Martin Gardner and Paperfolding
Martin Gardner is best known for his long-running monthly column of Mathematical Games in the venerable science magazine “Scientific American”. As his column developed it embraced many topics beyond the strict interpretation of its title but even then they did not by any means exhaust the whole of Martin’s wide-ranging interests. When David A Klarner edited a volume of mathematical recreations in tribute to Martin Gardner in 1981, he punningly gave it the title, "The Mathematical Gardner", but he hoped that it would later be accompanied by other volumes such as the Magical Gardner, the Literary Gardner, the Philosophical Gardner or the Scientific Gardner in tribute to Martin Gardner's wide-ranging interests. He did not suggest the “Origami Gardner", but it is a fact not often appreciated that paperfolding of a kind, though not strictly mainstream origami, was the origin of his column in “Scientific American”.
Some time during 1956, Martin Gardner submitted to Scientific American a short article about “Hexaflexagons”, curious folded paper devices, which had been discovered by chance by Arthur H. Stone, an English post-graduate student at Princeton University in 1939 and which were only now beginning to become more widely known. Martin’s article appeared in the issue of Scientific American for December 1956 and the editorial board of the journal was clearly impressed because Martin Gardner was asked for more articles. They followed without interruption from January 1957 until 1980. During 1981 he shared the column with Douglas Hofstadter, before retiring at the end of 1981. Hofstadter was the author of the very successful book “Escher, Gödel, Bach” which had appeared in 1979. Martin finally retired from his column at the end of 1981, apart from two further "surprise" columns in August and September 1983. 1956 was also the year of publication of Robert Harbin's "Paper Magic" and 1955 had seen the exhibition of the paperfolding art of Akira Yoshizawa arranged by Gershon Legman at the Stedelyjk Museum in Amsterdam. Suddenly paperfolding was in the air and the new initiatives quickly led to the formation by Lillian Oppenheimer of the Origami Center, in New York in October, 1958.
Yet Martin Gardner was not primarily a paperfolder, any more than he was a mathematician or a scientist. His first enthusiasm was conjuring and it was out of this that his other diverse recreational interests sprang, especially paperfolding.
Martin Gardner was born in Tulsa, Oklahoma on 21 October 1914. His father, a geologist, introduced him to magic when he taught him the "Paddle Trick" which employs a table knife and several pieces of paper. Martin describes several versions in his "Encyclopedia of Impromptu Magic". Before long he was inventing tricks of his own and he was only sixteen, and still at high school when he began contributing to the magical magazine, "The Sphinx". His first article was a "New Color Divination" in May 1930. Already, his contribution for the following August carried the somewhat precocious title "The Best Pocket Tricks of Martin Gardner". It included a version of the knife paddle move he had learnt from his father.
While a boy, Martin enjoyed Frank Rigney's jokes in “American Boy's Magazine”. Frank Rigney was also a conjuror, but he is best known to paperfolders as the illustrator and co-author with William D. Murray of "Fun with Paperfolding", which was published in 1928. For many years it remained the best introduction to paperfolding in English and influenced many paperfolders who later became well-known, including Lillian Oppenheimer. Far from being merely the illustrator of "Fun with Paperfolding", Frank Rigney was also the creator of several of the folds it included. Later, when Frank became the illustrator for Hugard's Magic Monthly for which Martin was contributing a regular column on magic, they became close personal friends. That was, however, in the future. Martin entered the University of Chicago where his principal studies were in philosophy. He was a resident of Hitchcock Hall and became a member of Phi-Beta-Kappa. He graduated as Bachelor of Arts in 1936 but for a time continued as a post-graduate student.
In November 1935, while he was still an undergraduate, Martin Gardner wrote “Match-ic” the first of his many publications. This was a slim booklet of tricks with matches. "Match-ic" contained nothing about paperfolding, but his next booklet, "After the Dessert", which was first issued in a mimeographed version in 1940, and published in a printed version in 1941, contained a number of paper-related tricks, not least of them the "Japanese Paper Bird" which Martin had found in Houdini's "Paper Magic" of 1922 and traced back to the supplement to the second edition of 1890 of Tissandier's "Scientific Recreations", translated from the original French. There is a silly but baffling trick, whereby a dollar bill was mysteriously turned upside-down, a trick which Martin Gardner included in several subsequent publications. He also included a stunt with a dinner napkin which he frankly described as an "Improvised Brassière". It was an old trick popular with conjurors and often politely bowdlerized and placed on the head as "Cat's Ears".
It appears that even at this age Martin Gardner envisaged a journalistic or literary career. He had already written a short story called "Thang" for his college literary magazine. After a short period of research, he took a job as a reporter for the Tulsa Tribune and also became a staff writer with the press-relations department of Chicago University. However, war intervened in 1941. He joined the United States Navy, and saw active service in a destroyer escort in the North Atlantic. More recently destroyer escorts have been reclassified as frigates. A destroyer escort was a small ship that could act as a scout for the fleet, searching out enemy submarines. With the constant threat from Nazi U-boats and the frequent storm-force winds of the North Atlantic, it was far from being a comfortable job. In his later book, "Whys and Wherefores", he gave a rare glimpse into his personal life when he wrote that his destroyer escort was a "ship small enough so that a sailor could really get to know the sea in a way quite different from that of the tourist who floats gently on the ocean in a huge hotel."
AFTER THE WAR
Following the War, Martin Gardner returned to Chicago to take up research again. But in 1946 he was aged 32 and scarcely a young post-graduate student. He resumed his writing and his short stories began to sell. Many appeared in "Esquire" magazine and he also wrote for "Humpty Dumpty", a magazine for children, for which he became contributing editor. In a short time he was able to devote himself to his freelance writing. In 1947 he moved to New York and for many years he lived in or near the city.
Martin Gardner married Charlotte Greenwald in 1952. They enjoyed 48 years together until she died in December, 2000. There are two children of the marriage, Jim, who became an assistant professor of educational psychology at the University of Oklahoma and Tom, who became a free-lance artist in Greenville, South Carolina
Martin’s book "New Mathematical Diversions", which was published in 1966, contains a cryptic dedication to his wife:
To L.R. AHCROF
In case anyone has tried unsuccessfully to translate it as Latin, the simple solution is that backwards it reads: "One more time for Charlotte my love." (It was the second book that Martin had dedicated to Charlotte, the first one being “Great Essays in Science”, a collection of classic scientific essays which he had edited.) Equally cryptic is the more straightforward dedication of his philosophical book "The Whys of a Philosophical Scrivener" of 1983:
"Why do I dedicate this took to Charlotte? She knows".
For a time the Gardners lived at Hastings-on-Hudson in New York State in a street very appropriately called Euclid Avenue. In 1982 they moved south to the warmer climate of Hendersonville in western North Carolina. Martin Gardner and a new generation of grandchildren have been able to share in his countless tricks and illusions with mutual delight.
Martin Gardner's interest in paperfolding is rooted firmly in his earlier passion for conjuring. The association between the two is a common one and countless leading paperfolders have also been magicians. There is a sort of mind which is attracted to puzzles, illusions, mathematical structures, patterns and unexpected transformations, linkages - and paperfolding. Martin Gardner has such a mind and intellectually it led him to wrestle with the awesome conundrums of philosophy. But he never lost touch with those trivial tricks and puzzles which lighten the burden of life.
One of the roots of modern paperfolding was in conjuring and it can be traced back well into the 19th century. Indeed, “The Magic Fan” or "Troublewit", as it is usually known in English, is a kind of paperfolding which goes back much further than that. The “Japanese Paper Bird” (now invariably known as “The Flapping Bird”), which Martin Gardner included in "After the Dessert" was apparently introduced to the West by travelling Japanese conjurors, probably in the1870s. There is still, however, some uncertainly about its true origin. In the 1920's, magicians like Will Blythe, Will Goldston and Houdini introduced magic using paper or "paper magic" into their acts. Towards the end of the Second World War the folding of dollar bills exercised a fascination for conjurors and instructions for tricks with dollar bills became a popular feature of the numerous magical magazines of the time. In his preface to Samuel Randlett's "The Best of Origami" (1963) Martin Gardner recalled attending a magic convention in the 1930s at which almost every magician present was wearing a finger ring with a large rectangular "jewel" that he had folded from a dollar bill.
Martin Gardner had always sought the company of other magicians. While still at home in Tulsa he had been a member of a group which included Logan Wait and Roger Montandon and as soon as he moved to Chicago, he was able to attach himself to a very rich fraternity of magicians. He joined the Chicago Magic Table and at Christmas took a job in a department store demonstrating magic sets. Magic shops are always meeting places for conjurors, anxious to find the latest tricks and Martin has told how he hung round Joe Berg's and Laurie Ireland's magic shops in Chicago. Laurie Ireland’s shop was later to pass through his widow to Jay Marshall, a good friend of origami, who renamed it Magic Inc. The shop later became a meeting place for CHAOS, the Chicago Area Origami Society. Another place of meeting for conjurors was the Nanking Chinese Restaurant. During his time at Chicago, Martin’s idol was Werner Dornfield and he later dedicated one of his books to "Dorny". Yet another feature of magical life in Chicago was the regular magical conventions. One wonders whether Martin had not chosen his University for the richness of its magical culture!
As soon as Martin moved to New York, he found a similar scene awaiting him. Lou Tannen's Magic Shop was one of the meeting places. Another was the apartment of Bruce and Bunny Elliott who frequently played host to meetings of magicians. Bruce was the editor of "The Phoenix", a magazine to which Martin Gardner became a frequent contributor. Indeed he was a frequent contributor to numerous magical magazines and inevitably became well-known to the conjurors of his generation. Only when other pressures built up did he relax his enthusiasm for practical magic, especially when his Scientific American column made increasingly voracious demands on him. It was, however, the move to North Carolina in 1982 that severely reduced his links with other magicians. Desirable though the move might have been for other reasons, there was, unfortunately, no community of conjurors in Hendersonville.
After his move to Hendersonville and the conclusion of his column for Scientific American, Martin Gardner took up again the fallacies perpetrated as science by self-appointed experts in many strange areas. His book "In the Name of Science" was first published in 1952 and in 1956 and was republished as a paperback under the name "Fads and Fallacies in the Name of Science" by Dover Publications. In it he brought his rational thinking to bear to expose the fallacies of such ideas as "Pyramidology" and the "Flat-Earth Theory”. He became a fellow of the Committee for the Scientific Investigation of Claims of the Paranormal and has written a regular column for its journal "The Skeptical Inquirer".
Martin Gardner's line of practical magic was the uncomplex, requiring the minimum of preparation. He eschewed elaborate mechanical illusions, and concentrated on tricks that made use of everyday objects, which were exemplified in his "Encyclopedia of Impromptu Magic". This “encyclopedia” originally appeared as a long series of short articles in Hugard’s Magic Monthly and was later published as a book by Magic Inc. of Chicago in 1978. It probably gives an unbalanced view of Martin's conjuring, for it excludes card tricks and tricks which require practised sleight of hand, but the book gives a good general view of his approach. Separate from this were Martin’s many articles on magic which constantly appeared in several other magical magazines and which were later collected together and published in 1993 in “Martin Gardner Presents”, another large book of the same size as the “Encyclopedia of Impromptu Magic”.
Paperfolding and paper tricks fitted in well with this scheme of things. Added to this, the basic geometry of folding paper fascinated Martin Gardner, the mathematician. His long series of contributions to Hugard's Magic Monthly began in 1948 and his contributions for February and September 1949 were on Dollar Bill Folds and on Stunts with Paper. For the Dollar Bill Folds, Martin included a Blowing Fish which could be made to blow out a candle, a method of reducing the size of a bill by folding, and "the Mushroom", two folds in a dollar bill which convert George Washington into a mushroom. It was a popular trick of Martin's for he has included it in several of his books. The article on Stunts with Paper was less concerned with paperfolding as such. It included a circle of paper-cut bunnies which was a simple device for apparently changing the expression of a face drawn on paper) named “Movies”. It also included a simple "Snapper", an animal’s head made from a small piece of card, which has the strength to pick up quite large objects in its "jaws".
Frankly, none of these tricks is very good paperfolding, or “Origami” as it was later to be known through the founding of the Origami Center in New York in October, 1958. Yet it was probably these articles that brought Martin Gardner to the attention of Gershon Legman. Gershon Legman was then still residing in New York before he moved to live in France in 1953. Legman had been avidly collecting information about paperfolding from all possible sources since 1945 and he wrote to anyone he thought might be able to tell him more about the subject or who might give him information for the list of books and papers about it which he was compiling. Gershon Legman was not himself a magician, but he was acquainted with many people who were conjurors, including Cy Enfield, the film producer, with whom he had been at school. It was Cy Enfield who taught Gershon Legman the Bow Tie or Lotus which set him off on his intensive quest for paperfolding in1945. (By a curious turn of events that amounted to a stroke of fate, it was Cy Enfield who introduced Robert Harbin to Gershon Legman.) But if Gershon Legman discovered that Martin Gardner was interested in tricks and paperfolding, we may be sure that through their acquaintance, Martin Gardner was introduced to the wider knowledge of paperfolding being built up by Gershon Legman.
There was a marked increase in Martin's paperfolding activity. In 1952 He contributed three items, not to a magic magazine, but to the “Children's Digest”. These were "How to Make a Paper Boat", "Loop the Loop" and "You'll Get a Bang out of This!” I have not seen these but the first two were probably some sort of boat and a flying device and the third was the traditional banger. Then in September 1952 Martin contributed a glider to "Parents' Magazine". But this was not quite the traditional paper glider folded from paper because it incorporated a modified nose which enabled it to be propelled by an elastic band. The refinement was, in fact, a discovery of Gershon Legman. Because “Children’s Digest” and “Parents’ Magazine were not magic magazines, these articles were omitted from the collection of Martin Gardner’s articles in “Martin Gardner Presents” and it appears that they have never been reprinted.
Gershon Legman published a preliminary edition of his bibliography in the magazine "Magicol" in May 1952, and his longer “Bibliography of Paper-folding” in a slender booklet later in the year. In this he listed Martin Gardner's two contributions to Hugard's Magic Monthly in February and September 1949 and his three contributions to "Children's Digest" between January and May 1952. The contribution of Gershon Legman's modified glider by Martin Gardner to Parents’ Magazine in September 1952 was pointedly not listed.
It would be wrong to imply that arising from these ublications there was an immediate intensification of Martin Gardner's interest in paperfolding. For him it remained, as it always had been, just an aspect of conjuring. We have noticed that some paperfolding was already included in Martin Gardner's early booklet "After the Dessert" (1940-1941). After the War, in 1949, he published another booklet in a similar vein called "Over the Coffee Cups". This, too, contained one or two paperfolding items. One was the notorious folding of a dollar bill which brought together the words "GAL TENDER AND PRIVATE" reputed to be used by a traveller who was asking for company in his room. Another was a trick by Samuel Berland. Once more, this was scarcely paperfolding, but it employed a method of folding one dollar bill to look like two bills, creating an illusion whereby the second bill was made to vanish.
Another favourite paper gimmick of Martin Gardner's was the Moebius Bands which he often presented under the mysterious name of the Afghan Bands. This name had been used by the famous Professor Hoffman as long ago as 1904, but its origin remained a mystery. A version of the Afghan Bands by Martin Gardner appeared in Hugard's Magic Monthly in December 1949. Moebius Bands are frequently made of paper, although they are not in essence paperfolding, and Gershon Legman did not think it appropriate to include the article in his Bibliography. Rather do the Afghan Bands portend the way Martin Gardner's interest would develop in the future. He was to include them in articles in Scientific American and in later books written by him.
Hugard's Magic Monthly quickly became a focus for much of Martin Gardner's work. After contributing some thirty-two articles between July 1945 and February 1951, he began a monthly column that ran from March 1951 until March 1958. The articles reflected Martin's general and uncomplicated approach to conjuring. He named the series the “Encyclopedia of Impromptu Magic”. I have not examined the individual copies of the magazine, but it seems that from the start they were envisaged as the basis for a comprehensive book of conjuring. As he came across them Martin made a practice of jotting down notes about conjuring tricks on filing cards which he kept in shoe boxes. When he wished to write an article, he rapidly recovered the information from the filing cards. Martin originally hoped to revise the compilation against published books in order to give acknowledgments to inventors of ideas and to polish it up generally. But time (as ever) would not allow and the collected articles were eventually published by Magic Inc. of Chicago in 1978. For the most part the articles were reprinted "raw". There are a few revisions and added notes but on the whole they are very limited and uneven.
Several of the headings in the book version of the Encyclopedia of Impromptu Magic are relevant to paperfolding including those on "Bill Folds", "Handkerchief Folding", "Magazine", "Newspaper" and a very long section headed simply "Paper". This last heading falls into various subsections, such as paper magic, paper stunts, paper cutting, paperfolding, paper work and geometrical curiosities. Much under this heading is related in some way to paperfolding in the wide sense of the term. Here, for instance are a long section on the Moebius Bands, a short one on geometrical folding, and a section on hexaflexagons. Many of the items are very familiar, and come from the international stock of traditional indoor recreations.
The section on Paper Folding, as such, is eight pages long and begins with a brief introductory history which mentions Japan, Unamuno and Froebel. There follows a short bibliography with tribute paid to Gershon Legman's longer bibliographies. It contains just sixteen items ranging from Will Blyth's "Paper Magic" of 1920 to Samuel Randlett's "The Best of Origami" of 1963. Obviously the list of books had either been updated or compiled specially for the book form of the Encyclopedia. Nevertheless it remains a very selective list.
Martin frankly describes the folded paper figures that he includes as only a selection, chosen because they can be animated in some amusing way. He freely acknowledges that figures of great realism and beauty may be found in the Oriental and Spanish works on paperfolding. (This statement is also dated in its curiously restricted reference to the Orient and Spain and it has clearly not been updated to include reference to the Western creations in Samuel Randlett's "Art of Origami" (1961) and "Best of Origami" (1963) or Robert Harbin’s “Secrets of Origami” (1963).
The models mentioned or reproduced include the bellows, the "hopping" frog, a boat that floats on water, a Pop Gun, the Paper Cup, the Kettle (in which water can be boiled) and the Salt Cellar in its various forms of "bug catcher" and "fortune-teller". They are all very interesting and Martin throws considerable light on each item. But what is significant is that it is merely a collection of existing folds, and apparently it does not include any folds of Martin’s own devising.
An interesting feature of "Hugard's Magic Monthly" and, therefore, of the "Encyclopedia of Impromptu Magic" is that it was illustrated as a labour of love by Frank Rigney, previously mentioned as the co-author and illustrator of "Fun with Paper Folding" by William D Murray and Francis J Rigney (1928). Martin pays a gracious tribute to him in his introduction to the book version of the Encyclopedia. It was an association that made them firm personal friends.
The instalments of "The Encyclopedia of Impromptu Magic" ran in Hugard's Magic Monthly until March 1958, although Martin continued to contribute a few articles until September 1961. By March 1958, however, changes were in the air. Another magician, Robert Harbin, had published his book, "Paper Magic" in England in 1956. This book, despite its title, was unambiguously about paperfolding and was not about paper magic in the sense of conjuring with paper. Harbin, too, had only recently become acquainted by correspondence with Gershon Legman. Then, In the summer of 1957, Mrs Lillian Oppenheimer, who had been greatly impressed on receiving a copy of “Paper Magic”, flew across the Atlantic to meet Robert Harbin in London, but she failed to meet Gershon Legman in France as she had hoped, because he was away from home. The world of Western paperfolding was suddenly set alight and in October of the following year Mrs Oppenheimer unexpectedly (perhaps not entirely unexpectedly) found herself the founder of the Origami Center of New York and the editor of a new journal called "The Origamian".
THE ORIGAMI CENTER
For such a spontaneous organization the Origami Center was remarkably well-organised. By the second issue of the “Origamian” in November, 1958 a long list of no less than thirty-five Hononary Members had been appointed. All of them were issued with special printed membership cards. Among the list appears "Martin Gardner - Paperfolder, Author".
Clearly, for him to be chosen, Martin Gardner had made a significant impact on the young world of modern Western paperfolding. Up to 1958 his total achievements as an author about any subject could scarcely be described as prodigious, although his published books included "In the Name of Science" (debunking pseudoscientific notions), "Mathematics Magic and Mystery" (on magical tricks using mathematics) and "Great Essays in Science" (a collection of classic essays of which he was merely the editor). He had not written a single book solely devoted to paperfolding. Lillian Oppenheimer was, perhaps, generous with her Honorary Memberships (she had a shrewd head for publicity) and we can infer that Martin Gardner had entered fully into that small circle which interchanged information about paperfolding during the months which preceded and followed the emergence of the Origami Center.
Martin Gardner was not the only magician to be chosen. Others were Robert Harbin, Paul Duke and Jay Marshall of Magic Inc. Several other Honorary Members were not described as magicians, but, like so many paperfolders, they carried magic wands in their knapsacks, including Guiseppi Baggi, Shari Lewis, Robert Neale and "Thok Sondergaard" of Denmark (who was no less than Thoki Yenn). Another honorary member was Lester Grimes of La Rochelle, New York, the doyen of paper magicians. Since before the Second World War he had been billed as "The Paper Wizard", performing his act in a paper costume and using only paper equipment and materials. He was surely a friend of Martin Gardner's. Lester Grimes played an active part in the early days of the Origami Center.
There is no record that Martin Gardner attended any of the early meetings of the Origami Center and he is not mentioned again in the five issues of the first volume of the Origamian which were published between November, 1958 and March 1959. In fact, apart from occasional references to his books, he is seldom again mentioned in the many later issues of the Origamian after it resumed publication in the summer of 1961. There is, however, a report that Martin Gardner attended the "Second Annual Origami Get-together at the Origami Center (meaning Lillian Oppenheimer's private apartment) on 2nd and 3rd November 1963, one of the earliest origami conventions to be held.
Another report in the Origamian for summer 1964 refers to Martin Gardner's short article on Origami in Encyclopaedia Britannica, which first appeared in the edition published earlier in that year. The Origamian reports that Martin was considerably upset that he had written the article as long ago as 1959, but publication had been delayed. Although he had repeatedly asked to revise the article, the editors had adamantly refused to agree to this. As a result, the article was out of date before it appeared and it contained no mention of the distinguished American and other Western paperfolders who had emerged since 1959, not least of them Fred Rohm and Neal Elias. No doubt the editors of Encyclopaedia Britannica had their reasons, but Martin Gardner felt very embarrassed.
As it is, the article is interesting for a number of reasons. It reveals that in 1959 Martin Gardner was considered a sufficient authority on origami to be invited to write the article. The article mentions Akira Yoshizawa, Miguel Unamuno and Vicente Solarzano Sagredo, the three most distinguished paperfolders before the formation of the Origami Center. Friedrich Froebel is mentioned in connection with the Kindergarten movement and the Bauhaus is mentioned in connection with training students in commercial design. Arthur H. Stone is given credit for the discovery of flexagons. Two modern folders who are named are George Rhoads and Guiseppe Baggi. Altogether the article is a remarkably comprehensive, but compact summary of paperfolding as it was in 1959. There is no mention of the Origami Center, but perhaps it was too soon for that. One wonders how Martin would have revised the article had he been allowed to do so in 1964, assuming that he would have been allocated no increase in space.
Not directly linked to the Origami Center, but certainly associated with it was the exhibition "Plane Geometry and Fancy Figures" held at the Cooper Union Museum, New York from the beginning of June in the summer of 1959. The exhibition had already been planned when the summer of 1958 suddenly brought Origami to the notice of the public in newspapers and on television. Because of this, Lillian Oppenheimer was invited to provide models for a section of the exhibition to be devoted to Origami. She gathered models from the United States, Europe and Japan. Martin Gardner was invited to contribute, but the catalogue lists only one model under his name, which was a flying bat. Although it was not apparent from the static exhibit, Martin has revealed that his bat had a secret. If its head was placed on one’s fingertip, it balanced horizontally. The secret was a penny concealed in the tip of each wing. Of course, Martin may have submitted other models which were not selected for display by the Museum authorities. As befitted a public museum, they were rigorous in their selection of models they considered suitable for exhibition and even included only a fraction of the models by Yoshizawa which had been submitted to them. Nevertheless, it seems fair to say that Martin Gardner's very limited contribution to the exhibition is confirmation that notwithstanding his very creative contributions to card tricks and puzzles, he was not a creative paperfolder.
If the world of paperfolding was beginning to change in 1956 with the publication of Robert Harbin's "Paper Magic", Martin Gardner's own world was changing. At the time it did not seem significant - just one more magazine article among so many. Yet in retrospect it was the turning point in his life. Martin Gardner can have had no idea of the consequences of that first article that he wrote for Scientific American in December 1956, of the delight it would bring to millions of people worldwide, of the fame (and, let it be said, financial reward) it would bring, of the broadening of his own interests or of the demanding challenge of writing a monthly column of such quality for twenty-five years. It is tempting to speculate about how many young people have been inspired by his column to take up mathematics as a career, or what advances in serious mathematics may have been stimulated by its regular disclosure of off-beat ideas in recreational mathematics.
The first of Martin Gardner’s articles for “Scientific American” appeared in the issue for December, 1956 with the title, "Flexagons". It was about hexaflexagons in particular. An article about Moebius Bands appeared in June, 1957 and another about Tetraflexagons in May, 1958. The article on "Origami" did not appear until July 1959. By then Martin had become so involved with his regular column in Scientific American, that he gave up his contributing editorship of “Humpty Dumpty”
Hexaflexagons straddled the two realms of paperfolding and mathematics. They had been discovered in 1939 by Arthur J Stone, then a twenty-three year old British research mathematician at Princeton University. He had trimmed the wider American file paper that he had bought, to fit into his narrower British files. Then he began to play with the excess strips of paper. After creasing them at 60 degree angles and interweaving them, he discovered that they formed flat hexagons that could be “flexed” to bring different faces of the paper successively into view. The Princeton mathematical department experienced a craze for what came to be named “flexagons” (obviously derived from word “hexagons”) and Arthur Stone became the focus of a small group of fellow students who were fascinated by the mathematics of the new devices. Arthur Stone was joined by Bryant Tuckerman, John W. Tukey and not least by Richard P. Feynman, a genius who later achieved fame as a brilliant physicist. Together they analysed the mathematics involved in flexagons and set out their theories in a comprehensive paper, which is said to have been a complete exposition of the subject. For some reason which has never been explained, this paper has never been published, and it has been left to others to publish analyses of their own. Martin Gardner's own account of hexaflexagons in his first article for Scientific American was not intended in any way to be comprehensive, but it is wonderfully informative and succinct.
Moebius Bands share an article with other curious topological models but the article has little to do with paperfolding. The bands are presented from the point of view of topology and as the basis of magical tricks.
Tetraflexagons are much less well-known than their cousins, the Hexaflexagons, but Arthur Stone was interested in them, too. In another article, Martin Gardner points out that they have been known as a “double-action hinge” for centuries and toys based on the principle were marketed in the 1890s. He also mentions a tetraflexagon in the form of a puzzle which was copyrighted in 1946 by Roger Montandon of The Montandon Magic Company" of Tulsa, Oklahoma. It was called "Cherchez la Femme", the puzzle being to find the picture of the young lady behind the facade of a grinning sailor. It was only in 1993, with the publication of the book, "Martin Gardner Presents" that it was disclosed that the originator of this puzzle was Martin Gardner himself. Perhaps his reticence about the puzzle and its publication is explained by the fact that when eventually the lady is found, she is discovered to be au naturel !
The article on Tetraflexagons also contains a full explanation and diagrams for a variant of the "Flexatube" puzzle in which a square tube of paper is turned inside-out by successive folding steps alone. It is revealed that this, too, was discovered by Arthur Stone while working on flexagons. No paperfolder fails to be fascinated by this magical folding device.
Martin Gardner’s article on Origami in “Scientific American” for July, 1959, gives a brief outline of the subject, describing it as "the ancient Japanese art of paper folding". In a few brief sentences it manages to mention Mrs. Oppenheimer, the Cooper Union Exhibition, the accomplishments of refined Japanese ladies, Lewis Carroll and Miguel Unamuno, the Spanish philosopher, who wrote a mock-serious treatise on paperfolding. These are followed by the pentagonal knot in a strip of paper which conceals within it a mystical pentagram and by the far-from-simple-scientific problem of why, when we fold a sheet of paper, the crease is a straight line. Although less related to classic paperfolding, Martin also demonstrates how a parabola may be formed by successively folding one edge of a square of paper to a selected point which becomes the focus of a curve formed by the creases. Martin Gardner could not resist concluding his article with instructions for the Flapping Bird. Written in 1956, the method was the old one of pre-creasing and crunching the points together, which was used by Tissandier and Houdini. Even though, at that time, Yoshizawa's scheme of different dotted lines to distinguish mountain and valley folds had not yet reached the West, nevertheless, Martin Gardner's diagrams are remarkably clear.
An article in Scientific American dated June 1960, which is headed "Paperfolding and Papercutting" is mainly about dissections, but it also touches on papercutting or "kirigami" and it includes the famous dissection puzzle usually known as "Heaven and Hell".
After 1960 there was a long wait before Martin Gardner included anything more related to paperfolding in his Scientific American column. His column was changing too. His earlier articles dealt with comparatively simple puzzles, tricks and phenomena which, even if they might conceal hidden mathematical mysteries, were within the scope of understanding of any reasonably educated person. However, his later articles began to delve deeper, reflecting the growing appreciation that the playful exploration inspired by recreational mathematics could occasionally open up new vistas in advanced mathematics which were entirely unexpected and yet which sometimes unexpectedly proved to be of great value in newly-emerging branches of science.
In April 1968 Martin reverted to play and wrote about "Puzzles and Tricks with a Dollar bill". For these he went right back to the beginning of this magical career, to include the trick of inverting a dollar bill and even the two folds that convert George Washington into a mushroom. There were also several mathematical tricks based on the serial number of a dollar bill. All of these were tricks which occur several times in different books of Martin Gardner's puzzles.
The following December Martin turned again to another of his favourite subjects, Moebius Bands. His article reproduces two prints by the Dutch artist, M C Escher and throws several new beams of light on an old subject, but there is little for paperfolders. Martin does, however, point out that a hexaflexagon is an interwoven Moebius band, something that is not immediately self-evident.
In May and September 1971 Martin Gardner introduced two new paperfolding topics. The article for May 1971 is on "The Combinatorial Richness of Folding a Piece of Paper". It reveals the unexpectedly difficult mathematical problem of determining the number of ways in which a map, or for that matter a single strip of stamps can be folded up. However, the article suddenly transforms itself into an account of the work of Robert Neale, a fellow magician and inventor of many paperfolding devices, most of which have an unusual "twist". Included are Robert Neale's Beelzebug Puzzle, an ingenious puzzle based on a tetraflexagon, and the famous "Sheep and Goats" paperfolding puzzle. Robert Neale's best-known trick "Bunny Bill"’ is merely mentioned, but the address from which it can be obtained is given as Magic Inc. of Chicago.
“Plaiting Polyhedrons”, which appeared in September 1971 outlines the absorbing method of folding the Platonic solids from strips of paper. It is a subject which has been investigated from various angles by several paperfolders and Martin Gardner's account whets the appetite. So far as is known a comprehensive book on this far-from-negligible subject remains to be written.
One of the new mathematical topics that has emerged since the Second World War is that of Fractals and one of Martin Gardner’s last articles in Scientific American related to paperfolding is about the Dragon Curve which is a kind of fractal. Apparently this article was included in a series of "Nine logical and illogical problems to solve" in November 1967, but I have not seen this. The part of the article about the Dragon Curve is reprinted in "Mathematical Magic Show", published ten years later in 1977. Martin Gardner demonstrates the method of creating the dragon curve by repeatedly folding a piece of paper in half. There are, of course, physical limitations which, in practice, restrict this process to about seven folds, but the general theory of the Dragon Curve as a fractal is not invalidated.
Virtually all of Martin Gardner's articles in Scientific American have been reproduced in his volumes of scientific recreations. Depending on which of Martin’s books are included in the list there are fifteen or sixteen collections of the Scientific American articles which were published over a period of 38 years by a variety of publishers in the United States and England. The first collection was “The Scientific American Book of Mathematical Recreations” which appeared in 1959. In England it was published in 1961 with the title “Mathematical Puzzles and Diversions from Scientific American”. The last volume in the series was “The Last Recreations” of 1997. Paperfolding is embedded in the books in just a few chapters. However, they demonstrate that just as Martin Gardner's own interests widened, so paperfolding has broadened its horizons, something that has been startlingly demonstrated by the recent explosion of interest in the mathematics of paperfolding in books and articles, in universities and by the three international conferences devoted to the mathematics and science of paperfolding which have taken place so far in Italy, Japan and California.
SOME OTHER INTERESTS
But Martin Gardner's interests have always spread far beyond conjuring, paperfolding and mathematics. Sometimes his books on the most unlikely subjects have overtones of paperfolding. He has remained a philosopher for all his life and his book, "The Whys of a Philosophical Scrivener" is an absorbing apology for his own personal philosophy. In it, he displays an unexpected appreciation of Miguel Unamuno, the great Spanish philosopher, poet and paperfolder, who died on New Year’s Eve, 1936 to 1937 at the beginning of the Spanish Civil War. Martin Gardner is said to have been influenced in his ideas on theism by Unamuno and we may wonder whether Martin Gardner and Unamuno shared a common way of thinking.
In the very different field of literary criticism, Martin Gardner annotated several popular classics including "The Ancient Mariner" and "The Night before Christmas". As might be expected, he was very attracted by the work of Lewis Carroll and made annotated editions of "Alice in Wonderland" and "Alice through the Looking Glass", which have since been combined in a single volume. In them, Martin did not fail refer to Lewis Carroll's own interest in paperfolding. However, he made sure that he took his information from Lewis Carroll's own diaries (which mention paper boats and bangers) and not to over-inflate Lewis Carroll into a "great and enthusiastic paperfolder” as some over-enthusiastic commentators have done.
When we come to assess Martin Gardner's place in the history of paperfolder we, too, must be careful not to exaggerate. He was not a creative folder and he did not write a single book solely about paperfolding. Had he not achieved fame through his column in Scientific American; our perception of his part in the growth of Western Origami is likely to have been much less.
Yet, Martin Gardner certainly did play an important part in the development of origami. In the 1930s and 1940s he was one of the magicians who helped to build up the popularity of paperfolding stunts and tricks. He played a part in the swelling interest in paperfolding in the West after 1957 when Gershon Legman, Robert Harbin and Lillian Oppenheimer linked up to form a firm international base for future development. Martin added a note of academic respectability to paperfolding through his article for Encyclopaedia Britannica, despite the undue delay in its publication.
Above all, it was Martin Gardner's handful of paperfolding articles in Scientific American which were later reprinted with additions in his subsequent books that brought paperfolding as a mixture of play, art and mathematics to the notice of a new audience and which demonstrated once and for all that paperfolding was much more than a children's pastime.
Martin Gardner's supreme achievement was his ability to communicate difficult and often profound subjects with a few deft, but human strokes of his pen. He removed fear from our approach to mathematics and science. We can be grateful that he grew up surrounded by the humble art of paperfolding and that he was able to show that it is not only fun to do, but that it, too, has its place in the greater world of mathematics and science and is not unworthy of our time and our interest.
15 February 1995.
Revised, 29th September, 2005.
© David Lister 1995, 2005.
This article first appeared first in 1995 in the late and much lamented private magazine FOLD.
I express my deep gratitude to Martin Gardner, to whom I have submitted the article. He has graciously given it his approval and has suggested some minor additions and corrections which I have incorporated in the revised edition.
I also thank Mick Guy for reviewing of this article and for suggesting corrections of some of the typographical errors of which I had inevitably fallen victim
I, alone, remain responsible for the content and for all inaccuracies.
I shall welcome any corrections to this article and also any further information or anecdotes about Martin Gardner’s involvement with paperfolding.
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