From synthetic geometry to dynamic geometry and back: the case of circular inversion

by Giulia Signorini, Michele Tocchet, both from “Liceo Filippo Buonarroti” in Pisa, and Anna Baccaglini-Frank, University of Pisa.

In 1871 Felix Klein published two papers, called “On the so-called non-Euclidean geometry”, in which he proposed to call the first type of geometry “elliptic geometry” (from the Greek ellipsis, that means omission) and the second type “hyperbolic geometry” (form the Greek hyperbola, that means excessive). A good model for elliptic geometry is the sphere.

Circle inversion provides an interesting example of geometric transformation that, unlike the affinities and isometries studied in high school, usually does not transform lines into lines (but into circles) and that can be presented in an elementary way since its properties can be explored with dynamic geometry software and easily proved in synthetic geometry. Indeed, some high school textbooks introduce circle inversion as an interesting topic of Euclidean geometry that can also be explored through dynamic geometry software.

Take a look at the GeoGebra geometry!

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Site of the Month October: The German Imaginary-Project

The Site of the month October ist The German Imaginary-Project.

IMAGINARY offers a platform for open and interactive mathematics with a variety of content that can be used in schools, at home, in museums, at exhibitions or for events and media activities. The main contents of IMAGINARY are its interactive programs and its picture galleries.
IMAGINARY was initiated at the Mathematisches Forschungsinstitut Oberwolfach (MFO), an institute of the Leibniz Association. The MFO is a shareholder of IMAGINARY.

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Site of the Month September: The Global Math Project

The site of the month September ist The Global Math Project.

The Global Math Project is a worldwide movement committed to inspiring educators everywhere to ignite and sustain in their students a love for learning mathematics. The ultimate goal of the Global Math Project is demonstrating, in a genuine and direct way, that classroom mathematics can and does, in and of itself, serve as a portal to a genuine, meaningful, and connected human experience. They want especially to prove that curriculum-relevant mathematics is uplifting for the mind and for the heart.

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Contribution of the month: Prizes in mathematics, 2020

by Nitsa Movshovitz-Hadar (Technion – Israel Institute of Technology)

Did you know? – Over 125,000 new items are added each year, to the international database managed by the American Mathematical Society called MathSciNet®1 A vast majority of these items contain new results, continuously enriching the ever-growing discipline known as Mathematics2. To celebrate these tremendous achievements, all to be credited to the incredible creativity of mathematicians, several valuable awards have been established. This vignette is about a few of the more prestigious ones. It opens with the two that nowadays are often described as the Nobel Prize in mathematics3: the Fields Medal and the Abel Prize.

The Fields Medal

The Fields Medal is one of two awards often described as the Nobel Prize of mathematics. It consists of a gold medal bearing the profile of Archimedes and a cash amount of 15,000 Canadian dollars (or roughly $11,540). It is awarded to 2-4 mathematicians every four years at the opening ceremony of the ICM – International Congress of Mathematicians, to recognize outstanding mathematical achievement for existing work and the promise of future achievement. A candidate’s 40th birthday must not occur before January 1st of the year of the Congress at which the Fields Medals are awarded. (Recall, Andrew Wiles, who proved the long-standing Fermat’s Conjecture, missed it as he was slightly over 40 when he and Taylor took care of the final polish of the proof.4)

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The Snapshots Project

The link of the month June is The Snapshots Project.

The Mathematics News Snapshots for High School (MNS) project aims to provide high school students with a glimpse into the exciting and dynamic world of contemporary mathematics. The project was founded in order to decrease the gap between the ever-growing nature of mathematics and the stagnated nature of school curricula. In this sense the project is exactly in line with the Klein-Project.

The Snapshots are designed and written by teams of experts in mathematics and mathematics education, directed by Prof. Nitsa Movshovitz-Hadar (Technion – Israel Institute of Technology).

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The International Day of Mathematics

The International Day of Mathematics (IDM) is a worldwide celebration.

Each year on March 14 all countries will be invited to participate through activities for both students and the general public in schools, museums, libraries and other spaces.

The International Day of Mathematics is a project led by the International Mathematical Union with the support of numerous international and regional organizations.

Take a look –> Mathematics is Everywhere

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A Richer Picture of Mathematics – The Göttingen Tradition and Beyond

March Book of the Month is A Richer Picture of Mathematics – The Göttingen Tradition and Beyond by David E. Rowe.

Historian David E. Rowe (Prof. em. for History of Natural Sciences at the University of Mainz) captures the rich tapestry of mathematical creativity in this collection of essays from the “Years Ago” column of The Mathematical Intelligencer. With topics ranging from ancient Greek mathematics to modern relativistic cosmology, this collection conveys the impetus and spirit of Rowe’s various and many-faceted contributions to the history of mathematics. Centered on the Göttingen mathematical tradition, these stories illuminate important facets of mathematical activity often overlooked in other accounts.

Six sections place the essays in chronological and thematic order, beginning with new introductions that contextualize each section. The essays that follow recount episodes relating to the section’s overall theme. All of the essays (one about “The young Felix Klein”) in this collection, with the exception of two, appeared over the course of more than 30 years in The Mathematical Intelligencer. Based largely on archival and primary sources, these vignettes offer unusual insights into behind-the-scenes events. Taken together, they aim to show how Göttingen managed to attract an extraordinary array of talented individuals, several of whom contributed to the development of a new mathematical culture during the first decades of the twentieth century.

See: https://www.springer.com/de/book/9783319678184

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Klein Concert

Maria Mannone from the University of Palermo is a theoretical physicist and a musician (composer and conductor). Her field of interest is mathematical music theory. She and her students have created music based on the Klein bottle. In May 2019 the first Klein Concert was performed at the Music Conservatory of Palermo. For more about the concert, including a video, see

“Klein Concert”: a Report on a Geometric Journey

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The Enormous Theorem

Originating author is Nitsa Movshovitz-Hadar.

Prolog

This article is an English translation with emendations of a Hebrew paper written in 2014 about two years after the untimely passing of the illustrious lecturer and group theory expert, Professor David Chillag. It is dedicated to commemorating him. He entrusted me with the transparencies that accompanied his lecture on the enormous theorem presented in 2011 at the Technion Mathematical Club, hoping to produce a paper for teachers and teachers’ teachers on the subject. Unfortunately, we did not manage to do it together.
I thank those of our mutual colleagues who helped me with this^1, and at the same time, of course, I take full responsibility for things that are not entirely clear or even incorrect.
Nitsa Movshovitz-Hadar

Introduction – What is this article all about?

The definition of a group in mathematics is an abstraction of the set of integers with the familiar operation of addition. A group is a set (such as the integers) with a binary operation which enables combining any two elements in the set and get as a result one element in the set (such as the sum of two integers), that has a few simple properties such as the existence of a unique “neutral element” (zero in the case of the integers). There is a large wealth of mathematical groups, differing from one another in composition and size. Some of them consist of a finite set (such as the set \{0,1,2,3,4,5,6,7,8,9,10,11\}), others consist of an infinite one (such as the integers). The binary operations vary as well (think, for example, about addition-modulo-12 defined on the finite set mentioned earlier^2, or about addition defined on the integers). The wealth of mathematical groups is so large that there is no chance of classifying even the finite ones of them. Fortunately, there is a special family of finite groups called simple finite groups, which can be classified. It turns out that this special family of finite groups is of great importance because, in some sense, all finite groups can be built from simple finite groups in a way that is somewhat analogous to the fact that all the positive integers can be built from the prime ones. The “enormous theorem,” the subject of this article, gives a complete classification of the simple finite groups. In this way, it provides tools for analyzing the structure and features of all finite groups. The theorem received its “enormous” nickname for some very good reason, to be explained below.
Let us meet the people behind the scene.

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Site of the Month: 3Blue1Brown

3Blue1Brown presents animated videos about mathematics. It was created by Grant Sanderson, a graduate student from Stanford University who worked for Khan Academy. He created a YouTube-channel with videos on calculus, linear algebra, geometry, topology,  and many special topics such as Fourier transformations or the Riemann hypothesis. He answers questions like “But why is a sphere’s surface area four times its shadow?” or “Ever wondered why slicing a cone gives an ellipse?”. The videos use 3D-animations and they explain mathematical problems from a visual perspective.

The channel name and logo reference the colour of Grant’s right eye, which has blue-brown sectoral heterochromia.

See also: https://en.wikipedia.org/wiki/3Blue1Brown

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