GEOMETRY / MINIMAL SURFACE / CATENOID MINIMAL SURFACE OBSERVATORY
- What to watch
- Separate equal coaxial rings and compare the two catenoid branches, their areas, and the limit of connected solutions.
- Lab signal
- Separate two coaxial rings and observe two catenoid branches, their areas, curvatures, and the limit of a connected minimal surface.
FRACTAL / FRACTAL OBSERVATORY
- What to watch
- Begin with recursive coastlines and zoom into the edge where shape keeps returning.
- Lab signal
- Zoom Mandelbrot, shift Julia parameters, and watch recursive coastlines unfold.
COMPLEX / COMPLEX MAPPING CHAMBER
- What to watch
- Switch functions and watch grids bend, flip, and wrap around the complex plane.
- Lab signal
- Switch complex functions and watch grids, circles, and points bend, flip, and wrap.
COMPLEX / RIEMANN SHEET NAVIGATOR
- What to watch
- Draw a loop around branch points and follow one function value as it continues onto another sheet.
- Lab signal
- Trace a closed path through analytic branches and observe how returning to the same z can land on a different sheet.
SPECTRUM / SPECTRAL DRUM
- What to watch
- Draw a membrane, solve its lowest eigenmodes, and watch geometry become a vibration spectrum.
- Lab signal
- Draw a membrane, solve its six lowest vibration modes, choose a strike point, and watch shape change the spectrum and motion.
PROJECTION / STEREOGRAPHIC PROJECTION FIELD
- What to watch
- Push a sphere onto a plane and notice how one pole stretches toward infinity.
- Lab signal
- Push points from a sphere onto a plane and watch the north pole stretch toward infinity.
HYPERBOLIC / HYPERBOLIC TESSELLATION CHAMBER
- What to watch
- Use the disk as a window into tiles that shrink toward an endless rim.
- Lab signal
- Warp a Poincare disk and watch hyperbolic tiles shrink toward an infinite rim.
HYPERBOLIC / HYPERBOLIC ISOMETRY STUDIO
- What to watch
- Apply two Poincare-disk isometries in both orders and watch noncommutativity appear while hyperbolic distance remains unchanged.
- Lab signal
- Apply two disk isometries in opposite orders, then compare geodesic paths, endpoints, trace classes, and preserved hyperbolic distance.
TROPICAL / TROPICAL CURVE BALANCER
- What to watch
- Build a max-plus corner locus beside its Newton subdivision, then select exact vertices, edges, and rays to see their dual relation.
- Lab signal
- Edit integer heights in a max-plus polynomial, build its tropical curve, and inspect exact vertices, weighted balancing, and the dual Newton subdivision.
LENS / GRAVITY LENS CHAMBER
- What to watch
- Bend a star field and read rings, arcs, and distortion as geometry in motion.
- Lab signal
- Bend starlight around a heavy center and watch rings, rays, and disks warp.
MODULAR / MODULAR FORMS FIELD
- What to watch
- Fold repeated domains back into a fundamental region and compare what stays aligned.
- Lab signal
- Shift, invert, and fold the upper half-plane into a domain.
GEOMETRY / ELLIPTIC CURVE GROUP LAW
- What to watch
- Place P and Q on a cubic, trace the chord or tangent, and reflect the third intersection to obtain their sum.
- Lab signal
- Place two points on an elliptic curve, trace the chord or tangent, and compare exact addition over rational presets and small finite fields.
RADON / RADON TRANSFORM SCANNER
- What to watch
- Scan projections from many angles and reconstruct the hidden shape from shadows.
- Lab signal
- Scan projections, stack a sinogram, and rebuild a rough image.
OPTICS / DISPERSION / PRISM DISPERSION OBSERVATORY
- What to watch
- Adjust incidence, prism angle, and material to watch wavelength-dependent rays split into an exit spectrum.
- Lab signal
- Adjust incident angle, prism apex angle, and material to observe wavelength-dependent refraction, exit spectra, angular spread, and total internal reflection.
GEOMETRY / FRACTAL VOLUME / SIERPINSKI TETRAHEDRON OBSERVATORY
- What to watch
- Open recursive tetrahedron layers and watch four scaled copies leave a measurable volume trace.
- Lab signal
- Observe a tetrahedron that keeps the same shape while its scale and volume shrink.
GEOMETRY / HARMONICS / SPHERICAL HARMONIC SURFACE OBSERVATORY
- What to watch
- Change spherical-harmonic l and m, then watch nodal lines and symmetry reshape the surface.
- Lab signal
- Adjust spherical-harmonic degree, order, amplitude, and phase to observe radial surface deformation, nodal lines, and symmetry on a sphere.
GEOMETRY / CURVATURE / DISCRETE CURVATURE MESH OBSERVATORY
- What to watch
- Read vertex angle defects, boundary geodesic curvature, Euler characteristic, and the Gauss–Bonnet residual on deterministic triangle meshes.
- Lab signal
- Read vertex angle defects, boundary geodesic curvature, Euler characteristic, and the Gauss–Bonnet residual on deterministic triangle meshes.
QUANTUM / HOLONOMY / BERRY PHASE HOLONOMY OBSERVATORY
- What to watch
- Carry an eigenstate around Bloch-sphere loops and compare Berry phase, solid angle, and curvature.
- Lab signal
- Carry a two-level eigenstate around closed loops on the Bloch sphere and compare its Berry phase with the enclosed curvature.
MODULAR / FAREY TESSELLATION / MODULAR GEODESIC TESSELLATION OBSERVATORY
- What to watch
- Read PSL(2,Z) reduction, Ford circles, Farey neighbors, and continued-fraction itineraries as one arithmetic tessellation.
- Lab signal
- Read PSL(2,Z) reduction, Ford circles, Farey adjacency, and continued-fraction itineraries as one exact upper half-plane observation.