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The Mobius Band

When made of paper a Mobius Band can be considered to be a folded paper object even though it does not contain any creases.

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c200 - 250 BC

The earliest known image of a Mobius Band is found in the central part of a mosaic from a Roman villa in Sentinum which can be dated to around 200–250CE. This Mobius Band is probably not a representation of an actual object. An untwisted band is found in other similar portrayals of this mythical scene. Information from 'Mobius strips before Mobius: Topological hints in ancient representations' by Julyan H. E. Cartwrigh and Diego L. Gonzalez available online at https://arxiv.org/pdf/1609.07779.pdf.

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1206

A pumping mechanism using Mobius Band topology is found in the 'Book of Knowledge of Ingenious Mechanical Devices' written by Al-Jazari in 1206. Information from 'Mobius strips before Mobius: Topological hints in ancient representations' by Julyan H. E. Cartwrigh and Diego L. Gonzalez available online at https://arxiv.org/pdf/1609.07779.pdf.

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1858

In 1858 Johann Benedict Listing (1808 - 1882), a German mathematician, made notes containing a description of a Mobius Band, which were not, however, published until 1861.

Also in 1858, but a few months later, another German mathematician, August Ferdinand Möbius, made a similar discovery. This was not published until 1865 (after his death).

Both men had studied under Carl Friedrich Gauss, the best German mathematician of the era, and it is possible that Gauss was the common factor in these discoveries. Information from 'History of the Möbius Band' https://sites.google.com/site/themobiusbandart/history-of-the-moebius-band.

I do not know if either of these discoveries involved paperfolding.

The Setinum mosaic pictured above has been in the possession of the Glyptothek Museum in Munich since 1828. It is therefore possible that either Gauss, Listing or Mobius might have seen this early depiction of a Mobius Band and have been influenced by it.

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1890

The Boy's Own paper no 589 of 26th April 1890 contained an article entitled 'A Glimpse of the 'Fourth Dimension'' by Rev. J. B. Bartlett which described how to make a Mobius Band (without naming it or giving its origin) and how to cut it to produce double length and linked bands.

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The Afghan Bands / Les Anneaux Mysterieuse

The Afghan Bands is a self-working magical effect in which three apparently identical long loops of paper (or sometimes cloth) are cut (or sometimes torn) lengthwise to produce first two separate loops, then a single double-length loop, and finally two interlocked loops. The first loop is a simple untwisted loop, the second a Mobius Strip and the third a fully twisted loop.

According to https://geniimagazine.com/wiki/index.php?title=Afghan_Bands 'The first professional magician to perform the effect was Felicien Trewey in the late 1800s' and 'Percy Selbit was the first to describe the Möbius strip as a method for a magic trick in the English language in 1901, coining the name 'Afghan Bands'.' I have not been able to find the original source for this information.

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1887

The first publication of the effect, under the name 'Les Anneaux Mysterieuse' appeared in 'La Nature' 709 of 1st January 1887, in an article headed 'Recreations Scientifique' and subheaded 'Les Anneaux de Papier'. This article is attributed to Dr Z...'. I do not know whose nom de plume this is. The effect subsequently appeared in the 5th Edition of 'Les Recreations Scientifiques' by Gaston Tissandier which was published in 1888.

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The Afghan Bands also appeared:

1890

As 'The Paper Rings', in 'Scientific Amusements' by Henry Frith, which was published by Ward, Locke and Co Ltd in London, New York and Melbourne in 1890.

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1896

As 'Los anillos de papel' in 'Repertorio Completo de Todos los Juegos', which was published in Madrid in 1896

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1899

In 'Le Livre des Amusettes' by Toto, which was published in Paris by Charles Mendel in 1899.

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1906

As 'Les Bandes de Papier' in 'Les Bon Jeudis' by Tom Tit, which was published in Paris in 1906.

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1907

In issue 1503 of the Boy's Own Paper of 2nd November 1907.

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1918

In 'Ciencia Recreativa' by Jose Estralella, which was published by Gustavo Gili in Barcelona in 1918.

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1920

In Will Blyth's 'Paper Magic, first published in London in 1920, as 'An Episode of Mere Man'.

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1922

In 'Houdini's Paper Magic' first published in New York in 1922, as 'Trewey's Paper Rings'.

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1928

In 'Fun with Paperfolding' by Murray and Rigney, published in New York in 1928, as 'The Mystery Loops'.

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1932

In 'Winter Nights Entertainments' by R M Abraham, which was first published by Constable and Constable in London in 1932, as 'Cutting the Paper Rings'. The effect is described but no illustration is given.

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1951

As 'La Cinta Magica' in the extended version of 'El Mundo de Papel' by Dr Nemesio Montero, which was published by G Miranda in Edicions Infancia in Valladolid in 1951.

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Flattening a Mobius Band into a Regular Hexagon

1893

Volume 3 of Tom Tit's 'La Science Amusante', published by Editions Larousse in Paris in 1893, contains a section showing how to flatten a Mobius Band into a regular hexagon by 'un coup de poing' (a punch).

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1903

This effect was also published in Issue 252 of the Buenos Aires edition of 'Caras y Caretas' on 1st August 1903.

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?

The way in which this is to be done is better illustrated by a trade card published by Chocolat Guerin-Boutron, which is undated but probably from the early years of the 20th Century.

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