Featured Book of the Month: Indra’s Pearls – The Vision of Felix Klein

Indra's_Pearls_book_coverAugust’s featured book of the month is “Indra’s Pearls: The Vision of Felix Klein” by David Mumford, Caroline Series and David Wright.

Wikipedia says:
“The book explores the patterns created by iterating conformal maps of the complex plane called Möbius transformations, and their connections with symmetry and self-similarity.

The book’s title refers to Indra’s net, a metaphorical object described in the Buddhist text of the Flower Garland Sutra. Indra’s net consists of an infinite array of gossamer strands and pearls. The frontispiece to Indra’s Pearls quotes the following description:

In the glistening surface of each pearl are reflected all the other pearls … In each reflection, again are reflected all the infinitely many other pearls, so that by this process, reflections of reflections continue without end.

The allusion to Felix Klein’s “vision” is a reference to Klein’s early investigations of Schottky groups and hand-drawn plots of their limit sets. It also refers to Klein’s wider vision of the connections between group theory, symmetry and geometry.”

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Featured Book of the Month: Mathematical Puzzles and Diversions


July’s featured book of the month is “Mathematical Puzzles and Diversions” by Martin Gardner.

This book, which was originally published by Simon & Schuster in 1959, and later by University of Chigaco Press in 1988, is the first of several collections of Martin Gardner’s column in the Scientific American. These books represent a very small proportion of the total writings of Martin Garnder — for a full bibliography go to Martin Gardner’s website.

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Site of the Month: Mathematikum


August’s site of the month: Mathematikum

This is the website of the Mathematics Museum in Gießen — it includes models and activities of interest to people of all ages.

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Site of the Month: Images des Maths


Verhulst dynamics, by Jean-François Colonna

July’s site of the month: Images des Maths

This site is affiliated to the French National Center for Scientific Research. It contains reports about the latest events in mathematical research, and up-to-date articles that contain interesting discussions and applications of mathematics (such as a modelling of the Ebola virus outbreak). The articles are written in concise and comprehensive fashion.

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The Revenge of the Infinitesimals

Originating author is Michèle Artigue.

Infinitesimals played an essential role in the emergence and development of differential and integral calculus. The evident productivity of this calculus did not prevent recurrent and fierce debates about the nature of these objects and the legitimacy of their use. At the end of the 19th century, when the construction of real numbers from integers and the modern definition of the concept of limit provided a solid foundation for differential and integral calculus, infinitesimals and the associated metaphysics was rejected and their use perceived synonymous with bygone and poorly rigorous practices. However, the language of infinitesimals continued to be used, for example in physics and even in mathematics. It never completely disappeared from the informal discourse and heuristic thinking of a number of researchers.

Is this language thus really incompatible with mathematical rigour? What does it offer that is interesting and specific, which explains its permanence? Non-Standard Analysis developed in the 20th century and provided answers to these questions and enabled infinitesimals to take their revenge.

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Featured Book of the Month: Elementary Mathematics from an Advanced Standpoint

From now on we will feature a different book every month that is likely to be of interest to secondary teachers wanting to know more about mathematics. We have made the decision to use this feature to bring older books to the attention of a new generation of teachers (rather than to add to recent book promotions). All books must therefore be older than 10 years.
For our first featured book we turn to Felix Klein’s original works that stimulated the Klein Project, his three volume work Elementary Mathematics from an Advanced Standpoint. Only two volumes have been published in English, although all three are available in German, and have been published in Portuguese. It is exactly ten years since Dover reprinted the English versions.
These books are essentially Klein’s own notes for a series of lectures he gave to graduates of mathematics preparing to become teachers in the gymnasium’s of the time. Of course, Klein’s books discuss mathematics that is more than 100 years old (they were first published in German in 1908), but remain extraordinarily relevant for today’s world.

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Site of the Month: Il Giardino di Archimede


This site of a museum in Florence not only encourages us to visit that beautiful city, but also contains materials for schools. It is easily navigated by those who do not speak Italian.
Il Giardino di Archimede

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Classifying objects

Originating author is Christiane Rousseau.
Mathematics offers tools for classifying objects. But is that of any practical use? More than we can imagine at first sight… It could allow us to conclude that a knot cannot be unknotted without cutting the rope, regardless how you move it in space. It could also tell you that you wrongly assembled your Rubik’s cube after dismantling it into pieces, and that there is no use trying to solve it.

In practice, classifying objects means grouping objects in classes of objects sharing some common properties. One very efficient way to do this is through an equivalence relation. Then, each object belongs to an equivalence class. But, this does not mean that we have an efficient way of describing an equivalence class! The mathematical notion of invariant offers an efficient way to do this. An invariant is some mathematical object (it could be simply a number) that is the same for all members of an equivalence class. Among the invariants we distinguish the complete invariants that characterize an equivalence class.

In this vignette, we will mostly work with examples and see how the notion of invariant is widely spread among mathematics, especially algebra and geometry. You will then be able to add your own examples.

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Site of the Month: NRICH

Starting in January, we began a “Site of the Month” feature.

eNRICHing mathematics

February’s site of the month is dedicated to: NRICH. NRICH is a team of mathematics teachers, who aim to enrich the mathematical experience for all. They provide engaging activities for students, as well as for teachers’ professional development. Take a look at NRICH’s page for upper-secondary students: NRICH


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The Hairy Ball Theorem

Originating author is João Pimentel Nunes

The Hairy Ball Theorem is from topology, that part of mathematics that is concerned with the form of spaces. For the most part, this result came from work at the end of the 19th century by Henri Poincaré[1], considered to be one of the founders of topology.

There are few mathematical results that are so familiar to us from everyday situations: many readers are faced every morning with the hairy ball theorem when they try to comb their hair and find a persistent whorl at the top of their heads. Stated simply, the Hairy Ball Theorem says that it is impossible to comb a spherical ball covered in hair so that there are no whorls.

Check out this cool video explanation of the theorem
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