LATTICE / BAND GAP / PHONONIC BAND GAP OBSERVATORY
- What to watch
- Change alternating masses and select a Bloch mode to compare acoustic and optical branches and the gap between them.
- Lab signal
- Change alternating masses to open a frequency gap. Select a Bloch wave and observe its acoustic or optical motion in a periodic chain.
TERRITORY / VORONOI TERRITORY CHAMBER
- What to watch
- Move sites and watch territory borders redraw around nearest distance.
- Lab signal
- Move sites around and watch nearby space split into shifting territories.
TOPOLOGY / TOPOLOGY FLOOD
- What to watch
- Edit a point cloud and raise its distance threshold to watch components merge and H1 intervals appear.
- Lab signal
- Edit a point cloud, compute H0/H1 persistence, and sweep epsilon to watch components merge and one-dimensional intervals appear and disappear.
HODGE / HODGE DECOMPOSITION WORKBENCH
- What to watch
- Edit a cell complex and its edge field, then watch topology determine the harmonic subspace while reconstruction and orthogonality remain visible.
- Lab signal
- Decompose a discrete edge field on a square-cell complex into gradient, rotational, and harmonic parts, and see how holes determine the harmonic dimension.
OPTIMIZATION / OPTIMAL TRANSPORT WORKBENCH
- What to watch
- Move weighted supply and destination points, then compare a globally optimal transport plan with a source-order Greedy baseline.
- Lab signal
- Move supply and destination points, change their masses, and solve the globally optimal way to carry all mass at minimum total cost.
MONODROMY / POLYNOMIAL ROOT BRAID COMPARATOR
- What to watch
- Trace the same two coefficient loops in opposite orders and watch computed roots form different braids and final arrangements.
- Lab signal
- Trace the same two coefficient loops in opposite orders and compare the computed root paths, three-dimensional braids, and final root arrangements.
STRUCTURE / P ADIC LIFT TREE
- What to watch
- Lift polynomial roots from mod p through powers of p, and watch each branch continue uniquely, split, die, or fail to begin.
- Lab signal
- Lift polynomial roots from mod p through powers of p, and watch each branch continue uniquely, split, die, or fail to begin.
STRUCTURE / LATTICE REDUCTION FORGE
- What to watch
- Change the basis with exact 2D Gauss steps while the lattice, determinant, and area stay fixed.
- Lab signal
- Reduce a 2D integer lattice basis while keeping the lattice, determinant, and fundamental area unchanged.
SYMMETRY / SYMMETRY ORBIT CHAMBER
- What to watch
- Copy points through mirrors and rotations until a symmetry orbit appears.
- Lab signal
- Choose rotations or reflections and watch copies orbit around axes and fixed points.
GAP / PRIME GAP CONSTELLATION
- What to watch
- Scan prime gaps as a star map of voids, twins, density, and rhythm.
- Lab signal
- Scan primes and watch long gaps, twin pairs, and density drift become a star map.
KNOT / KNOT INVARIANT CHAMBER
- What to watch
- Rotate a loop and inspect crossings that keep the knot from coming apart.
- Lab signal
- Rotate a closed loop and inspect crossings that keep the knot from coming apart.
DIMENSION / DIMENSIONAL PROJECTION ROOM
- What to watch
- Rotate higher-dimensional forms and watch their shadows fold across the screen.
- Lab signal
- Rotate higher-dimensional shapes and watch their shadows fold across the screen.
STRUCTURE / GRAPH DIFFUSION / GRAPH SPECTRAL DIFFUSION FIELD
- What to watch
- Observe heat diffusion on a weighted graph through Laplacian eigenmodes, spectral gap, heat trace, and community mixing.
- Lab signal
- Observe heat diffusion on a weighted graph through Laplacian eigenmodes, spectral gap, heat trace, and community mixing.
STRUCTURE / DENSITY GEODESIC / OPTIMAL TRANSPORT DENSITY GEODESIC OBSERVATORY
- What to watch
- Move between two one-dimensional densities and observe monotone transport, displacement geodesics, W2 cost, entropy, and barycenters.
- Lab signal
- Observe a balanced one-dimensional density transport plan through monotone coupling, displacement geodesics, W2 cost, entropy, and barycenters.
STRUCTURE / DIFFRACTION / QUASICRYSTAL DIFFRACTION OBSERVATORY
- What to watch
- Move the phason and window to inspect accepted points, local coordination, eightfold symmetry, and S(k) diffraction peaks.
- Lab signal
- Observe an Ammann–Beenker quasicrystal from a 4D cut-and-project window: accepted points, local coordination, eightfold rotation, and S(k) diffraction peaks.
CRITICAL POINT / LEVEL SET / MORSE LANDSCAPE OBSERVATORY
- What to watch
- Raise a scalar level and watch finite sublevel cells, Euler changes, and critical-point events enter in order.
- Lab signal
- Read critical points and sublevel-set changes in a finite two-dimensional scalar landscape.
NOETHER / CONSERVATION / NOETHER SYMMETRY CONSERVATION OBSERVATORY
- What to watch
- Track translation, rotation, and time symmetry through conserved charges, then expose drift with explicit breaking.
- Lab signal
- Observe how translation, rotation, and time symmetry carry a charge, and how an explicit breaking term makes its balance drift.
STRUCTURE / RANDOM GRAPH / RANDOM GRAPH GIANT COMPONENT OBSERVATORY
- What to watch
- Vary mean degree and seed in a finite random graph, then watch its giant component emerge without importing a real network.
- Lab signal
- Observe the emergence of a giant connected component in an Erdős–Rényi random graph as the mean degree c crosses the finite-size threshold near 1.
DEFECT / WINDING / TOPOLOGICAL DEFECT WINDING OBSERVATORY
- What to watch
- Read local vortex and anti-vortex fixtures through closed-loop winding and enclosed charge; the model does not claim a global single vortex.
- Lab signal
- Observe vortices, anti-vortices, winding, current, and energy in a 64×64 periodic XY phase field.
STRUCTURE / CATENARY EQUILIBRIUM / CATENARY CHAIN OBSERVATORY
- What to watch
- Observe a uniform flexible chain at static equilibrium, then compare catenary span, sag, and support tension.
- Lab signal
- Observe the static catenary of a uniform flexible chain and read how span, chain length, linear density, and gravity set its sag and support tension.
STRUCTURE / CAPILLARY BALANCE / CAPILLARY RISE OBSERVATORY
- What to watch
- Observe Jurin rise or depression as surface tension balances hydrostatic pressure in a static tube.
- Lab signal
- Observe Jurin-law capillary rise or depression as surface tension balances hydrostatic pressure in a deterministic tube model.