Manifold
Visual Differential Geometry. After Tristan Needham, every proof here is a picture you can drag — and, where it's honest, a thing you can hear. Carry an arrow around a loop and watch the shape of space turn it. From the curvature you can feel to the gravity it becomes.
Needham wrote it as "a mathematical drama in five acts." Each act becomes a sequence of lessons built on a single draggable gesture — and the gestures compound: the arrow you transport in Act II is the same arrow that becomes gravity in Act V.
What space is, before any formula
Why no flat map of a sphere can be true, and how the very failure to flatten becomes a ruler for curvature — a circle that comes up short, a triangle that runs over.
- The map that must lie live
- Curvature you can feel live
Distance, maps & the other curvature
The metric as the surface's secret rulebook: conformal & stereographic maps, and the mirror-world of negative curvature — the pseudosphere and the hyperbolic plane, where circles run long and triangles go thin.
The two bends, and the global law
Principal curvatures and the shape operator; geodesics as taut threads; then the leap from local to global — add the bending over a whole closed surface and out drops a rigid integer.
The arrow that comes home turned
Carry an arrow around a loop and the shape of space turns it — holonomy that is the enclosed curvature, the keystone gesture, deepening into Riemann's tensor and the tide between neighbouring geodesics.
Fields of slabs, and gravity itself
A 1-form as a stack of sheets you pierce, dω as a boundary you watch, Stokes as one loop — and the payoff: Cartan's moving frames turn parallel transport through curved spacetime into gravity, with Maxwell falling out of dF=0.
The Shape of Distance
The Riemann sphere as living light: hundreds of motes spiral up loxodromes and fall as logarithmic spirals on the floor, light-painted into long-exposure trails with bloom — while an evolving, slightly inharmonic drone climbs as you push toward the pole. Stereographic projection, the conformal map, made mesmerizing and audible.
The Other Curvature
Flip the sign of curvature and enter the mirror-world. In a glowing Poincaré disc a "straight line" bows, circles run long, and triangles come up short; spin the pseudosphere, the real trumpet that holds this geometry. Then hear it — the surplus detunes the beat sharp, the exact opposite of the sphere.
Curvature You Can Feel
Grow a geodesic circle on a luminous world and watch it fall short of 2πr — flatten the cap and a wedge is missing. Drag a triangle whose angles overshoot 180° by exactly its area. Then hear it: the shortfall detunes two tones into a beat, so a flat world is silent and a curved one pulses.
Play the Curvature
Draw loops on a glowing world; each one fences off an area of the sphere — its holonomy — and that area becomes a pitched, colored, sustained voice. Stack loops into a drone, spin the constellation in 3D. Curvature, made audible and luminous.
All thirteen lessons and the instrument are live — the complete drama, every act, mapped onto Needham's true five. The method never changed: see it, drag it, and where it's honest — hear it. Each animation now light-paints its own flow and sings its own evolving voice.