# 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.

## Proof

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## 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

- 1926: Henry Ernest Dudeney:
*Modern Puzzles*... (previous) ... (next): Solutions: $120$. -- The Ribbon Pentagon - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$ - 1992: David Wells:
*Curious and Interesting Puzzles*... (previous) ... (next): Henry van Etten: $126$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$