Trefoil Knot in Paper forms Pentagon
Jump to navigation
Jump to search
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
This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
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): $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$