Trefoil Knot in Paper forms Pentagon

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Theorem

Take a strip of paper with parallel edges.

Tie a simple trefoil knot in it.

Flatten the knot and pull the paper tight.

The knot will be in the shape of a regular pentagon.

The edges of the paper strip trace out a pentagram.


PentagramKnot.png


Proof




Historical Note

This construction first occurred in Urbano d'Aviso's Trattato della Sfera, 2nd ed. of $1682$.

It appears to have been added as an afterthought, as the rest of the work does not depend upon the construction of polygons.

This construction regularly crops up in puzzle anthologies.


Sources