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Polyhedral surfaces and flat foldings

A strip of paper can form a flattened Möbius strip in the plane by folding it at 60 ∘ {\displaystyle 60^{\circ }} angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral triangles, folded at the edges where two triangles meet. Its aspect ratio – the ratio of the strip's length to its width – is 3 ≈ 1.73 {\displaystyle {\sqrt {3}}\approx 1.73}, and the same folding method works for any larger aspect ratio. For a strip of nine equilateral triangles, the result is a trihexaflexagon, which can be flexed to reveal different parts of its surface. For strips too short to apply this method directly, one can first "accordion fold" the strip in its wide direction back and forth using an even number of folds. With two folds, for example, a 1 × 1 {\displaystyle 1\times 1} strip would become a 1 × 1 3 {\displaystyle 1\times {\tfrac {1}{3}}} folded strip whose cross section is in the shape of an 'N' and would remain an 'N' after a half-twist. The narrower accordion-folded strip can then be folded and joined in the same way that a longer strip would be.

The Möbius strip can also be embedded as a polyhedral surface in space or flat-folded in the plane, with only five triangular faces sharing five vertices. In this sense, it is simpler than the cylinder, which requires six triangles and six vertices, even when represented more abstractly as a simplicial complex. A five-triangle Möbius strip can be represented most symmetrically by five of the ten equilateral triangles of a four-dimensional regular simplex. This four-dimensional polyhedral Möbius strip is the only tight Möbius strip, one that is fully four-dimensional and for which all cuts by hyperplanes separate it into two parts that are topologically equivalent to disks or circles.

Other polyhedral embeddings of Möbius strips include one with four convex quadrilaterals as faces, another with three non-convex quadrilateral faces, and one using the vertices and center point of a regular octahedron, with a triangular boundary. Every abstract triangulation of the projective plane can be embedded into 3D as a polyhedral Möbius strip with a triangular boundary after removing one of its faces; an example is the six-vertex projective plane obtained by adding one vertex to the five-vertex Möbius strip, connected by triangles to each of its boundary edges. However, not every abstract triangulation of the Möbius strip can be represented geometrically, as a polyhedral surface. To be realizable, it is necessary and sufficient that there be no two disjoint non-contractible 3-cycles in the triangulation.