Steme

Steme is a hands-on topology laboratory for exploring quotient spaces — the surfaces you get by gluing the edges of a square together. Choose how the four edges are identified and instantly ⚡ build 🔨 a torus, Klein bottle, projective plane, sphere, cylinder or Möbius band, then feel how each surface wraps. Everything runs as real mathematics on your device, not a set of static pictures. A quotient-space engine builds the cell structure of your identified polygon, computes the Euler characteristic (V − E + F), works out orientability, genus or crosscap number and the boundary circles, and classifies the resulting surface live as you edit the gluing. The lab has three connected views: • GLUE — pick one of six ready-made gluings or edit the edge identifications yourself, with a live fundamental-polygon diagram 📊 drawing every identification arrow. • EXPLORE — drag ✋ a token across the square and watch 👀 it reappear on the opposite edge, flipped when the identification reverses orientation; or fire a straight geodesic and watch the "snake" wrap around, counting laps and detecting when the path closes. • CLASSIFY — read the full breakdown 🧾 (χ, genus or crosscap, orientability, boundary) beside a schematic diagram 📊 and a χ–genus quadrant chart that places your surface among the classic ones. Save 💾 any surface you build 🔨 to your own local collection 📚, attach a reference photo 📸 — a paper model, a sketch, a whiteboard — from your camera or gallery, then reopen and compare surfaces later. A built-in glossary explains the topology terms, and a settings screen lets you tune haptics, motion and the grid, or reset all your data whenever you like. With Steme there are no accounts and no ads, and the core work — every calculation and every wraparound — happens right on your device 📱. A connection is used only to check ✅ connectivity at start 🚀-up (a retry screen appears if that connection is unavailable) and to open the Privacy Policy and Terms in an in-app web view. Whether you are a student meeting these surfaces for the first time ⏱️, a teacher demonstrating edge identifications, or just curious how a doughnut and a coffee cup can be "the same," Steme turns abstract quotient-space topology into something you can build 🔨, touch and keep.