Surfaces with only one side

Give a paper strip a half-twist and glue the ends. The result has one edge and one side — an ant walking the surface visits 'both sides' without ever crossing the edge. In four-dimensional space, the same trick on a tube produces the Klein bottle: a closed surface with no inside or outside. Toys, art, and the foundations of topology.

Surface

Möbius & Listing, 1858. χ = 0, one side, one edge.

Auto-spin

Speed

0.45 rad/s

Width w

0.320

Number of half-twists

1 = classical Möbius (one-sided). 2 = ordinary two-sided cylinder. 3, 5 are non-orientable like 1; 4 is orientable like 2.

Cut ratio (from centre)

v = ±0.000 · w

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