We will soon open a new website for the Klein project. It will contain all the existing material, and some new material. We seek your input into the existing material, new material, and how you think this website should be arranged. You are invited to click on the link to the anonymous Google Docs survey. It will take 2-3 minutes to complete. Thank you for your help.

Klein Project Development Survey


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Site of the Month: Mathematics Today

October’s site of the month: Mathematics Today

Mathematics Today is a magazine published by the UK Institute of Mathematics and its Applications.

It is a general interest magazine about mathematics intended for those interested in the subject. Much material is UK-specific, however it contains many general articles that are downloadable from the website.



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Featured Book of the Month: Cuentos Con Cuentas

CuentosOctober’s Book of the Month is Cuentos Con Cuentas by Miguel de Guzman (Labor, Barcelona, 1984).

This is also published in English under the title The Countingbury Tales, translated by Jody Doran and published by World Scientific (2000).

Have a look inside.

Miguel de Guzman wrote other similar books, the most well known being Aventuras Matematicas which is also published in Chinese, Finnish, French and Portuguese.

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Trying to predict a floating leaf: chaos and predictions

Jahreszeit__Herbst__Blatt__Baum__rot__farbig__Wasser__schwimmenOriginating author is César R. de Oliveira, Universidade Federal de São Carlos.

What path will a leaf follow floating down a turbulent stream? Is it even possible to make a mathematical model that will predict such motion? Is this the same sort of problem as predicting the path of planets as they move round the sun? Even when we know all the rules governing the motion of an object, and can determine precisely the initial conditions, it turns out that some motions can be predicted and some cannot. And it is not just a matter of complexity: we can model unpredictable systems with very simple equations. In this Vignette we illustrate mathematically the existence of chaotic dynamical systems using the decimal form of real numbers. You will see how the unpredictability can be simply generated. One of the main goals of theoretical models is to make (good) predictions. However, there are deterministic dynamical systems that in practice are unpredictable; they are the so-called chaotic systems. The aim of this text is to discuss how this unpredictability is generated, and the main tool here will be the decimal representation of real numbers.

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How to get rid of quantifiers?

Quantifiers_5Originating authors are Reinhard Oldenburg and Michele Artigue.

How do computer packages do abstract algebraic problems such as proving statements “for all x” or finding whether a Real Number x with certain conditions exists?

Recent advances draw on theorems in mathematical logic, as well as improvements in computing.

High school students learn how to solve problems such as the following: « For what values of the real number c, does the polynomial P(x)=x^2+cx+c have two distinct real roots ? », and algorithmically get the answer: c<0 or c>4. Doing so, they have in some sense, found a way of transforming the sentence expressing the problem (\exists x_1 \, \exists x_2 \, (x_1\not= x_2 and P(x_1)=0 and P(x_2)=0) involving the two existential quantifiers ‘there exists x_1’ and ‘there exists x_2’ into the sentence c<0 or c>4 which no longer includes quantifiers. For what kind of problems is this possible, theoretically but also practically with an effective computer program? In 1938, thanks to a theorem of elimination of quantifiers proved by the logician Alfred Tarski, a decisive step was achieved regarding these questions, but this was not at all the end of the story… READ MORE

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Featured Book of the Month: Mathematical Models

Mathematical Models

September’s featured book of the month is “Mathematical Models” by H.M. Cundy and A.P. Rolett.

This classic was first published in 1952 by Oxford University Press, but was republished in paperback by Tarquin in 1981.

As well as nets polyhedra, it has a wide variety of linkages and dissections, as well as several mechanical models.

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Site of the Month: Bridges

Bridges Org

September’s site of the month: Bridges

Bridges is an organisation that oversees the annual Bridges conference on Mathematics and Art. It contains images and resources of many different kinds of artistic representations, from poetry to models, from dance to origami, from juggling to painting. The -Resources- link on the homepage contains links to other websites of interest to teachers.

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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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