What is the way of packing oranges? — Kepler’s conjecture on the packing of spheres

Originating author is Christiane Rousseau.
What is the densest packing of spheres? Kepler conjectured that it was the one you observe with oranges at the fruit shop, and which is called the face-centered cubic lattice (Figure 1). At the International Congress of Mathematicians in 1900, David Hilbert gave a very famous lecture in which he stated 23 problems that would have deep significance for the advance of mathematical science in the 20-th century. The problem of the densest packing of spheres, also called Kepler’s conjecture, is part of Hilbert’s 18-th problem. Kepler’s conjecture was only proved in 1998 by Thomas Hales, and the details of the proof were published in 2006.

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Higher Dimensions

Figure 1: Illustration of a Calabi-Yau-manifold (Important for the description of higher dimensional models in superstring-theory).

Originating authors are Markus Ruppert and Hans-Georg Weigand.
1. Looking for the next dimension
Does our world really have more than three dimensions? If so, do objects in higher dimension have a relation to the world around us? Is it possible to get a perception of these objects or do they withdraw any representation? The Theory of Relativity uses four dimensions to explain the concept of space-time, six dimensions are necessary to describe the bending of space-time and different string theories even use representations in up to 26 dimensions (e.g. L. Botelho, R. Botelho, 1999). Another current domain of application for higher dimensional objects and their three-dimensional representations is the study of non-periodic structures in modern crystallography. Within the concept of quasicrystals projections of higher dimensional point-sets (such as the integer-lattice in dimension 5) to three dimensional space are supposed to be good models for non-periodic crystalline structures (see section 5 below).

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Benford’s law: learning to fraud or to detect frauds?

Originating author is Christiane Rousseau.
It is very risky to change too many numbers in some fi nancial statements if one does not know some mathematics. Indeed, most often the numbers appearing in fi nancial statements follow some strange mathematical rule, called Benford’s law, or law of the fi rst signi ficant digit. If one forgets to follow the rule, then the numbers will fail some statistical tests and are likely to be scrutinized with care. Benford’s law claims that if you collect numbers at random and calculate the frequencies of their fi rst signifi cant digits, the numbers with fi rst signifi cant digit 1 should appear around 30% of the time, while the numbers with fi rst signifi cant digit 9 appear only 4.5% of the time. This rule is observed in many other sets of numbers, like powers of 2 or Fibonacci numbers.

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Map colouring and Gröbner Bases

This picture is a property of mathscareers.org.uk, who kindly granted permission to use it in this work.

Originating author is Marcelo Escudeiro Hernandes.
By the famous “Four Colour Theorem”, only four colours we need to colour a map so that no bordering regions have the same colour. Using polynomial equations and Gröbner bases we can determine if three colours are sufficient for a particular map.

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Symmetry Step by Step

Originating author is Ana Cannas da Silva. Symmetry has always fascinated and served humankind in architecture, arts, engineering and science. Over thousands of years symmetric patterns have been used to create fabrics, baskets, floors, wallpapers and wrapping papers, and so on.
At the end of the 19th century, the Russian mathematician and mineralogist Yevgraf Fyodorov established that there are 17 types of symmetry for patterns in the plan [WPG]. That is, we can have exactly 17 different wallpapers in terms of replications of symmetry, and no more! Notably, all these types of symmetries can be found in decorative arts in antiquity.

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Recurrence and induction

Originating authors are Michèle Artigue and Ferdinando Arzarello.
Given a square grid, it is easy to draw squares whose vertices are intersections of the grid lines. But is it possible to do so for other regular polygons, for instance an octagon ? The answer is : « No » and it can be proved, for the octagon, as follows (Payan, 1994) :

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موقع الشهر: “الرياضيات هنا الآن” Math Here and Now

موقع الشهر: “الرياضيات هنا الآن” Math Here and Now
نشر يوم 6 يناير 2015 من قبل سارة سبونمان Sarah Spoenemann
موقع شهر ديسمبر 2014 : “الرياضيات هنا الآن”
http://www.nctm.org/resources/content.aspx?id=17109
يعتبَر هذا الموقع التابع للمجلس القومي لمدرسي الرياضيات (الأمريكي) National Council of Teachers of Mathematics مصدرا ثريًا للمدارس الثانوية. إنه يحتوي على مواضيع مُحَيَّنَة تستعرض أمثلة من رياضيات عالمنا. يتضمن كل مثال من هذا القبيل عنوانا فرعيا، يحمل اسم “الرياضيات” “The Maths”، يذهب بعيدا في التفاصيل الرياضية.

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الكتاب المتميّز لهذا الشهر: “ابتهاج الرياضياتيين” Mathematicians’s Delight

51IT40traAL._SY344_BO1,204,203,200_الكتاب المتميّز لهذا الشهر: “ابتهاج الرياضياتيين” Mathematicians’s Delight
نشر يوم 6 يناير 2015 من قبل سارة سبونمانSarah Spoeneman

كتاب شهر ديسمبر هو بعنوان “ابتهاج الرياضياتيين” Mathematicians’s Delight لصاحبه ولتر وارويك سيير Walter Warwick Sawyer الذي أصدرته دار دوفر Dover للنشر عام 2007 (وكان أول ناشر له دار بنغوين للكتب Penguin Books عام 1943).
هذا هو أول كتاب لولتر سيير، والهدف منه “تبديد الخوف من الرياضيات”. حسب موقع ولتر سيير
http://www.wwsawyer.org فمن المحتمل أن يكون هذا الكتاب أنجح كتاب أُلِّف في الرياضيات وعَرف عديد الطبعات و 10 ترجمات إلى لغات أخرى. وقد بيعت منه أزيد من 500 ألف نسخة. يقدم الكتاب الرياضيات مركّْزا على سياق الموضوع. “قمْ بكذا، اصنعْ كذا، لاحظْ كذا، رتبْ كذا، وعندئذ فقط نستوضح السبب”.

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Mathematicians’s Delight

51IT40traAL._SY344_BO1,204,203,200_December’s Book of the Month is Mathematician’s Delight by W. W. Sawyer, Dover Publications 2007 (originally Penguin Books, 1943).

This is Sawyer’s first book, and was written with the aim “to dispel the fear of mathematics.” According to the W. W. Sawyer website, it is probably the most successful math book ever written, going through numerous editions, translations into 10 languages, and selling more than 500,000 copies. It introduces mathematics with an emphasis on context: “Do things, make things, notice things, arrange things, and only then reason about things.”

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Math Here and Now

December’s Site of the Month: Math Here and Now

This is one of the High School resource pages of the National Council of Teachers of Mathematics. It contains up to date instances of mathematics in our world. Each example has a sub-section entitled “The Math” that goes into more mathematical detail.

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