volume fraction follows 4 × (1/2)^3 per layer; the 3D view is a browser model.

SPHERICAL HARMONIC SURFACE OBSERVATORY

Spherical Harmonic Surface Observatory

Observe

A spherical surface deformed by degree l and order m, with nodal lines and symmetry visible in the shape.

First touch

Change l, m, amplitude, and phase, then rotate the sphere to compare harmonic patterns.

Worth watching

Nodal lines and repeated lobes reveal symmetry without turning the surface into a physical measurement.

DISCRETE CURVATURE MESH OBSERVATORY

Discrete Curvature Mesh Observatory

Observe

Local angle defects and boundary curvature adding up to a global Gauss–Bonnet identity on a finite triangle mesh.

First touch

Change a mesh preset, drag a vertex or rotate the view, then read angle defects, boundary curvature, and the residual.

Worth watching

The residual stays near zero for this finite mesh identity; this is not a smooth-surface estimator or physical measurement.

BERRY PHASE HOLONOMY OBSERVATORY

Berry Phase Holonomy Observatory

Observe

A two-level eigenstate transported around Bloch-sphere loops, with Berry phase, solid angle, and local curvature compared.

First touch

Choose CAP, REVERSE, RETRACE, or FIGURE 8, press TRACE, then use STEP and switch bands.

Worth watching

The phase follows oriented enclosed area; reversing the band changes curvature sign while the finite gap remains visible.

MODULAR GEODESIC TESSELLATION OBSERVATORY

Modular Geodesic Tessellation Observatory

Observe

An exact PSL(2,Z) arithmetic state connecting upper-half-plane reduction, Ford circles, Farey neighbors, and continued fractions.

First touch

Choose DOMAIN, FORD, GEODESIC, or ITINERARY, then use REDUCE STEP and EXTEND to expose the same integer word.

Worth watching

Determinant-one Farey contacts and canonical continued-fraction L/R choices remain synchronized across every lens.