Pentagonal knot in a strip of paper

Geometry, division, tiling etc.
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Edwin Corrie
Posts: 80
Joined: Tue Dec 06, 2011 11:40 am

Pentagonal knot in a strip of paper

Post by Edwin Corrie » Wed Aug 29, 2012 10:44 pm

The pentagonal knot made from a strip of paper is well known, but I recently became interested in it after finding references in several places to the fact that it was first described (in the West – I believe it was also discovered independently in Japan) by Urbano d’Aviso in 1682.

A bit of internet research revealed a link to the Google Books scanned version of “Sfera astronomica” (1690) and the following information (which, if you look carefully at the clues, seems to come from our friend Roberto Morassi):

E' la ristampa del 1690 denominata "Sfera Astronomica", attribuita direttamente a Bonaventura Cavalieri, stampata dal libraio Antonio Manara e dedicata all'Eccellentissimo e Illustrissimo Signore Don Carlo Maria Carafa di Branciforte, Principe di Botera, della Roccella, del Sacro Romano Imperio e Grande di Spagna di Prima Classe. Quella originale, sotto il nome del D'Aviso (che a sua volta la attribuisce al Cavalieri) fu stampata del Mascardi nel 1682 e dedicata al Sig. Cardinale Michelangelo Ricci.
http://it.hobby.enigmi.narkive.com/g2a0 ... a-striscia

Basically this says that the original edition was in 1682 but it was reprinted in 1690 under the pseudonym of Bonaventura Cavalieri. I also found another rather long article by Ferdinando Jacoli which not only examines the mathematics behind the pentagonal knot but also discusses the authorship and publication history of the book.

Anyway, the Google Books version is here (page 255):
http://www.google.it/books?id=3MvpXzNO6 ... #PPA255,M1

The text is as follows:

Con l'occasione di questo disegnare le figure, ti voglio dare il modo di descrivere, e formare mechanicamente un Pentagono, che è una delle piu difficili figure da disegnare, e pure è la più facile che si facci in natura, perche non è altro, che un semplice nodo. Prenderai per tanto, una striscia di carta della larghezza, che tu vorrai, e che habbi li lati paralleli, e con quella procura di fare un nodo, come se fosse una corda, avertendo pero che la carta resti sempre stesi nelle piegature, che stringendola tanto che resti ben tirata, se tagliarai con le forbici li capi che avanzano, haverai fatto un Pentagono giustissimo.
Farai anco la figura Esagona se prenderai due striscie di carta di eguale larghezza, e con li lati paralleli, e procurarai di fare con esse un nodo, facendo che le punte dell'incuruatura che haverai fatta di una striscia, passino per l'aperto dell'incuruatura dell'altra, che stringendole adattatamente, e che mantenghino sempre la loro larghezza, tagliando l'avanzi delle punte, haverai fatto un Esagono perfettissimo.


Translation:

I want to show you how to describe and form mechanically a Pentagon, which is one of the most difficult figures to draw, and yet the easiest to make in nature, because it is nothing more than a simple knot. Take a strip of paper of whatever width you like, with parallel sides, and with it try to make a knot, as if it was a piece of cord, but making sure that the paper always remains stretched in the folds, and if you tighten it so that it remains taut, if you cut off the excess ends with a pair of scissors you will have made a very accurate Pentagon.
You can also make a Hexagon if you take two strips of paper of the same width, with parallel sides, and try to make a knot with them, making sure that the ends of the loop which you have made with one strip pass through the open part of the loop made with the other strip, and tighten it suitably so that they maintain their width, and if you cut off the excess ends you will have made a perfect Hexagon.

There are of course many references in books on puzzles, as well as the incredible “Knotology” work of Heinz Strobl. I have found various articles about the pentagonal knot and other polygonal knots (hexagonal, heptagonal etc.), mainly in serious mathematical books and journals, and I can post them here if anyone is interested.

JGimeno
Posts: 33
Joined: Sat Dec 03, 2011 6:47 pm

Re: Pentagonal knot in a strip of paper

Post by JGimeno » Thu Aug 30, 2012 7:57 am

Miguel de Unamuno, made ​​a poem about the knot pentagonal in 1928, which was published posthumously in 1953 in "Cancionero".
"Dios jugando con los dobles
cinco dedos de ambas manos
anudó cinta de yerba;
de cinco puntas fue el lazo.
De donde sacó la estrella
pentagonal, que sus brazos
dió a las blancas frescas alas
de la rosa del garbanzo."

[Translation:
"God playing with double
five fingers of both hands
knotted ribbon grass;
five-point was the tie.
Where did the star
pentagonal, his arms
White gave fresh wings
Chickpea rose."
]
I am attaching the scanned original manuscript
Attachments
Unamuno - Nudo pentagonal.jpg
Unamuno - Nudo pentagonal.jpg (127.28 KiB) Viewed 4031 times

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