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This month's topics:

Fibonacci on the wall

"The Fibonacci sequence discovered on the facade of a church in Pisa" was the headline in the Florence edition of La Repubblica for September 19, 2015. The story, by Giovanni Gagliardi and Laura Montanari, tells how Pietro Armienti, a professor of Earth Sciences at Pisa, discovered "a message that no one has read for more than eight hundred years. A message encoded in the perfect geometries of the lunette of the church of San Nicola, in Pisa, and which for the readers of Dan Brown and of his Da Vinci Code has a familiar name: the Fibonacci sequence. The formula due to the Pisan mathematician is represented by a series of figures including circles and squares in the marble decoration of a small church in the center of the Tuscan city."

San Nicola lunette

The entrance to the church of San Nicola, in Pisa, with an enlargement of the lunette in question. Image from Wikimedia Commons. A beautiful set of pictures of the lunette and its surroundings has been posted by Getty images.

Prof Armienti's discovery is explained in more detail in an article he wrote for the Journal of Cultural Heritage. In "The medieval roots of modern scientific thought. A Fibonacci abacus on the facade of the church of San Nicola in Pisa" he states: "The intarsia [image formed with inlaid stone] reveals the direct influence of the great Pisan mathematician Leonardo Fibonacci due to the presence of circles whose radii represent the first nine elements of the Fibonacci's sequence and which were arranged to depict some properties of the sequence. Moreover, the tiles [in the background of the lunette] can be used as an abacus to draw sequences of regular polygons inscribed in a circle of given radius."

San Nicola

The design of the intarsia in the San Nicola lunette contains circles of radius 1, 2, 3, 5, 8, 13, 21, 34 and 55, the first nine elements of the Fibonacci sequence. Background image courtesy of Pietro Armienti.

At the end of his article, Prof Armienti addresses the conspicuous placement of a mathematical artifact on an ecclesiastical facade. "Symptomatic of the cultural climate of the [Pisan] Republic, which at the time controlled the trade in the Mediterranean, the lunette could well be said to represent a milestone in the history of scientific thought of the Christian West. ... The position of the intarsia, on the main entrance of the church, is an expression of the thesis supported by Thomas Aquinas in his Summa Theologiae, i.e. that knowledge is a gateway to the divine, and rational truth and revealed truth cannot contradict one another. 'Veritas: Adaequatio intellectus ad rem. Adaequatio rei ad intellectum. Adaequatio intellectus et rei.'"

On the other hand: "At the end of this exploration, which began one summer morning in front of the church while grumbling about the delay of my wife, I am left with the belief that it is always worth waiting for the preparations of a beautiful woman. ... "

Topology and the mechanics of knots

"Untangling the Mechanics and Topology in the Frictional Response of Long Overhand Elastic Knots" appeared in Physical Review Letters for the week ending September 11, 2015. The authors, M. K. Jawed, P. Dieleman, and P. M. Reis (MIT) and B. Audoly (Paris 6) begin their paper with a familiar example:

  • "Shoelaces are commonly tied using the reef knot, which comprises two trefoil knots: the first is left handed and the other right handed. Mistakenly tying two consecutive left-handed trefoil knots leads to the mechanically inferior granny knot, whose lower performance illustrates the important interplay between topology and mechanics."

In the authors' words: "we perform a systematic investigation of elastic knots under tension and explore how their mechanical response is influenced by topology." The knots they investigated, through experiment and analysis, are the overhand knots with unknotting number 1 up to 10.

overhand knots

Overhand knots with unknotting number $n = 1, 2, 3, 4$. The unknotting number is the number of crossings that need to be eliminated to make the string unknotted. If the two ends are joined, the overhand knot with unknotting number $n$ becomes a topological knot with $2n+1$ crossings. Braid and loop are parts of the knot treated separately in the authors' analysis.

The team's experiments consisted of tying overhand knots in Nitinol (nickel-titanium) rods of circular cross section, and clamping one extremity of the rod. The other end was displaced slowly, so as to tighten the knot; the experimenters recorded the relation between the tensile force and the length of the knot (the braid plus the loop).

Here is how the theoretical part of the paper is explained by Michael Shirber, writing in the American Physical Society online feature Physics.

  • "The researchers first modeled the braid as a helix whose pitch (twist steepness) is closely related to the friction. The tighter the helix, the greater the friction. The loop, on the other hand, resembles a classic mechanics problem of an elastic rod being bent while holding one end fixed. The team took the well-known solution to this problem and combined it with their helix model. The result was a single mathematical relation between the topology ($n$) and the mechanical variables--friction, bending stiffness, and tension."

As the researchers report in their Abstract, "Excellent agreement is found between theory and experiments for overhand knots over a wide range of crossing numbers."

[A QuickTime movie, Discrete elastic rods, made in 2008 by Audoly and another team: M. Bergou, S. Robinson and E. Grinspun (Columbia) and M. Wardetzky (Freie Universität Berlin) investigates different aspects of a similar medium and is definitely worth a look. -TP]

A maths colouring book for adults

Alex Bellos posted "Nested fish and golden triangles: adult colouring and the beauty of maths" on The Guardian's website, September 17, 2015. "When people say that maths is 'beautiful' it is usually meant in the abstract sense, such as to describe a theorem whose power, depth and concision provoke feelings of awe. Bertrand Russell called this 'a beauty cold and austere ... sublimely pure, and capable of a stern perfection'. Yet humans have traditionally also found aesthetic, sacred beauty in mathematics. Islamic and Hindu cultures, for example, are rich in stunning images based on geometric design. It was reflecting on the role of maths as a meditative and contemplative medium that I decided to compile a colouring book.

No mathematical knowledge is required or assumed to colour in the ... images in the book ... . But by colouring them in you will be engaging with mathematical ideas, some of which are millennia old and some of which are recent discoveries."

Bellos' article contains (downloadable) four of the 80 patterns printed in his book, to be available also in the U.S. as "Patterns of the Universe," starting December 1. Here are some sample full-scale extracts (the plates are 10.5in square) with examples of colorings.

fish fish+c
tridoku tridoku+c

Full-scale details of two of the patterns from Alex Bellos' adult maths colouring book, with suggested realizations. Fish: © David Bailey, "Tridoku": © Edmund Harriss.

The inheritance of math anxiety

Don't try to help your kids with their math homework if you suffer from math anxiety yourself. This is the lesson to be learned from Jan Hoffman's posting on the New York Times "Well" blog, August 24, 2015. "Children of highly math-anxious parents learned less math and were more likely to develop math anxiety themselves, but only when their parents provided frequent help on math homework, according to a study of first- and second-graders, published in Psychological Science." Hoffman links to the study, by a team of five psychologists from Chicago, UCLA and Temple: "Intergenerational Effects of Parents' Math Anxiety on Children's Math Achievement and Anxiety." From the abstract:

  • "We found that when parents are more math anxious, their children learn significantly less math over the school year and have more math anxiety by the school year's end--but only if math-anxious parents report providing frequent help with math homework. Notably, when parents reported helping with math homework less often, children's math achievement and attitudes were not related to parents' math anxiety. Parents' math anxiety did not predict children's reading achievement, which suggests that the effects of parents' math anxiety are specific to children's math achievement."


Tony Phillips
Stony Brook University
tony at

Math Digest Math Digest
On Media Coverage of Math

Math Digest includes posts throughout each month, with summaries of math stories and unique insights (and occasionally videos, interviews and podcasts) on math-related topics recently covered by the media.

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On mathemagician Art Benjamin, by Samantha Faria

Arthur Benjamin

Arthur Benjamin's love of math grew with every Martin Gardner puzzle he completed. Each problem was creative, stimulating and engaging. Unfortunately, he explains, "both the fun and the explanations are often missing from math instruction in today's schools." A proclaimed "mathemagician," Benjamin astonishes his audiences not just with his super-speedy calculations but with his ability to captivate their attention and explain topics in interesting ways. The Harvey Mudd professor admits that he "learned how to be a good teacher through [his] early experiences as a magician. My approach to teaching has always been, 'How do I make this material entertaining?'” In his new book, The Magic of Math, Benjamin hopes to do just that. (Photo of Arthur Benjamin courtesy of Harvey Mudd College.)

See "From Poof to Proof: Inside the Mind of a Mathemagician," by Liz Logan, Smithsonian Magazine, September 8, 2015

--- Samantha Faria

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