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Impossible shapes reveal topology in action. Impossible shapes reveal topology in action.

Shapes that break intuition

By Peter Teoh, Science Writer

Some shapes cannot exist in ordinary 3D space without self-intersection. In mathematics, they still make sense and reveal deep ideas about surfaces and topology.

Explainer: Mobius strips and beyond

Focus: A Mobius strip has only one side and one edge, which you can prove by tracing a line along its surface. A Klein bottle goes further, forming a surface with no inside or outside. In 3D it must intersect itself, but in 4D it can exist without crossing.

These shapes help explain orientation, boundaries, and how surfaces can be classified. They show that geometry is about rules, not just what we can build.

Summary of Key Ideas:

  • Topology studies properties that survive stretching.
  • Mobius strips have one side and one edge.
  • Klein bottles are non-orientable surfaces.

Side Notes

  • The Penrose triangle is an optical illusion, not a true surface.
  • Non-orientable surfaces reverse left and right.
  • Topological art and sculpture.
  • Visual proofs for surface properties.

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