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.


Sources