Introduction
Chapter 1 - Entries
Chapter 2 - Definitions
Summary
Appendix 1 - At-a-Glance Chronological Listing
Appendix 2 - Frequency Analysis
References (Basic, without meta detail)
INTRODUCTION
Sometimes (albeit rarely) seen in association with the Möbius strip is a feature called a ‘paradromic ring’. I must admit, before the Möbius study, I was assuredly unfamiliar with the term (and possibly most other (true) mathematicians as well?). So be it. To this end, as in the study in the round, I first surveyed the literature, including books, magazines, journals and newspapers. The premise is very much a historical one, as to the introduction of the term and by whom. The premise then expanded to include its definition. Like much of the study in the round, this is not as straightforward as may otherwise be thought. Some references in effect repeat in various ways. A case in point is provided by Rouse Ball’s Mathematical Recreations, and further complicated by many editions. The book is often quoted, or perhaps more accurately, appears in searches. Some are of just the phrase alone, without further indication. Some are in passing, whilst others are given a more considered treatment.
Although the term ‘paradromic’ is not unknown, it can hardly be described as in everyday use. Indeed, it is far from it! As can be seen from the inventory, with just 42 entries, up to the present day, found in extensive searching, the term can hardly be said to be in everyday use! Its heyday, relatively speaking, was in the 1880s–1890s, but even so, that was of just nine entries. Indeed, whole decades can be seen without any entry! Furthermore, some of the entries are most lightweight indeed, of a reference without further addition (such as Sykes). The term was first used (paradromen) in a mathematical context by Johann Benedikt Listing in Vorstudien zur Topologie in 1848. However, it is doubtful whether this refers to a Möbius strip, as I cannot find it. However, Rouse Ball, in Mathematical Recreations (p. 53) and Keith Kendig in Never a Dull Moment (p. 104) assert that it does. Therefore, the discrepancy might be due to my own oversight. I will have to look at this again. Certainly, it is not used in Der Census, of 1861. Whatever, it did not immediately catch on (in relative terms). Indeed, it was essentially ignored for 34 years! Not until 1882 in The Encyclopedia Britannica, in connection with an entry on knots, leaning heavily on Listing, was it used again. Then again, more notably, by Peter Guthrie Tait, in 1883 and 1884, where he credits Listing. Other usages, in sequence, include W. W. Rouse Ball, 1892, Cardiff Naturalists' Society, 1893, W. B. Croft, 1893, The Echo, 1893, and W. H. F. Patterson, 1894. Interestingly, there is one reference titled as Listing’s paradromic rings (Croft, English Mechanic and World of Science and Art 1893, 1894), with no reference being made to Möbius.
The precise definition of a paradromic ring remains unclear, as definitions seem to vary among sources; see below. ‘Paradromic’ is not clear as to intention. As such, there seems to be no consensus on this. Indeed, some authors, I believe, simply use the term as a synonym for a Möbius strip! F. Hyde Maberley, in 1932, is a case in point. In short, it appears to be cutting down the centre a ‘many twisted’ Möbius strip. The intricacies here overwhelm me, and so given that these are way beyond my understanding, I see little point in ‘investigating’ as such, contenting myself with a ‘brief survey’, to at least acknowledge the premise.
As such, the study evolved in extent. Unlike previously, where I undertook the history of the Möbius strip with a remit of up to 1930 (with a few notable exceptions), as otherwise it would be impractical, this has included most modern-day instances, up to and including the present day. Originally, I was intending to be more restrictive, in line with that study, but with relatively few distinct instances, ‘all references’ became a practical proposition, although I was fully aware that much of the latest accounts would repeat previous authors. However, by omission, I would forever be wondering if a vital reference would be overlooked, and so rather than let this situation be, I thus decided to do an ‘all-encompassing’ study, leaving no room for doubt. Of course, this has to be judged as against the time required, which, although it was a little more than I would have liked, it was not over-excessive. As anticipated, much of the recent literature repeats, to no real benefit. Whatever, it is now done.
Although I could continue a little longer, it is starting to become too time-consuming beyond reason. The law of diminishing returns is becoming evident, with far too much time required to wheedle out a new instance.
For once in the investigation, searching is straightforward (in contrast with Möbius/Moebius strip/band in all combinations). There are only a few searchable terms possible: paradromic, paradromic rings, paradromic winding (rare), and combinations thereof with Listing. And that’s about it! The best resources were the Internet Archive, Google Books, and the British Newspaper Archive.
CHAPTER 1
Entries
The entries are accompanied by ‘notes’, which in effect form part of the study per se, rather than a bare-bones listing. These are all added to the main essay listing. Some bibliographic details remain open, to be resolved at a later date. Not all are conducive to easy documenting.
* When I say ‘not illustrated’, I am referring to paradromic rings specifically; the text may contain other illustrations not strictly relevant to the paradromic aspect.
1882 (1)
The Encyclopedia Britannica. A Dictionary of Arts, Sciences, and General Literature. Ninth Edition. American Reprint, Vol. XIV. Philadelphia: J. M. Stoddart & Co. 1882, pp. 128–129. Not illustrated*
KNOT. In the scientific sense, a knot is an endless physical line which cannot be deformed into a circle. A physical line is flexible and inextensible, and cannot be cut, -so that no lap of it can be drawn through another.
The founder of the theory of knots is undoubtedly Listing. In his "Vorstudien zur Topologie" (Göttinger Studien, 1847); a work in many respects of startling originality, a few pages only are devoted to the subject. He treats knots from the elementary notion of twisting one physical line (or thread) round another, and shows that from the projection of a knot on a surface we can thus obtain a notion of the relative situation of its coils. He distinguishes "reduced"...
It is to be noticed that, although every knot in which the crossings are alternately over and under is irreducible, the converse is not generally true. This is obvious at once from fig. 6, which is merely the three crossing knot with a doubled string-what Listing calls "paradromic."
I am getting more than a little confused with all these Encyclopedia Britannica volumes, editions, and reprints. This is stated as an American reprint (the UK edition appears unavailable). This here begins with the letter K, specifically on Kaolin to L on Lons-le-Saulnier, so it’s an L-K.
In an entry on the knot, has one reference only to paradromic, quoting Listing’s usage and the Vorstudien.
…what Listing calls “paradromic”…
Illustrated* with six figures, but not paradromic rings as such, but rather knots, with substantial text. As the title suggests, this is strictly on (mathematical) knots, and not on the Möbius strip as such. Listing and Tait are mentioned extensively, but not Möbius. Other mentions include Reimann and Gauss, and four dimensions.
Who wrote the entry is not made clear; some entries have initials at the end and others not, as here.
Peripherally, it includes the juggler’s trick of putting a knot on an endless unknotted band assertion. I thought this originated with Tait, but I misremembered (he does not use the word ‘jugglers’), although he using a similar sentiment. P. G. Tait's 1883 obituary of Listing in Nature:
This remark of Listing's forms the sole basis of a work which recently had a large sale in Vienna: - showing how, in emulation of the celebrated Slade, to tie an irreducible knot on an endless string!
The same text and illustrations are found in later editions of 1890, 1891, 1893, 1894, 1895, 1902, 1903, 1910, 1911.
879 pp, 230MB download!
Printed
1883 (1)
Peter Guthrie Tait. ‘Johann Benedict Listing’ (Obituary). Nature. February 1, Volume 27, 1883, pp. 316–317. Not illustrated
ONE of the few remaining links that still continued to connect our time with that in which Gauss had made Göttingen one of the chief intellectual centres of the civilised world has just been broken by the death of Listing….
The consideration of double-threaded screws, twisted bundles of fibres, &c., leads to the general theory of paradromic winding. From this follow the properties of a large class of knots which form "clear coils." A special example of these, given by Listing for threads, is the well-known juggler's trick of slitting a ring-formed band up the middle, through its whole length, so that instead of separating into two parts, it remains in a continuous ring. For this purpose it is only necessary to give a strip of paper one half-twist before pasting the ends together. If three half-twists be given, the paper still remains a continuous band after slitting, but it cannot be opened into a ring, it is in fact a trefoil knot. This remark of Listing's forms the sole basis of a work which recently had a large sale in Vienna: - showing how, in emulation of' the celebrated Slade, to tie an irreducible knot on an endless string!
On the face of it, with Tait referring to ‘winding’ and not ‘rings’ in association with paradromic, one could justifiably query this inclusion. However, later in the passage, talk turns to what is a Möbius strip, and so there appears to be a connection of sorts that justifies the inclusion.
Paradromic is mentioned just the once.
Printed
1884 (1)
Peter Guthrie Tait. ‘Prof. Tait on Listing’s Topologie’. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Series 5, Issue 103, Volume 17, 1884, pp. 37–38. Not illustrated*
SOME of you may have been puzzled by the advertised title of this Address. But certainly not more puzzled than I was while seeking a title for it…
(8) The last statement hints at a subject treated by Listing, which he calls paradromic winding. Some of his results are very curious and instructive.
Take a long narrow tape or strip of paper. Give it any number, m, of half-twists, then bend it round and paste its ends together.
If m be zero, or any other even number, the two-sided surface thus formed has two edges, which are paradromic. If the strip be now slit up midway between the edges, it will be split into two. These have each rn/2 full twists, like the original, and (except when there is no twist, when of course the two can be separated) are m/2 times linked together.
But if m be odd, there is but one surface and one edge; so that we may draw a line on the paper from any point of the original front of the strip to any point of the back, without crossing the edge. Hence, when the strip is slit up midway, it remains one, but with m fulll twists, and (if m > 1) it is knotted. It becomes, in fact, as its single edge was before slitting, a paradromic knot, a double clear coil with m crossings.
[This simple result of Listing's was the sole basis of an elaborate pamphlet which a few years ago had an extensive sale in Vienna; its object being to show how to perform (without the usual conjurer's or spiritualist's deception) the celebrated trick of tying a knot on an endless cord.]
The study of the one-sided autotomic surface which is generated by increasing indefinitely the breadth of the paper band, in cases where m is odd, is highly interesting and instructive. But we must get on. (ALL)
A lengthy treatment. Three references to paradromic. I now understand better after reading Bruns.
Zenodo
Printed
1888 (1)
Although Édouard Lucas’s L'Arithmétique Amusante is dated 1895, a ~1888 date can be assigned to a paradromic reference, pp. 222–223. See Lucas’s entry under the 1895 date.
1892 (2)
W. W. Rouse Ball. Mathematical Recreations and Problems. London Macmillan and Co. and New York, Second edition, 1892, pp. 53–54. (First Edition February 1892, not seen.) Not illustrated.
Paradromic Rings. The fact just stated is illustrated by the familiar experiment of making paradromic rings by cutting a paper ring prepared in the following manner.
Take a strip of paper or piece of tape, say, for convenience, an inch or two wide and at least nine or ten inches long, rule a line in the middle down the length AB of the strip, gum one end over the other end B, and we get a ring like a section of a cylinder. If this ring is cut by a pair of scissors along the ruled line we obtain two rings exactly like the first, except that they are only half the width. Next suppose that the end A is twisted through two right angles before it is gummed to B (the result of which is that the back of the strip at A is gummed over the front of the strip at B), then a cut along the line will produce only one ring. Next suppose that the end A is twisted once completely round {i.e. through four right angles) before it is gummed to B, then a similar cut produces two interlaced rings. If any of my readers think that these results could be predicted off-hand, it may be interesting to them to see if they can predict correctly the effect of again cutting the rings formed in the second and third experiments down their middle lines in a manner similar to that above described.
The theory is due to J. B. Listing* who discussed the case when the end A receives m half-twists, that is, is twisted through m pi, before it is gummed to B.
If m is even we obtain a surface which has two sides and two edges, which are termed paradromic. If the ring is cut along a line midway between the edges, we obtain two rings, each of which has m half-twists, and which are linked together ½m times.
If m is odd we obtain a surface having only one side and one edge. If this ring is cut along its mid-line, we obtain only one ring, but it has 2m half-twists, and if m is greater than unity it is knotted.
* Vorstudien zur Topologie, Die Studien, Göttingen, 1847, part x.
The fourth known reference to ‘Paradromic rings’, after Listing, Encyclopedia Britannica, and Tait.
This is a trying book to unravel, with many editions, a title change and latterly revised by H. S. M. Coxeter, at least since 1939 (11th edition). This is further exacerbated by the limited variety online; for example, the first edition (1892) is not available. However, in practice, not having seen the first edition is not a major drawback. I have the second edition (also of 1892), in which it is stated that no material changes have been made. So the paradromic ring entry is taken at face value as the same as the first edition. This is all text. However, a later edition (date unclear, but possibly introduced by Coxeter, gives illustrations, as well as minor differences in the text.
A lengthy treatment. Given that this is more important than most, one of the earliest historical references (1892) to paradromic rings, I show the relevant text in full above. Of note is that Listing (and the Vorstudien) is mentioned, whilst Möbius is not. What we now know as the Möbius strip is clearly described. This Ball credits to Listing in the Vorstudien, part x. I am struggling to find it!
Also of note is that on his Wikipedia page. Ball is said to be an amateur magician and a founding president of one of the world’s oldest magic societies (I was unaware of this until 2 October 2024). This possibly explains his interest from that viewpoint.
Bio. Walter William Rouse Ball (1850–1925), known as W. W. Rouse Ball, was a British mathematician, lawyer, and fellow at Trinity College, Cambridge, from 1878 to 1905. He was also a keen amateur magician, and the founding president of the Cambridge Pentacle Club in 1919, one of the world's oldest magic societies. Wikipedia
https://archive.org/details/mathematicalrecr00ball/page/80/mode/2up?q=listing (1914 edition)
1905, 1922, and 1926, editions are not illustrated. 1962 illustrated.
Chas T. Whitmell. Cardiff Naturalists' Society. Reports & Transactions. Cardiff, Volume XXV (25); 1892–93, pp. 6–33, see p. 20. Not illustrated
Space and its Dimensions.
PARADROMIC RINGS.
I conclude my remarks upon surface geometry with a brief reference to what are called Paradromic Rings. Take a strip of paper, bend it round into a ring and gum the ends together. We get thus a short hollow cylinder, or ring, with two surfaces (the inside and the outside), and two edges. Now take another strip, and give the paper a half twist before gumming the ends together. We thus produce a twisted ring with only one surface and only one edge. If the flat strip was 10 inches long, we shall find that the single edge of the twisted ring is 20 inches long, and that we can draw along its surface a continuous line of the same length, the end of such a line coinciding with its beginning. Let the line be drawn along the middle of the strip.
When we have travelled half-way we shall have reached a point exactly opposite (through the paper) to the starting point, and yet we have not crossed the edge of the paper. This curious result might have seemed impossible, for we cannot get to the other side of a flat sheet of paper unless we go round the edge of it. But the half twist reduces the two surfaces and edges into one of each. When we have thus travelled 10 inches we are as far as we can be from the starting point, and yet in another sense we are almost in contact with it. For, if the paper be considered to be a surface only, we have only surface between the two points. Move on 10 inches further and we reach once more the starting point.
A remarkable geometry, that of single elliptic space, is suggested by the twisted ring, but I cannot enter upon it here, nor can it really be illustrated by diagrams or models. In this space, every shortest or straightest line is re-entrant, as it is on a sphere, and yet two such lines intersect in only one point. A complete plane in this space is finite, not infinite, and its "two sides" are not distinct, as we can pass continuously from a point on one side of it to the same point on what is called the "other" side of it, that is to say we do not need to go over an edge. Quite apart from single elliptic space, the subject of paradromic rings, the results obtained by twisting and cutting them, and the links and knots they exhibit, are very fascinating. (ALL)
A considered treatment. ‘Paradromic rings’ appear to be used as a synonym for the Möbius strip. No mention is made of the background. Part illustrated, but not on the Möbius strip (paradromic rings).
Bio. Charles Thomas Whitmell (1849–1919) was an English astronomer, mathematician and educationalist. …In July 1879 Whitmell was appointed as one of Her Majesty's Assistant Inspectors of Schools, at first working in the area around Sheffield, and in September 1883 was promoted to Her Majesty's Inspector of Schools (HMI) for the South Wales region, centred on Cardiff….In both Cardiff and Leeds Charles Whitmell played a leading role in the burgeoning scientific societies. In Cardiff he was an active member of the Chemical and Geological Societies and was a President of the Cardiff Naturalists' Society….He was a regular correspondent of the popular journal, the English Mechanic, wherein he provided detailed answers to a wide variety of queries on physics, astronomy and mathematics, posed by its readers.
Wikipedia
Printed
1893 (3)
The Echo (London). Tuesday, September 8, 1893. p. 1. Not illustrated
Listing’s Paradromic Rings:
Advert for English Mechanic and World of Science. Inconsequential in itself. The entry is one of a long list of articles documenting the day’s contents. The first in a series of references of an identical nature in The Echo of the day (8–9, 12, 13 September; 7, 10 October 1893), and so are not listed, in relation to the English Mechanic and World of Science.
Interesting in that Listing is credited, rather than Möbius. Incidentally, this is the only such reference in the BNA archive (ignoring the ‘repeats’).
Syndicated, 1/6. 1893 (6)
BNA
Printed
W. B. Croft. English Mechanic and World of Science. September 8, 1893, pp. 66–67. Not illustrated
LISTING’S PARADROMIC RINGS.
[35372.] - AN interesting geometrical idea was put forward by Listing in “Topologie" (Göttingen, 1847). Simple forms of it have been given in recent years in scientific papers, but I am not aware of any popular key to the general principles. The present account is suggested by some figures in a paper published by F. Dingeldey a few years back in Vienna…
An extensive, in-depth discussion. Illustrated. Of note is that Croft associates Listing with the paradromic rings, and cites the Vorstudien. Mentions Listing and Dingledey (albeit just each once) but not Möbius. This is not an easy read due to the small print! See the minor correction in his article below. Also see the W. H. F. Patterson entry, a follow-up to this post.
Bio. Croft, a mathematician at Winchester College, is (unfairly) little known. Nature has the only known obituary on him. No mention is made of his Möbius strip interest in it.
Nature, Google Books
Printed
https://www.nature.com/articles/121624a0.
————. English Mechanic and World of Science. October 6, 1893, p. 157. Not illustrated
Listing’s Paradromic Rings
[35430] In my letter, published on Sept. 8, there is a wrong expression in the paragraphs relating to Fig. 6. It is clear that, in that figure, one cross-over from inner to outer passes on one side of the middle ring, while the other cross-over passes on the other side; so this middle ring comes out a separate ring, not quite free, but linked. The other cases may be inferred from this example. Sometimes there is double linking, as when both ends of two figures of eight are linked.
Winchester College, Sept. 28. W. B. Croft. (ALL)
A minor correction to his article above. Not illustrated.
Google Books
Printed
1894 (1)
W. H. F. Patterson. English Mechanic and World of Science, No. 1505, January 26, 1894, p. 511. Not illustrated
Some Experiments with Paper Rings.
It was only the other day that my attention was called to an article in the ENGLISH MECHANIC of Sept. 8, 1893 (No. 35372, pp. 66–67) entitled “Listings Paradromic Rings.” About three years ago I made some experiments in this direction with a view to deducing a formula by which the result of dividing in any degree a ring, with any number of half-twists (I, like your correspondent, considered one half-turn as unity) could be obtained. I found that letting...[Formulas given]
A lengthy and considered treatment.This builds on W. B. Croft’s earlier posting (September 8, 1893, pp. 66–67) to the English Mechanic, which he cites, albeit not crediting Croft by name. Patterson is a new name to me. I have not been able to find any biographical details of him. He is obviously a mathematician of some kind, as evidenced by the mathematical nature of his post.
Possibly he may be:
MR. W. H. F. PATTERSON. The death took place yesterday of Mr. William Hugh Ferrar Patterson, V/arren Hoad, Donaghadee, a director of the firm of Robert … (BNA)
Google Books
Printed
1895 (1)
Édouard Lucas. L'Arithmétique Amusante, Paris, Gauthier-Villars et fils. 1895, pp. 222–223.Not illustrated
Les hélices paradromes
Translation
3. Paradrome propellers. [Möbius‑type twisted strips] — This theory was treated for the first time by Listing in the work entitled: Vorstudien zuir Topologie, published in Gottingen, in 1848. The very simple results obtained by the author were the sole foundation of a laborious Memoir which sold a lot in Vienna a few years ago; the purpose of this Memoir was to show how one can perform, without using the deceptive process of the conjurer or the spiritualist, the famous amusing physics trick which consists of tying a knot with an endless rope [without end].
Let's take a long narrow ribbon or a strip of paper; if we glue the two ends together, without twisting, we form a sort of roll, like a napkin ring; the surface of this roller has two faces, one interior, the other exterior; the roll has two edges or two ridges that match the two edges of the tape. If we cut the strip of paper with a pair of scissors, through a longitudinal slit equidistant from the edges, it is obvious that we will obtain two other rolls similar to the first, but half the height.
Instead of bringing the two ends together without twisting, let us turn one of the ends of the ribbon half a turn by applying a half twist; let's bend it and glue its ends together; we thus form a singular surface;
Page 223
this surface has only one face and one edge, so that we can trace on the paper, from any point on the old front to any point on the old back, a continuous line which does not meet the single edge formed by the edges of the original strip (1).
Consequently, when the strip is split longitudinally in the middle, using a pair of scissors, we do not obtain two distinct strips, but only one, and this strip is affected by two half-twists, while the primitive band was assigned only one.
More generally, if we have made n half-twists before joining the ends of the ribbon, there are two cases to consider:
When n is an even number, the surface still has two faces and two edges each forming a closed circuit; if we cut the ribbon along its length, it is divided into two others each having n half-twists, like the original ribbon; but the two half-ribbons cannot be separated from each other and are tied - ½ n times.
This reference is significant in that it is one of the earliest French references, although how early is imprecise. Édouard Lucas passed in tragic circumstances at the early age of 49, in 1891, and so that date is a start point (and not the date of the posthumous publication, 1895). However, there is an even earlier date documented, 1888, in one of a series of three notebooks. Unfortunately, the Möbius strip reference is in a non-dated notebook, and cannot be said to be before or after 1888. So awarding a date here is problematic. But the reference is obviously before the publication date. With all the uncertainty in mind, I will give a date of ~1888.
https://archive.org/details/larithmetiqueam00lucarich/page/n235/mode/2up
1902 (1)
The Globe and Traveller. Saturday Evening, July 12, 1902, p. 1. Not illustrated
‘Knots’.
In the hey-day of the yachting season…
…tell the difference between a “paradromic” and an “irreducible” knot and would be at a loss to say whether it could or could not “exist in the space of four dimensions”....
The premise here is largely non-mathematical, a substantial piece on knots per se in the context of yachting, of a whole column. Interesting here is the use of four dimensions, in vogue of the day.
BNA
Printed
The Savannah Morning News (US), December 27, 1902, p. 3. Not illustrated
Nothing But Knots
What they are Named and How They are Tied…
…tell the difference between a “paradromic” and an “irreducible” knot and would be at a loss to say whether it could or could not “exist in the space of four dimensions”....
From the London Globe
A repeat, duly credited in the text, of The London Globe account. As an aside, this is the only US newspaper reference I have been able to find.
Chronicling America
Printed
https://www.loc.gov/resource/sn89053684/1902-12-27/ed-1/?sp=3&q=%22nothing+but+knots%22
1906 (1)
Linlithgowshire Gazette, Friday, March 2, 1906, p. 3. Not illustrated
Those who wish……Paradromic Rings…
A mention in passing with the promotion of Rouse Ball’s Mathematical Recreations and Essays, where Rouse Ball discusses paradromic rings. Inconsequential.
BNA
Printed
1907 (1)
Wellington College Natural Science Society. Thirty-Eighth Annual Report of the Wellington College Natural Science Society, 1907/1908, pp. 21–22. Not illustrated
Minutes of Open Meetings. Saturday, July 6th
A Conversazione was held in the New Buildings. The following is the list of the exhibits:
…18. H. R. Pollock and J. H. Hay.
Some Optical Illusions.
Principle of Soaring of Birds.
Paradromic Rings.
Transparent Fused Silica.
X Ray Photographs.
A simple report of a ‘conversazione’. Nothing more is given as to the occasion. The term ‘conversazione’ was new to me. I now find that it is:
A conversazione is a "social gathering [predominantly] held by [a] learned or art society" for conversation and discussion, especially about the arts, literature, medicine, and science. Wikipedia
A cursory look for Pollack and Hay bore no fruit.
Printed
1908 (2)
Paul Carus (ed.) The Open Court. Open Court Publishing Company, Chicago. Vol. XXII, No. 629, October 1908. Not paginated. Not illustrated
Devoted to the Science of Religion, the Religion of Science, and the Extension of the Religious Parliament idea.
Advert (two-page spread) for the William White book. They both have the same publisher.
https://archive.org/details/opencourt_oct226292caru/mode/2up?q=%22paradromic+rings%22
William Frank White. A Scrap-Book of Elementary Mathematics; Notes, Recreations, Essays. Open Court Publishing Company, 1908, p. 117. Not illustrated
Paradromic rings.* A puzzle of a very different sort is made as follows. Take a strip of paper, say half as wide and twice as long as this page; give one end a half turn and paste it to the other end. The ring thus formed is used in theory of functions to illustrate a surface that has only one face: a line can be drawn on the paper from any point of it to any other point of it, whether the two points were on the same side or on opposite sides of the strip from which the ring was made. The ring is to be slit — cut lengthwise all the way around, making the strip of half the present width. State in advance what will result. Try and see. Now predict the effect of a second and a third slitting.
* The theory of these rings is due to Listing, Topologie, part 10. See Ball's Recreations, p. 75-6.
Not illustrated. The text appears to be derived from Rouse Ball, whom he quotes.
The book is a collection of mathematical problems and puzzles aimed at elementary school students. The book covers a wide range of topics, including arithmetic, geometry, algebra, and trigonometry.
Publication. White makes it clear early in this work that this is not a resource guide or a textbook. Rather, White has created a volume consisting primarily of puzzles, notes, and intriguing principles of mathematics. Essentially, this is the fun side of math. Amazon
Bio. William F. White was a professor of mathematics at State Normal School in New Paltz, New York and held a PhD from Colgate University.
Printed
1909 (2)
Benjamin E. Smith (superintendent). The Century Dictionary Supplement. The Century Co., Volume 12, 1909, p. 942. Not illustrated
Gives a dictionary entry definition of paradromic Running parallel: as a paradromic winding. Possibly the first dictionary entry?
Werner Encyclopaedia. A Standard Work of Reference in Art, Literature, Science, History, Geography, Commerce, Biography, Discovery and Invention, Vol. 14. Akron, Ohio, The Werner Company, 1909, pp. 128–129. Not illustrated
It is to be noticed that, although every knot in which the crossings are alternately over and under is irr reducible, the converse is not generally true. This is obvious at once from fig. 6, which is merely the three crossing knot with a doubled string — what Listing calls “paradromic.”
This repeats the Encyclopedia Britannica entry of 1882 on knots with much on Listing and Tait! Documented as ‘seen and noted’. 172 MB download!
https://archive.org/details/wernerencyclopae14unse/page/n9/mode/2up?q=paradromic
1914 (1)
V. E. Johnson. ‘Tricks for the Young Entertainer. Some Simple and Diverting Magical Mysteries’. The Boy's Own Annual, Volume 37, 1914, pp. 165–166. Illustrated
IV. THE CHINESE BANDS, OR THE PARADROMIC RINGS.
This is a very pretty and puzzling drawing-room trick, if well presented. The only necessary apparatus is a pair of good scissors, and three or four strips of plain white paper, some five or six feet long...
Presented as a magic trick. No reason or explanation is given for the term ‘paradromic’, it being used just once in the title and caption. No mention is made to Möbius.
Also of note for the (presumed) first use of ‘Chinese Bands’ to add to ‘far east ‘Afghan, Indian and Hindu references! I have not been able to find other such references, hindered by being such a popular term.
Bio. Johnson was a magician, and author of Chemical Magic.
Google Books
Printed
1922 (2)
Mabel Sykes and Clarence Elmer Comstock. Solid Geometry. Rand, McNally & Company, Chicago, New York, [1922], p. 207. Not illustrated
Paradromic Rings. Ball, Mathematical Recreations and Essays.
A brief mention of Rouse Ball in the references, but nothing more in the main text. Inconsequential.
Bio 1. Mabel Sykes (1868–1938) was a high-school mathematics teacher in Chicago at the start of the twentieth century. She authored textbooks and journal articles and served the greater mathematical community while teaching for almost four decades in the public school system. MAA
Bio 2. Little is known of Comstock, beyond being a professor of mathematics and a collaborator with Sykes.
Internet Archive
Printed
https://archive.org/details/solidgeometry01comsgoog/page/n218/mode/2up?q=%22paradromic+rings%22
‘The Mathematical Clubs of Weseley College, Wellesley, Mass.’, The American Mathematical Monthly, Volume 29, 1922, p. 419. Not illustrated
The Mathematics Club of Wellesley College was started in the spring of 1921, with member-
ship limited to the members of the junior and senior classes taking elective mathematics…
The programs of the other meetings were as follows:
November 18, 1921: "The angle bisector and the use of two right angles in the solution of the
cubic" by Miss Marion Stark, instructor; "Paradromic rings" by Eleanor Johnson '23;
"How to draw a straight line" by Margaret Merrell '22; …
A mention in passing in the context of mathematical clubs.
JSTOR
Printed
1923 (1)
G. T. Bennett. ‘Paradromic Rings’. Letter in Nature, June 30, 1923, p. 882. Not illustrated
Paradromic Rings. PROF. C. V. BOYS, in his letter “A Puzzle Paper Band” in NATURE of June 9, p. 774, gives scant credit to the geometers. Forty years ago [1883] they described the endless band of paper with a half-turn twist in it, and found that if cut down the middle line it gave a single endless band with four half-twists. But they were so obsessed, he says, with the consequence of cutting down the middle line that they missed the result he now describes. This consists in taking a band with four half-twists and converting it by manipulation into a half-twist band of double thickness….
Bennett taking C. V. Boys to task for lack of credit in reply to the former's letter of 9 June 1923! Also see Anne Betts.
Curiously, mentions Listing, but not Möbius. Was Bennett ignorant of Möbius, even at such a late date of 1923?
Bio. Geoffrey Thomas Bennett OBE (1868–1943) was an English mathematician, professor at the University of Cambridge. Wikipedia
Nature
Printed
https://www.nature.com/articles/111882b0
1925* (1)
Harold Thayer Davis and Malcolm G. McCown. A Course in General Mathematics. Principia Press of Trinity University. 1962. Originally published 1925, p. 275. Not illustrated
10. Paradromic rings.The rather astonishing results which can be obtained from comparatively simple problems are illustrated in the following simple experiment:
Let three strips of paper be prepared about an inch in width and ten inches long...
Note that I have only seen the 1935 edition. One of eleven recreations. Describes the Afghan Bands trick, but not by name! Possibly ‘paradromic’ is used here as a synonym for a Möbius strip.
Bio. Harold Thayer Davis was a mathematician, statistician, and econometrician, known for the Davis distribution. Wikipedia
HaithiTrust
Printed
https://babel.hathitrust.org/cgi/pt?id=uc1.$b423548&seq=1
1928 (2)
The Daily News, Tuesday, January 10, 1928, p. 3. Not illustrated
Breakfast-Time Problem, by Henry E. Dudeney.
No. 321
A Curious Paradox. Colonel Crackham perplexed…Is it possible to have a strip of paper… that has only one side?
Although paradromic is not directly mentioned here, it is included as the source that led to the following relevant reference..
BNA
Printed
The Daily News, Wednesday, January 11, 1928, p. 3. Not illustrated
Breakfast Time Problem, by Henry E. Dudeney.
No. 322
Answer to problem 321
Let the paper be in the form of a ribbon of conventional length…have nowhere turned over the edge it is clear that it has only one side. Most readers are probably aware that this is a paradromic ring, and that if you cut all along the line you have made you will get, to your surprise, one large ring. If you give the...
Answer to the above. A very interesting reference for Henry E. Dudeney. He gives the impression here that the term is obvious, and furthermore, to a general readership! This is surely overstating the case of the day. No mention is made of Möbius or Listing, with the impression that paradomic is used here as synonym for a Möbius strip.
Printed
1931 (1)
F. W. Westaway. Craftsmanship In The Teaching Of Elementary Mathematics. Blackie and Son Ltd. 1931, 1937, p. 615. Not illustrated
Chapter XLIV
Mathematical Recreations
2. Geometrical problems and paradoxes. Shunting and ferry-boat problems. Paradromic rings.
A mention in passing. Inconsequential.
Bio. Westaway, Frederick William (1864–1946), school inspector and writer on science, Oxford Dictionary of National Biography.
Internet Archive
https://archive.org/details/in.ernet.dli.2015.209482/page/n637/mode/2up?q=paradromic (41 MB download)
1932 (2)
Frank Hyde Maberly. Nature, Volume 129, No. 3252, February 27, 1932, p. 317. Not illustrated
A Symbol of the Space-Time Continuum
If a strip of stamp edging is taken, given a half twist, and the ends joined, we have formed one of the well-known paradromic rings, having only one surface and one edge.
The figure occupies three dimensions in space, but like time has the feature of endlessness, or like Einstein’s finite space returns on itself. I suggest, therefore, that it may well stand for a symbol of the space-time continuum.
F. HYDE MABERLY.
Royal Geographical Society,
Kensington Gore, London, S.W.7 (All)
Again, a reference to the ‘well-known’ paradromic rings. No mention is made of Möbius or Listing. The impression given is that the Möbius strip as an entity is titled as a paradomic ring.
Bio. Biographical details are somewhat sketchy, but Hyde appears to have been an Irish doctor.
Internet Archive
Printed
https://archive.org/details/dbc.wroc.pl.034925/page/316/mode/2up?q=%22paradromic+rings%22
The Children's Newspaper (edited by Arthur Mee). January 16, 1932, not paginated. Not illustrated.
A One-Sided Ribbon.
If you ask a friend whether it is possible for a piece of ribbon to have both length and breadth but only one side to it, he will probably tell you that it is not.
To show him that lie is wrong take a ribbon of paper and paste the two ends together, but before doing so turn one end over so that the paper ring has a twist in it. Now if you start at any point to draw a pencil line along the length of the ribbon you can keep straight on without lifting your pencil until you have covered the whole surface.
This is called a paradromic ring, and if you cut along the pencil line you will find that instead of the paper falling into two parts you will get one large ring. (All)
Also see a much later reference of the same title in 1943, in which the last paragraph repeats!
Publication. The Children's Newspaper was a long-running newspaper published by the Amalgamated Press (later Fleetway Publications) aimed at pre-teenage children founded by Arthur Mee in 1919. It ran for 2,397 weekly issues before being merged with Look and Learn in 1965. Wikipedia
Syndicated, 1/9. 1932 (1), 1933 (6), 1939 (2)
Internet Archive
Printed
1933 (6)
Morecambe Guardian, Friday, February 10, 1933, p. 11. Not illustrated
Ribbon with One Side
If you ask a friend whether it is possible for a piece of ribbon to have both length and breadth but only one side to it he will probably tell you that it is not.
To show him that he is wrong...This is called a paradromic ring, and if you cut along…
No mention is made to Möbius.
Syndicated, 2/9. 1932 (1) 1933 (6), 1939 (2)
BNA
Evening Despatch, Saturday, February 11, 1933, p. 6. Not illustrated
Saturday Oddments
By Uncle Mac
…To show him that he is wrong ...This is called a paradromic ring, and if you cut along…
As detailed above.Children's section, although not stated as such. The ‘Ribbon with One Side’ title is omitted.
Syndicated, 3/9. 1932 (1), 1933 (6), 1939 (2)
BNA
Printed
Buchan Observer and East Aberdeenshire Advertiser (Scotland), Tuesday, August 8, 1933, p. 6. Not illustrated
Ribbon with One Side
…To show him that he is wrong ...This is called a paradromic ring, and if you cut along…
As detailed above.
Syndicated, 4/9. 1932 (1), 1933 (6), 1939 (2)
BNA
Printed
Cornish Guardian, Thursday, August 10, 1933, p. 8, 9. Not illustrated
Ribbon with One Side
…To show him that he is wrong ...This is called a paradromic ring, and if you cut along…
As detailed above.
Syndicated, 5/9. 1932 (1), 1933 (6), 1939 (1)
BNA
Printed
The Newark Herald, Saturday, August 12, 1933, p. 2. Not illustrated
The Kiddies’ Corner
Ribbon with One Side
…To show him that he is wrong ...This is called a paradromic ring, and if you cut along…
As detailed above.
Syndicated, 6/9. 1932 (1). 1933 (6), 1939 (1)
BNA
Printed
Westerham Herald, Saturday, August 26, 1933, p. 3. Not illustrated
Our Children’s Corner Conducted by Uncle Ted
Ribbon with One Side
…To show him that he is wrong ...This is called a paradromic ring, and if you cut along…
As detailed above.
Syndicated, 7/9. 1932 (1) 1933 (6), 1939 (2)
BNA
Printed
1937 (1)
Anon. The Society of the Engineers. 1937 London, England, pp. 9, 12. Not illustrated
(17) In conclusion the lecturer showed the paradox of the paradromic rings
Ackerman… paradromic rings problem
Details are scant here, of a snippet view only. From what I have seen I will take it at face value as a true intention.
1939 (1)
Buchan Observer and East Aberdeenshire…(Scotland), Tuesday, August 8, 1939. Not illustrated
Ribbon with One Side
…To show him that he is wrong ...This is called a paradromic ring, and if you cut along…
As detailed above
Syndicated, 8/9. 1932 (1). 1933 (6), 1939 (2)
BNA
Cambridge Independent Press, Friday, August 11, 1939, p. 6 or 8. Not illustrated
Ribbon with One Side
…To show him that he is wrong ...This is called a paradromic ring, and if you cut along…
As detailed above.
Syndicated, 1/9. 1932 (1) 1933 (6), 1939 (1)
BNA
Printed
1943 (1)
The Children's Newspaper. December 4, 1943, p. 8. Not illustrated.
Paradromic Ring.
Take a ribbon of paper and paste the two ends together, but before doing so turn one end over so that the paper ring has a twist in it. Now if you start at any point, to draw a pencil line along the length of the ribbon, preferably in the middle, you can keep straight on without lifting your pencil until you have marked both sides of the paper.
This is called a paradromic ring, and if you cut along the pencil line you will find that, instead of the paper falling into two parts you will get one large ring. (All)
The text is partly repeated from a 1932 reference, of the same title.
Internet Archive
Printed
https://archive.org/details/The_Childrens_Newspaper_1289_1943-12-04/page/n7/mode/2up?q=paradromic
1947 (1)
Scientific American, September 1947, p. 138. Not illustrated.
Our Book Corner.
paradromic rings
A mention in passing in the context of an ad for the latest edition of Rouse Ball’s book.
Inconsequential
1953 (1)
Howard Whitley Eves. An Introduction to the History of Mathematics. New York, Rinehart [1953]: Fourth Edition. Not illustrated.
15.24 Paradromic Rings
Discuss the procedure of the witch doctor who gave advice to couples wondering if they should get married. If he wished to prophesy a future breakup in the proposed marriage, he would split an untwisted band; if he wished to prophesy that the couple would quarrel but still stay together, he would split a band having a full twist; if he wished to prophesy a perfect marriage, he would split a Möbius strip.
A bit strange! Whatever, inconsequential.
Bio. Howard Whitley Eves (1911–2004) was an American mathematician, known for his work in geometry and the history of mathematics. Wikipedia
Internet Archive
https://archive.org/details/introductiontohi0000eves_v2m4_4thedi/page/576/mode/2up?q=paradromic&view=theater (half-hour loan)
1962 (1)
W. W. Rouse Ball. Revised by H. S. M. Coxeter. Mathematical Recreations and Essays, The Macmillan Company, 1962, pp. 125–126. Illustrated (Note the subtle title change from the 1892 edition).
Page 125
Paradromic Rings. The difficulty of mentally realizing the effect of geometrical alterations in certain simple figures is illustrated by the familiar experiment of making paradromic rings by cutting a paper ring prepared in the following manner. [This paragraph is essentially the same, with only minor changes, as in the 1892 text]
Take a strip of paper or piece of tape, say, for convenience, an inch or two wide, and at least 9 or 10 inches long, rule a line in the middle down the length AB of the strip, gum one end over the other end B, and we get a ring like a section of a cylinder. If this ring is cut by a pair of scissors along the ruled line, we obtain two rings exactly like the first, except that they are only half the width. Next suppose that the end A is twisted through two right angles before it is gummed to B (the result of which is that the back of the strip at A is gummed [This paragraph is the same as in the 1892 text]
Page 126
over the front of the strip at B), then a cut along the line will produce only one ring. Next suppose that the end A is twisted once completely round (i.e. through four right angles) before it is gummed to B, then a similar cut produces two interlaced rings. If any of my readers think that these results could be predicted off-hand, it may be interesting to them to see if they can predict correctly the effect of again cutting the rings formed in the second and third experiments down their middle lines in a manner similar to that above described. [This paragraph is the same as in the 1892 text]
The theory is due to Listing and Tait,* who discussed the case when the end A receives m half-twists—that is, is twisted through mπ—before it is gummed to B. [This paragraph is the same as in the 1892 text, save for the addition of Tait’s name]
If m is even, we obtain a surface which has two sides and two edges. If the ring is cut along a line midway between the edges, we obtain two rings, each of which has m half-twists, and which are linked together ½ m times. [paradomic is omitted after ‘two edges’]
If m is odd, we obtain a surface having only one side and one edge. If this ring is cut along its mid-line, we obtain only one ring, but it has 2m + 2 half-twists, and if m is greater than unity it is knotted. If the ring, instead of being bisected, is trisected,† we obtain two interlocked rings: one like the original ring (coming from the middle third), and one like the bisected ring. The manner in which these two rings are interlocked is illustrated by the following diagrams (of the cases m = 3 and m = 5).
[after If the ring, instead… is omitted]
[Diagram]
Only minor changes (by Coxeter?) to the 1982 text have been made. The most significant changes are the two illustrations and the inclusion of Tait. Above, I detail the differences. No mention is made of Möbius or Listing. The impression given is that the Möbius strip as an entity is still titled as a paradomic ring.
* Vorstudien zur Topologie, Géttinger Studien, 1847, part x
† This remark is due to J. M. Andreas. [Andreas is not known}
Internet Archive
1965 (1)
George D. Bishop. Teaching Mathematics in the Secondary School. Collins, 1965, pp. 137–138.
Not illustrated.
Paradromic rings. Take a strip of paper or piece of tape, 1 in. or 2 in. wide, at least 9 or 10 in. long. Rule a line in the middle down the length AB of the strip, gum one end over the other end, B, so as to get a ring like the section of a cylinder. If this ring is cut by a pair of scissors along the ruled line the result is 2 rings exactly like the first except only half the width. But if A is twisted through 2 right angles before it is gummed to B (back of strip at A is gummed over front of strip at B) then a cut along the line will produce only one ring. Next if the end A is twisted once completely round (i.e. through 4 right angles) before it is gummed to B then a similar cut will produce 2 interlaced rings. (All)
The impression given is that the Möbius strip as an entity is still titled as a paradomic ring.
Internet Archive
Bio. Little is known of Bishop. He was a senior lecturer at the University of the West Indies. https://archive.org/details/teachingmathemat0000geor/page/138/mode/2up?q=%22paradromic+rings%22 (one hour loan)
1970 (1)
Belfast Telegraph, Saturday, January 31, 1970, p. 6. Not illustrated
A piece of paper with only one side!
Look Around for Younger Readers
Have you ever seen a piece of paper with one side?..... That's why figures of this kind are called MOBIUS [sic] STRIPS. Another name for them is PARADROMIC RINGS. Paradomic simply means “running side by side”...George McBride.
Children’s section. A relatively lengthy treatment for a children’s section. It’s perhaps a little surprising, given the context, that the term would be used here, albeit the background is made.
BNA
Printed
1973 (1)
William L. Schaaf. A Bibliography of Recreational Mathematics. NCTM, Vol. 3. 1973, p. 155. Not illustrated
Glossary.
Paradromic rings. A modification of the familiar Möbius band, given m half-twists (i.e., when one end of the original strip is turned through an angle of rπ] radians). When m is even, a surface with two sides and two edges is obtained; when cut along the center line, this surface yields two rings each having m half-twists and linked together m/2 times. When m is odd, the surface obtained has only one side and one edge; when cut along its center this surface yields only one ring, which has 2m + 2 half-twists (if m > 1, the ring is knotted). (ALL)
This now makes sense after reading Bruns. Are parts of this derived from Tait? Certainly, the text is alike.
https://archive.org/details/ERIC_ED087631/page/n167/mode/2up?q=paradromic
Printed
1977 (1)
Adrien L. Hess. Mathematics Projects Handbook. Reston, Va.: National Council of Teachers of Mathematics, 1977, p. 16. Not illustrated
A Topological Study of Paradromic Curves
Variation of Mobius Strip and Problem Other Problems of Topology
Minimalist text. The same text is repeated in a 1999 updated edition.
Internet Archive
https://archive.org/details/mathematicsproje0000hess_q0u3/page/16/mode/2up?q=paradromic
One-hour loan
1994 (1)
Steven Schwartzman. The Words of Mathematics: An Etymological Dictionary of Mathematical Terms used in English. Washington, DC: Mathematical Association of America. 1994, p. 158. Not illustrated
paradromic (adjective): the first element is from Greek para “alongside,” from the Indo-European root per originally “forward, in front,” but with many extended meanings. The second element is from Greek dromos “a running,” from the Indo-European root der “to run.” Suppose the end of a strip of paper is twisted a certain number of times and then joined to the opposite end of the strip; if you now use a pair of scissors to cut completely around the strip parallel to and a certain distance from an edge, the resulting rings are said to be paradromie. Before being cut they “ran alongside” each other. The most familiar paradromic rings are the ones that result from cutting a Moebius strip. [165, 38]
Definition, alongside other mathematical terms. A nice, readable treatment.
Bio. Steven Schwartzman is a former math teacher. National Assocation of Scholars.
Internet Archive
Printed
https://archive.org/details/wordsofmathemati0000schw/page/4/mode/2up?q=paradromic
1999 (1)
Eric W. Weisstein. CRC Encyclopedia Of Mathematics. 1999, p. 1180. Illustrated
Cutting a Möbius strip, giving it extra twists, and reconnecting the ends produces unexpected figures called Paradromic Rings (Listing and Tait 1847, Ball and Coxeter 1987) which are summarized in the table below.
Möbius strip and paradromic ring discussion combined.
Internet Archive
One-hour loan.
2004 (1)
David J. Darling. The Universal Book of Mathematics: from Abracadabra to Zeno's Paradoxes.
Hoboken, N.J.: Wiley, 2004, pp. 208–210. Not illustrated
A simple and wonderfully entertaining two-dimensional object, also known as the Mobius strip, that has only one surface and one edge. It is named after the German mathematician and theoretical astronomer August Ferdinand Mobius (1790–1868), who discovered it in September 1858, although his compatriot and fellow mathematician Johann Benedict Listing (1808–1882) independently devised the same object in July 1858…
The Mobius band has a lot of curious properties. If you cut down the middle of the band, instead of getting two separate strips, it becomes one long strip with two half-twists in it. If you cut this one down the middle, you get two strips wound around each other. Alternatively, if you cut along the band, about a third of the way in from the edge, you will get two strips; one is a thinner Mobius band, the other is a long strip with two half-twists in it. Other interesting combinations of strips can be obtained by making Mobius bands with two or more flips in them instead of one. Cutting a Mobius band, giving it extra twists, and reconnecting the ends produces unexpected figures called paradromic rings.
The association is clearly made.
Internet Archive
https://archive.org/details/universalbookofm0000darl/page/210/mode/2up?q=paradromic (one hour loan)
2005 (1)
Catherine A. Gorini. Facts on File Geometry Handbook. New York, N.Y.: Checkmark Books, 2005, pp. 104, 116. Not illustrated
Möbius band A nonorientable one-sided surface obtained by gluing the ends of a rectangle to each other after a half-twist. It is also called a Möbius strip. (p. 104)
paradromic rings Rings of equal width that are produced by cutting a strip lengthwise that has been given any number of half twists and then attached at its ends. (p. 116)
Gives definitions.
Internet Archive
https://archive.org/details/factsonfilegeome0000gori_s5m2/page/104/mode/2up?q=mobius
One-hour loan
2006 (1)
Cliff Pickover. The Möbius Strip: Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology. New York: Thunder's Mouth Press, 2006, pp. 65–66. Not illustrated
Paradromic rings
Researchers from the late 1800s to the present day have catalogued the effects of extra twists to pieces of paper before joining the ends to form Möbiuslike [sic] strips. The surprising results are called paradromic rings. Some of the possible paradromic ring structures are listed in the following table.
A good treatment, where Pickover lists the possibilities.
Internet Archive
https://archive.org/details/mbiusstripdrau00pick
2009 (1)
Timothy V. Craine and Rheta Rubenstein (ed.). Understanding Geometry for a Changing World. Reston, VA: National Council of Teachers of Mathematics, 2009, pp. 33–48. Illustrated
Möbius Concepts Strip and Tori. Yuichi Handa, David A. James, Thomas Mattman.
call paradromic rings, which are studied by knot theorists (see Ball and Coxeter [1987, pp. 127–28]), including one of the authors of this article.
A limited preview, hence the omission of the first line of text. A nice treatment nonetheless with a table to fill in.
Internet Archive
https://archive.org/details/understandinggeo0000unse_g9d6/page/48/mode/2up?q=paradromic&view=theater (one hour loan)
2012 (1)
Runzhu Fan and Li Xiang, ‘The Structure and Topological Properties of Möbius Strip Dissection’, Shing-Tung Yau High School Mathematics Awards. 2012, pp. 327+. Illustrated.
Taking a rectangular strip of paper, giving it half-twists and reconnecting the ends of the
strip produces figure called a Paradromic ring[10,11,2].... A Paradromic ring is a generation of a Möbius strip. Paradromic ring is one-sided if is odd, two-sided if is even twisted paper rings…
Largely academic. Has 59 references to paradromic rings!
Internet Archive
Printed
2014 (1)
Gianni A. Sarcone. Impossible Folding Puzzles and other Mathematical Paradoxes. Mineola: Dover Publications, 2014.
Page 29
Paradromic rings
Fold a sheet of paper…
Page 50
Any topologist, origami expert, or high school geometry teacher will tell you that cutting an interlaced strip of paper that forms a perfect trefoil knot from a flat single piece of paper is physically impossible, and has been known to be so for centuries. Yet, with this trick you can prove the contrary! Another way to form a trefoil in a piece of paper is to cut a ‘paradromic ring’ with three half-twist lengthwise (see “Paradromic rings’ on page 29).
Discusses both the Möbius strip (pp. 10, 30) and paradromic rings (pp. 29, 50).
Bio. Gianni A. Sarcone (1962–) is a visual artist and author who collaborates with educational publications, writing articles and columns on topics related to art, science, and mathematics education…Considered a leading authority on visual perception by academic institutions, Sarcone was invited to serve as a juror[4] at the Third Annual "Best Illusion of the Year Contest" held in Sarasota, Florida (USA). Wikipedia
Internet Archive
https://archive.org/details/impossiblefoldin0000sarc/page/30/mode/2up
One-hour loan
2017 (1)
Carson J. Bruns and J. Fraser Stoddart. The Nature of the Mechanical Bond: From Molecules to Machines. John Wiley & Sons Ltd, 2017, p. 71. Illustrated
2. 2. 3
The Möbius Approach
The Möbius strip is an iconic object… The bands with an even number of half twists have the more intuitive two-surface situation common to ribbonlike objects, and are know as paradromic rings…. paradromic rings…. Another interesting property, exclusive to the Möbius strips (n = odd), is that an off-center cut parallel to the edge will afford (Figure 2.8c) a link comprising a large ring with (2n+2) half-twists and a smaller ring ...
Academic. At last (12 July 2024), a description of paradromic rings I can understand (in an academic chemistry book)! I’ve long been held back by this. Now I understand. It also has an excellent diagram. Why can’t others explain it as well as this? Odd twists = Möbius strip; even twists = Paradromic ring. Simple!
Bio. I am a researcher in the interdisciplinary field of molecular nanotechnology. I direct the Emergent Nanomaterials Laboratory at the ATLAS Institute of the ATLAS Institute of the University of Colorado Boulder.
Google Books
Printed
https://www.google.co.uk/books/edition/The_Nature_of_the_Mechanical_Bond/xShUDQAAQBAJ?hl=en&gbpv=1&dq=Paradromic+rings&pg=PA71&printsec=frontcover (Limited preview)
2018 (2)
Peter Prevos. The Möbius Strip in Magic – A Treatise on Afghan Bands. Third Hemisphere Publishing, Kangaroo Flat, Australia (eBook), 2018, pp. 8–9. Not illustrated
Topology
Topology is one of the most…
The children of Möbius strips are called paradromic rings, which means rings that run parallel to each other….
A nice, considered treatment. This implies that all odd twisted strips are to be considered paradromic.
Printed
Mark Blacklock. The Emergence of the Fourth Dimension: Higher Spatial Thinking in the Fin de Siècle. Oxford University Press, 2018, p. 44. Not illustrated.
... paradromic winding. From this follow the properties of a large class of knots which form “clear coils.” The mathematics of knots suggested itself as a route…
Merely quoting Tait.
Google Books
https://www.google.co.uk/books/edition/The_Emergence_of_the_Fourth_Dimension/nrNSDwAAQBAJ?hl=en&gbpv=1 (Limited preview)
2020 (1)
Nancy Ho, J. Godzik, Jennifer Jones, T. Mattman, Dan Sours. ‘Invisible Knots and Rainbow Rings’. Mathematics Magazine, Vol. 93, No. 1, February 2020, pp. 4–18. Illustrated
Möbius strip experiments are surefire triggers of Aha! experiences, even in very young audiences. Maybe you don’t remember the first time someone challenged you to color one side blue and the other red, or asked you to guess the result of cutting a Möbius strip in half, but you surely recall the outcome. (If not, we encourage you to put aside the magazine for a moment, gather up some paper, tape, and scissors, and remind yourself what a bisected Möbius strip looks like. See Figure 1)...
You have just created examples of paradromic rings, which we’ll denote P(m, n). (We first learned of these constructions from the delightful book of Ball and Coxeter [2].)
Paywalled, but an ArXiv version is available. The figures are at the end of the paper.
Academic, with occasional popular discussions. 26 references to paradromic!
Semantic Scholar
https://www.semanticscholar.org/reader/ded420e08dfcd66d84104150889b23b7f8b7ce6b
?
Elran, Yossi and Ann Schwartz. Should We Call Them Flexa-Bands?’. In The Mathematics of Various Entertaining Subjects, pp. 249–261.
... A paper on half-twisted band and paradromic ring notation…
https://www.jstor.org/action/doBasicSearch?Query=%22paradromic+ring%22&so=rel
————. Wolfram (Weisstein)
Paradromic Rings
Rings produced by cutting a strip that has been given m half twists and been re-attached into n equal strips (Ball and Coxeter 1987, pp. 127–128).
Merely quoting Rouse Ball.
Wikipedia. Möbius strip
Cutting a Möbius strip along the centerline with a pair of scissors yields one long strip with four half-twists in it (relative to an untwisted annulus or cylinder) rather than two separate strips. Two of the half-twists come from the fact that this thinner strip goes two times through the half-twist in the original Möbius strip, and the other two come from the way the two halves of the thinner strip wrap around each other. The result is not a Möbius strip, but instead is topologically equivalent to a cylinder. Cutting this double-twisted strip again along its centerline produces two linked double-twisted strips. If, instead, a Möbius strip is cut lengthwise, a third of the way across its width, it produces two linked strips. One of the two is a central, thinner, Möbius strip, while the other has two half-twists.[6] These interlinked shapes, formed by lengthwise slices of Möbius strips with varying widths, are sometimes called paradromic rings.[17][18]
Wikipedia. J. B. Listing.
Listing first introduced the term "topology" to replace the older term "geometria situs" (also called sometimes "Analysis situs"), in a famous article published in 1847, although he had used the term in correspondence some years earlier.[1] He (independently) discovered the properties of the Möbius strip, or half-twisted strip, at the same time (1858) as August Ferdinand Möbius, and went further in exploring the properties of strips with higher-order twists (paradromic rings). He discovered topological invariants which came to be called Listing numbers.[2]
CHAPTER 2
Definitions
Upon having completed a ‘general survey’, with printouts and ‘analysis’ (as above), as far as ‘understanding’ what a paradromic ring was, I was still very much uncertain. To this end, I have decided to compile what I term ‘definitions’, where the author is clearly making some attempt, taken from the text. Subsumed within 25 pages, it made for trying reading; a dedicated chapter will make the task as easy as can be.
Seemingly as ever wth the broader study, though, complications ensue. Although there are indeed strict definitions in a dictionary-type sense, most of the discussion is within ‘broad commentary’ amid related text. At times, I have had to be selective in selection.
Infuriatingly, even with this dedicated compilation, I am essentially no further forward! The term is described in the vaguest of terms, and more often than not, I am confused if this term is used as a synonym for a Möbius strip, or half-twist models, or full twist models, never mind the cutting aspect!
Peter Guthrie Tait. ‘Johann Benedict Listing’ (Obituary). Nature. February 1, Volume 27, 1883, pp. 316–317. Not illustrated
Take a long narrow tape or strip of paper. Give it any number, m, of half-twists, then bend it round and paste its ends together.
If m be zero, or any other even number, the two-sided surface thus formed has two edges, which are paradromic. If the strip be now slit up midway between the edges, it will be split into two. These have each rn/2 full twists, like the original, and (except when there is no twist, when of course the two can be separated) are m/2 times linked together.
But if m be odd, there is but one surface and one edge; so that we may draw a line on the paper from any point of the original front of the strip to any point of the back, without crossing the edge. Hence, when the strip is slit up midway, it remains one, but with m fulll twists, and (if m > 1) it is knotted. It becomes, in fact, as its single edge was before slitting, a paradromic knot, a double clear coil with m crossings.
W. W. Rouse Ball. Mathematical Recreations and Problems. London Macmillan and Co. and New York, Second edition, 1892, pp. 53–54. (First Edition February 1892, not seen.) Not illustrated.
Paradromic Rings. The fact just stated is illustrated by the familiar experiment of making paradromic rings by cutting a paper ring prepared in the following manner.
Take a strip of paper or piece of tape, say, for convenience, an inch or two wide and at least nine or ten inches long, rule a line in the middle down the length AB of the strip, gum one end over the other end B, and we get a ring like a section of a cylinder. If this ring is cut by a pair of scissors along the ruled line we obtain two rings exactly like the first, except that they are only half the width. Next suppose that the end A is twisted through two right angles before it is gummed to B (the result of which is that the back of the strip at A is gummed over the front of the strip at B), then a cut along the line will produce only one ring. Next suppose that the end A is twisted once completely round {i.e. through four right angles) before it is gummed to B, then a similar cut produces two interlaced rings. If any of my readers think that these results could be predicted off-hand, it may be interesting to them to see if they can predict correctly the effect of again cutting the rings formed in the second and third experiments down their middle lines in a manner similar to that above described.
The theory is due to J. B. Listing* who discussed the case when the end A receives m half-twists, that is, is twisted through m pi, before it is gummed to B.
If m is even we obtain a surface which has two sides and two edges, which are termed paradromic. If the ring is cut along a line midway between the edges, we obtain two rings, each of which has m half-twists, and which are linked together ½m times.
If m is odd we obtain a surface having only one side and one edge. If this ring is cut along its mid-line, we obtain only one ring, but it has 2m half-twists, and if m is greater than unity it is knotted.
The key passage here is:
If m is even we obtain a surface which has two sides and two edges, which are termed paradromic.
This to me implies that a paradromic ring is not a Möbius strip!
Chas T. Whitmell. Cardiff Naturalists' Society. Reports & Transactions. Cardiff, Volume XXV (25); 1892–93, pp. 6–33, see p. 20. Not Illustrated
Space and its Dimensions.
PARADROMIC RINGS.
I conclude my remarks upon surface geometry with a brief reference to what are called Paradromic Rings. Take a strip of paper, bend it round into a ring and gum the ends together. We get thus a short hollow cylinder, or ring, with two surfaces (the inside and the outside), and two edges. Now take another strip, and give the paper a half twist before gumming the ends together. We thus produce a twisted ring with only one surface and only one edge. If the flat strip was 10 inches long, we shall find that the single edge of the twisted ring is 20 inches long, and that we can draw along its surface a continuous line of the same length, the end of such a line coinciding with its beginning. Let the line be drawn along the middle of the strip.
This to me implies that a paradromic ring is a Möbius strip!
William Frank White. A Scrap-Book of Elementary Mathematics; Notes, Recreations, Essays. Open Court Publishing Company, 1908, p. 117. Not illustrated
Paradromic rings.* A puzzle of a very different sort is made as follows. Take a strip of paper, say half as wide and twice as long as this page; give one end a half turn and paste it to the other end. The ring thus formed is used in theory of functions to illustrate a surface that has only one face: a line can be drawn on the paper from any point of it to any other point of it, whether the two points were on the same side or on opposite sides of the strip from which the ring was made. The ring is to be slit — cut lengthwise all the way around, making the strip of half the present width. State in advance what will result. Try and see. Now predict the effect of a second and a third slitting.
* The theory of these rings is due to Listing, Topologie, part 10. See Ball's Recreations, p. 75-6
This to me implies that a paradromic ring is a Möbius strip with multiple cuts.
Benjamin E. Smith (superintendent). The Century Dictionary Supplement. The Century Co., Volume 12, 1909, p. 942. Not illustrated
Running parallel: as a paradromic winding. Gives a dictionary definition. Possibly the first dictionary entry?
G. T. Bennett. ‘Paradromic Rings’. Letter in Nature, June 30, 1923, p. 882. Not illustrated
Paradromic Rings. PROF. C. V. BOYS, in his letter “A Puzzle Paper Band” in NATURE of June 9, p. 774, gives scant credit to the geometers. Forty years ago [1883] they described the endless band of paper with a half-turn twist in it, and found that if cut down the middle line it gave a single endless band with four half-twists. But they were so obsessed, he says, with the consequence of cutting down the middle line that they missed the result he now describes. This consists in taking a band with four half-twists and converting it by manipulation into a half-twist band of double thickness….
This to me implies that a paradromic ring is a Möbius strip with multiple cuts, but it’s so convoluted….
The Daily News, Wednesday, January 11, 1928, p. 3. Not illustrated
Breakfast Time Problem, by Henry E. Dudeney.
No. 322
Answer to problem 321
Let the paper be in the form of a ribbon of conventional length…have nowhere turned over the edge it is clear that it has only one side. Most readers are probably aware that this is a paradromic ring, and that if you cut all along the line you have made you will get, to your surprise, one large ring. If you give the…
This to me implies that a paradromic ring is a Möbius strip!
Frank Hyde Maberly. Nature, Volume 129, No. 3252, February 27, 1932, p. 317. Not illustrated
A Symbol of the Space-Time Continuum
If a strip of stamp edging is taken, given a half twist, and the ends joined, we have formed one of the well-known paradromic rings, having only one surface and one edge.
Synonym.
The Children's Newspaper (edited by Arthur Mee). January 16, 1932, not paginated. Not illustrated.
A One-Sided Ribbon.
If you ask a friend whether it is possible for a piece of ribbon to have both length and breadth but only one side to it, he will probably tell you that it is not.
To show him that lie is wrong take a ribbon of paper and paste the two ends together, but before doing so turn one end over so that the paper ring has a twist in it. Now if you start at any point to draw a pencil line along the length of the ribbon you can keep straight on without lifting your pencil until you have covered the whole surface.
This is called a paradromic ring, and if you cut along the pencil line you will find that instead of the paper falling into two parts you will get one large ring.
Synonym.
George D. Bishop. Teaching Mathematics in the Secondary School. Collins, 1965, pp. 137–138.
Not illustrated.
Paradromic rings. Take a strip of paper or piece of tape, 1 in. or 2 in. wide, at least 9 or 10 in. long. Rule a line in the middle down the length AB of the strip, gum one end over the other end, B, so as to get a ring like the section of a cylinder. If this ring is cut by a pair of scissors along the ruled line the result is 2 rings exactly like the first except only half the width. But if A is twisted through 2 right angles before it is gummed to B (back of strip at A is gummed over front of strip at B) then a cut along the line will produce only one ring. Next if the end A is twisted once completely round (i.e. through 4 right angles) before it is gummed to B then a similar cut will produce 2 interlaced rings. (All)
Synonym.
Belfast Telegraph, Saturday, January 31, 1970, p. 6. Not illustrated
A piece of paper with only one side!
Look Around for Younger Readers
Have you ever seen a piece of paper with one side?..... That's why figures of this kind are called MOBIUS [sic] STRIPS. Another name for them is PARADROMIC RINGS. Paradomic simply means “running side by side”...George McBride.
Synonym.
William L. Schaaf. A Bibliography of Recreational Mathematics. NCTM, Vol. 3. 1973, p. 155. Not illustrated
Glossary.
Paradromic rings. A modification of the familiar Möbius band, given m half-twists (i.e., when one end of the original strip is turned through an angle of rπ] radians). When m is even, a surface with two sides and two edges is obtained; when cut along the center line, this surface yields two rings each having m half-twists and linked together m/2 times. When m is odd, the surface obtained has only one side and one edge; when cut along its center this surface yields only one ring, which has 2m + 2 half-twists (if m > 1, the ring is knotted). (ALL)
One of the few dedicated definitions. That said, I’m still unsure of the intention here!
Steven Schwartzman. The Words of Mathematics: An Etymological Dictionary of Mathematical Terms used in English. Washington, DC: Mathematical Association of America. 1994, p. 158. Not illustrated
paradromic (adjective): the first element is from Greek para “alongside,” from the Indo-European root per originally “forward, in front,” but with many extended meanings. The second element is from Greek dromos “a running,” from the Indo-European root der “to run.” Suppose the end of a strip of paper is twisted a certain number of times and then joined to the opposite end of the strip; if you now use a pair of scissors to cut completely around the strip parallel to and a certain distance from an edge, the resulting rings are said to be paradromic. Before being cut they “ran alongside” each other. The most familiar paradromic rings are the ones that result from cutting a Moebius strip. [165, 38]
One of the few dedicated definitions. Gives a very nice derivation.
Carson J. Bruns and J. Fraser Stoddart. The Nature of the Mechanical Bond: From Molecules to Machines. John Wiley & Sons Ltd, 2017, p. 71. Illustrated
2. 2. 3
The Möbius Approach
The Möbius strip is an iconic object… The bands with an even number of half twists have the more intuitive two-surface situation common to ribbonlike objects, and are known rather as paradromic rings. Another interesting property, exclusive to the Möbius strips (n = odd), is that an off-center cut parallel to the edge will afford (Figure 2.8c) a link comprising a large ring with (2n+2) half-twists and a smaller ring...
At last, a description of paradromic rings I can understand (in an academic chemistry book)! Why can’t others explain it as well as this? Odd twists = Möbius strip; even twists = Paradromic ring. Simple! But do others follow this?
Peter Prevos. The Möbius Strip in Magic – A Treatise on Afghan Bands. Third Hemisphere Publishing, Kangaroo Flat, Australia (eBook) 2018, pp. 8–9. Not illustrated
Topology
Topology is one of the most…
The children of Möbius strips are called paradromic rings, which means rings that run parallel to each other….
A nice, considered treatment. This implies that all odd twisted strips are to be considered paradromic.
Printed
Summary
As such, the essay admirably documents the usage of paradromic rings, but it must be admitted, to little purpose in understanding the intricacies, as the term is used in a loose sense. One author will say one thing, and another something else! And then sometimes it appears to be used as a synonym (as an ‘upgrade’?) for a Möbius strip. What can you do? Even now, at the nominal end of the study, I am unclear. I say nominal, in that there will always be more to do, with further searches. Be that as it may, I do not unduly begrudge the study, albeit it has taken two distinct visits, over eleven days, which is strictly hard to justify. Indeed, it is a ridiculous amount of time, out of all proportion as to worth. However, it had to be done to properly study, rather than leave vast swathes untouched, all with the danger of having forgotten the intricacies upon a later-day return.
No matter, it can be regarded as just another brick in the wall of documenting the progress in the study of the Möbius strip in the round, and which it contributes notably so. Undoubtedly, omissions and (minor) errors remain, which can be addressed after, benefiting from a distance in time.
APPENDIX 1
At-a-Glance Inventory (excised from the references)
On occasion, there are various concerns and quibbles, indicated by superscripts, addressed in the footnotes.
18481. Johann B. Listing. Vorstudien zur Topologie, pp. 45–47
1882. The Encyclopedia Britannica. A Dictionary of Arts, Sciences, and General…, pp. 128–129
1883. Peter G. Tait. ‘Johann Benedict Listing’. Nature. February 1, Volume 27, p. 317
1884. Peter G. Tait. ‘Prof. Tait on Listing’s Topologie’. The London, Edinburgh…, pp. 37–38
1892. W. W. Rouse Ball. Mathematical Recreations and Problems of…, 1892
1892. Chas. T. Whitmell. ‘Space and its Dimensions’. In Cardiff Naturalists' Society, p. 20
1893. Anon. ‘Listing’s Paradromic Rings’. The Echo. Tuesday, 8 September, 1893, p. 1
1893. W. B. Croft. ‘Listing’s Paradromic Rings’. English Mechanic and Mirror of Science, pp. 66–67
1893. W. B. Croft ‘Listing’s Paradromic Rings’. (Correction) English Mechanic and Mirror…, p. 157
1894. W. H. F. Patterson. ‘Some Experiments with Paper Rings’. English Mechanic, January, p. 511
1902. Anon. ‘Knots’. The Globe and Traveller. Saturday Evening, July 12, 1902, p. 1
1906. Anon. Paradromic Rings. Linlithgowshire Gazette, Friday, March 2, 1906, p. 3
1907. Thirty-Eighth Annual Report of the Wellington College Natural Science Society, p. 22
1908. William F. White. A Scrap-Book of Elementary Mathematics; Notes, Recreations…, p. 117
1908. Paul Carus (ed.). The Open Court. Open Court Publishing Company, (advert) Not paginated.
1909. The Century Dictionary Supplement, p. 942
19092. Werner Encyclopaedia. A Standard Work of Reference in Art…, pp. 128–129
1914. V. E. Johnson. ‘Tricks for the Young Entertainer’. The Boy's Own Annual, pp. 165–166
[1922]. Mabel Sykes et al. Solid Geometry, p. 207
1922. The Mathematical Clubs of Weseley College, Wellesley, Mass., p. 419
1923. G. T. Bennett. ‘Paradromic Rings’. Letter in Nature, June 30, 1923, p. 882
19253. Harold T. Davis et al. A Course in General Mathematics, p. 275
1928. Henry Dudeney. ‘Breakfast Time Problem’, No. 322. The Daily News, January 1, p. 3
1931. F. W. Westaway. Craftsmanship In The Teaching Of Elementary Mathematics, p. 615.
1932. Frank H. Maberly. ‘A Symbol of the Space-Time Continuum’. Nature, p. 317
1937. Anon. The Society of the Engineers, pp. 9, 12
1943. Anon. The Children's Newspaper. December 4, p. 8
1947. Scientific American, September, p. 138
[1953]. Howard W. Eves. An Introduction to the History of Mathematics.
1965. George D. Bishop. Teaching Mathematics in the Secondary School. pp. 137–138
1970. George McBride. ‘A piece of paper with only one side!’ Belfast Telegraph, January 31, p. 6
1973. William L. Schaaf. A Bibliography of Recreational Mathematics. Vol. 3, p. 155
1977. Adrien L. Hess. Mathematics Projects Handbook. NCTM, p. 16
1994. Steven Schwartzman. The Words of Mathematics: An Etymological Dictionary of…, p. 158
1999. Eric W. Weisstein. CRC Encyclopedia Of Mathematics. p. 1180
2004. David J. Darling. The Universal Book of Mathematics: From Abracadabra to…, pp. 208–210
2005. Catherine A. Gorini. Facts on File. Geometry Handbook,pp. 104, 116
2006. Clifford Pickover. The Möbius Strip: Dr. August Möbius's Marvelous Band in…, pp. 65–66
2009. Timothy V. Craine et al (ed.). Understanding Geometry for a Changing World, pp. 33–48
2014. Gianni A. Sarcone. Impossible Folding Puzzles and other Mathematical Paradoxes, pp. 29, 50
2017. Carson J. Bruns et al. The Nature of the Mechanical Bond: From Molecules to Machines, p. 71
2018. Mark Blacklock. The Emergence of the Fourth Dimension: Higher Spatial …, p. 44
2018. Peter Prevos. The Möbius Strip in Magic – A Treatise on Afghan Bands. T 2018, pp. 8–9
1 Listing refers to ‘paradromen’ five times with no obvious connection to (paradromic) rings (twice in connection to ‘spiral lines’). However, as the term is then subsequently used by others arising from this reference, I have decided to include.
2 This repeats the text in The Encyclopedia Britannica, 1892
3 Presumed date; I have only seen a later edition, of 1935.
APPENDIX 2
Frequency Analysis
As can be seen, the term is seldom used, with many years not having a single recorded instance, with gaps at times of a decade! And even when recorded, typically, there is just one or two references! Note that as some of the newspaper references have been syndicated, I only include the first instance here, as this will otherwise distort the frequency.
1848 (1)
1882 (1)
1883 (1)
1884 (1)
1892 (2)
1893 (3)
1894 (1)
1902 (1)
1906 (1)
1907 (1)
1908 (2)
1909 (2)
1914 (1)
1922 (2)
1923 (1)
1925 (1)
1928 (1)
1931 (1)
1932 (1)
1937 (1)
1943 (1)
1947 (1)
1953 (1)
1965 (1)
1970 (1)
1973 (1)
1977 (1)
1994 (1)
1999 (1)
2004 (1)
2005 (1)
2006 (1)
2009 (1)
2014 (1)
2017 (1)
2018 (2)
REFERENCES
(Basic)
Anon. The Society of the Engineers. 1937 London, England, pp. 9, 12.
Ball, W. W. Rouse (Walter William Rouse). Mathematical Recreations and Problems. London Macmillan and Co. and New York, Second edition, 1892, pp. 53–54. (First Edition February 1892, not seen.)
Bennett, G. T. ‘Paradromic Rings’. Letter in Nature, June 30, 1923, p. 882.
Bishop, George D. Teaching Mathematics in the Secondary School. Collins, 1965, pp. 137–138.
Blacklock, Mark. The Emergence of the Fourth Dimension: Higher Spatial Thinking in the Fin de Siècle. Oxford University Press, 2018.
Bruns, Carson J. and J. Fraser Stoddart. The Nature of the Mechanical Bond: From Molecules to Machines. John Wiley & Sons Ltd, 2017, p. 71.
Carus, Paul (ed.) The Open Court. Open Court Publishing Company, Chicago. Vol. XXII, No. 629, October 1908. Not paginated.
Craine, Timothy V. and Rheta Rubenstein (ed.). Understanding Geometry for a Changing World. Reston, VA: National Council of Teachers of Mathematics, 2009, pp. 33–48.
Croft, W. B. ‘Listing’s Paradromic Rings’. English Mechanic and Mirror of Science. September 8, 1893, pp. 66–67.
————. ‘Listing’s Paradromic Rings’. English Mechanic and Mirror of Science. October 6, 1893, p.
157.
Darling, David J. The Universal Book of Mathematics: from Abracadabra to Zeno's Paradoxes.
Hoboken, N.J.: Wiley, 2004, pp. 208–210.
Davis, Harold Thayer and Malcolm G. McCown. A Course in General Mathematics. Principia Press of Trinity University. 1962. Originally published 1925, p. 275.
Dudeney, Henry E.. ‘Breakfast Time Problem’ No. 321. The Daily News, Tuesday, January 10, 1928, p. 3.
————. ‘Breakfast Time Problem’, No. 322 (Answer). The Daily News, Wednesday, January 11, 1928, p. 3. Answer to problem 321
Elran, Yossi and Ann Schwartz. ‘Should We Call Them Flexa-Bands?’. In The Mathematics of Various Entertaining Subjects, pp. 249–261.
Eves, Howard Whitley. An Introduction to the History of Mathematics. New York, Rinehart [1953]: Fourth Edition
Fan, Runzhu and Li Xiang, ‘The Structure and Topological Properties of Möbius Strip Dissection’, Shing-Tung Yau High School Mathematics Awards. 2012, pp. 327+.
Gorini, Catherine A. Facts on File Geometry Handbook. New York, N.Y.: Checkmark Books, 2005, pp. 104, 116.
Hess, Adrien L. Mathematics Projects Handbook. Reston, Va.: National Council of Teachers of Mathematics, 1977, p. 16.
Ho, Nancy, J. Godzik, Jennifer Jones, T. Mattman, Dan Sours. ‘Invisible Knots and Rainbow Rings’. Mathematics Magazine, Vol. 93, No. 1, February 2020, pp. 4–18.
Johnson, V. E. ‘Tricks for the Young Entertainer. Some Simple and Diverting Magical Mysteries’. The Boy's Own Annual, Volume 37, 1914, pp. 165–166.
Linlithgowshire Gazette, Friday, March 2, 1906, p. 3.
Maberly, Frank. Hyde. ‘A Symbol of the Space-Time Continuum’. Nature, Volume 129, No. 3252, February 27, 1932, p. 317.
McBride, George. ‘A piece of paper with only one side!’ Belfast Telegraph, Saturday, January 31, 1970, p. 6.
Patterson, W. H. F. ‘Some Experiments with Paper Rings’. English Mechanic, No. 1505, January 26, 1894, p. 511.
Pickover, Cliff. The Möbius Strip: Dr. August Möbius's Marvelous Band in Mathematics, Games, Literature, Art, Technology, and Cosmology. New York: Thunder's Mouth Press, 2006, pp. 65–66.
Sarcone, Gianni A. Impossible Folding Puzzles and other Mathematical Paradoxes. Mineola: Dover Publications, 2014, pp. 29, 50.
Schaaf, William L. A Bibliography of Recreational Mathematics. NCTM, Vol. 3, 1973, p. 155.
Schwartzman, Steven. The Words of Mathematics: An Etymological Dictionary of Mathematical Terms used in English. Washington, DC: Mathematical Association of America. 1994, p. 158.
Scientific American, September 1947, p. 138.
Smith, Benjamin E (superintendent). The Century Dictionary Supplement. The Century Co., Volume 12, 1909, p. 942.
Sykes, Mabel and Clarence Elmer Comstock. Solid Geometry. Rand, McNally & Company, Chicago, New York, [1922], p. 207.
Tait, Peter Guthrie. ‘Johann Benedict Listing’. Nature. February 1, Volume 27, 1883, p. 317.
Tait, Peter Guthrie. ‘Prof. Tait on Listing’s Topologie’. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Series 5, Issue 103, Volume 17, 1884, pp. 37–38.
The Children's Newspaper (edited by Arthur Mee). January 16, 1932, not paginated. (7 October 2024).
The Echo. Tuesday, September 8, 1893. p. 1.
The Encyclopedia Brittanica. A Dictionary of Arts, Sciences, and General Literature. Ninth Edition. American Reprint, Vol. XIV. Philadelphia: J. M. Stoddart & Co. 1882, pp. 128–129.
The Globe and Traveller. Saturday Evening, July 12, 1902, p. 1.
‘The Mathematical Clubs of Weseley College, Wellesley, Mass’, The American Mathematical Monthly, Volume 29, 1922, p. 419.
Weisstein, Eric W. CRC Encyclopedia Of Mathematics. 1999, p. 1180.
————. Wolfram
Wellington College Natural Science Society. Thirty-Eighth Annual Report of the Wellington College Natural Science Society, 1907/1908, p. 22.
Werner Encyclopaedia. A Standard Work of Reference in Art, Literature, Science, History, Geography, Commerce, Biography, Discovery and Invention, Vol. 14. Akron, Ohio, The Werner Company, 1909, pp. 128–129.
Westaway, F. W. Craftsmanship In The Teaching Of Elementary Mathematics. Blackie and Son Ltd. 1931, 1937, p. 615.
White, William Frank. A Scrap-Book of Elementary Mathematics; Notes, Recreations, Essays. Open Court Publishing Company, 1908, p. 117.
Whitmell, Chas. T. ‘Space and its Dimensions’. In Cardiff Naturalists' Society. Reports & Transactions. Cardiff, Volume XXV (25); 1892–93, pp. 6–33, see p. 20.
Wikipedia. Möbius strip
Wikipedia. J. B. Listing.
Page Created: 28 January 2026.
A resumption of adding aspects of my Möbius Strip studies as webpages after a break, with the material cut and pasted from the source (essay), with, on occasion, modifications more suitable for the webpage (e,g. excluding 'notes' personal to self, of no interest to the viewer). As with all these studies, there is a certain lag between the (mature) essay and its appearance as a webpage.
Essay Text: 30 September; 1–4, 7 October 2024 (Version 1), 11–15 August 2025 (Version 2)