Some compact manifolds chosen for elegance and practicality.
2D
- Equilateral triangle: 1 degree of freedom in shape, namely size
- Rectangle: 2 degrees of freedom. Rectangles of various aspects ratios are useful practically.
- Circle (actually disc): 1 degree of freedom
- Sphere surface: 1 degree of freedom
- Box surface: 3 degrees of freedom
- Rectangular tiling (flat torus): 2 degrees of freedom
- Regular hexagonal tiling: 1 degree of freedom
Others: various special triangles, regular and irregular polygons, ellipse, other tilings, surfaces of regular and irregular polyhedra, ellipsoids, torus in 3 space, hemisphere, cylinder side (truncated so as not to be infinite), cone, two cones joined at bases, sphericon.
3D
We limit to Euclidean space for practicality.
- Regular tetrahedron
- Box
- Sphere (actually ball)
- Box honeycomb
Others not already listed above: various special tetrahedra, square pyramids, Kelvin cell honeycomb, rhombic dodecahedron honeycomb.
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