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Everything Is Number

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Everything Is Number Everything is Number Stimulating excursions to beautiful mathematics Paintings by Eugen Jost Thun, Switzerland Texts by Peter Baptist Bayreuth, Germany The story behind the exhibition Eugen Jost’s pictures have titles like “Hardy’s Taxi III”, “A Taste of Pi”, “Good Luck”, “Blue Stars”, and “Prime Time”. Actually their style is different, but they have a background common to all of them that one doesn’t expect behind these titles: mathematics. The paintings tell stories, they stimulate interest for mathematical results and relationships just as for the persons who were engaged in these topics. Mathematical theories and problems are not only a matter of mind, they also have an effect on sensibilities and aesthetic feelings, comparable to artistic activities. The famous British number theorist Godfrey Harold Hardy (1877 – 1947) pointed out: “A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” Eugen Jost’s paintings convince by their diversity. They contain elementary and more complex problems, they attract kids, students, and adults with and without mathematical background. Often people equate mathematics with arithmetic and focus on computational skills. But mathematics involves more than computation. The paintings of the exhibition – and in this booklet – clearly show: Mathematics is a study of patterns and relationships, a way of thinking and a science that is characterized by order and internal consistency, a language that uses carefully defined terms and symbols, a tool that helps to explain the world. At the beginning there was a calendar for the year 2008 that combined mathematics and arts. Especially for this purpose Eugen Jost created paintings with mathematical motifs or references to mathematics. For further mathematical information there were explanatory texts on the backside of each calendar sheet. Meanwhile the calendar has been history for a long time. It was a real bestseller. Two more calendars and math & arts books followed. We have had dozens of exhibitions of original paintings and many more of digital prints of the paintings. The idea to get an access to mathematics with the help of art works is still vivid and popular. There are still unabated requests for our exhibition. For this Athens exhibition Eugen Jost has created a considerable number of new paintings. The texts by Peter Baptist provide information, hints, remarks, and questions, to encourage reflection on mathematics. View the paintings, discover patterns, configurations, and relationships. Enjoy the exhibition and the beauty of mathematics. Be curious about an initially unfamiliar, but really stimulating approach to mathematics and history of mathematics. Arts helps to show that mathematics is much more than mere computing, mathematics is part of our culture. Hardy’s Taxi III Hardy’s Taxi III A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. … The mathematician’s patterns, like the painter’s or poet’s, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics. … • This quotation comes from Godfrey Harold Hardy (1877 – 1947), an outstanding Cambridge mathematician and eccentric. • Many mathematicians derive aesthetic pleasure from their work. They describe – like Hardy - mathematics as a creative activity and as an art form. Comparisons are made with painting, music, and poetry. For Eugen Jost, mathematics is a huge flower garden and he makes us discover the most beautiful flowers in his paintings. • Our first flower is the number 1729. A taxi with this number took Hardy to a hospital for visiting his ill colleague Srinivasa Ramanujan (1887 – 1920). • Ramanujan was an Indian natural mathematical genius. Hardy recognized the brilliance of his work, and he asked him to come to Cambridge. • For Hardy 1729 was a boring number, but Ramanujan immediately realized that this was a very special number. 1729 is the smallest number that can be written as the sum of two cubes in two different ways (1729 = 93 + 103 = 13 + 123). • By the way, the sum of the digits of 1729 is a divisor of 1729. Do you know other numbers with this property? Another interesting feature of this “taxi number”: By multiplying the sum of the digits (1+7+2+9) with its mirror number 91 we get 1729! • A (3x3)-magic square forms the background of the painting. It can be recognized only faintly. Another one is on the bottom right. Its cells are filled with digits, dot patterns and one question mark. • Tributes to Pythagoras are the triple (3,4,5), the corresponding square numbers, and the generalization with semicircles on the sides of a right triangle. • Additional number flowers are o Square numbers o Triangular numbers o Fibonacci numbers o Prime numbers o Number � o Euler’s number e " " " " " • e is the value of the infinite series + + + + + … #! "! &! '! (! • Another infinite series with finite value is " " " " + + + + … ") &) ') () This sum had puzzled mathematicians for a long time. In 1736 Leonhard Euler proved *) that the value of this sum of reciprocals of the squares of positive integers equals . + • Euler’s name is also connected with the problem of the bridges of Königsberg. River Pregel divides this former Prussian city in several parts that are connected by seven bridges. Euler was intrigued by the question of whether a route through the city is possible that would traverse each of the seven bridges exactly once? His negative resolution laid the foundation of graph theory. • A remarkable formula in mathematics is Euler’s identity �-* + 1 = 0. This equation that was proven by Euler combines important mathematical constants e, �, i (the imaginary unit which satisfies i2 = -1). • Once more Leonhard Euler. His polyhedron formula V + F – E = 2 shows the relation between the numbers of vertices (V), faces (F), and edges (E) of simple polyhedrons. • An example for a simple polyhedron is a cube. Next to the polyhedron formula we see the net of a cube. Do you know how many different nets of a cube exist? • The symbol of the Pythagorean identity is the pentagram. This shape of a five-pointed star belongs to our mathematical flowers as well as the Star of David. In comparison to the pentagram it has one additional vertex. This hexagram can be viewed as a compound of two equilateral triangles. • The number 153 inside the sketch of a fish refers to a miraculous catch of fish that is reported in the Bible (Gospel of John 21,11). Up to now no symbolic significance of the number 153 has been found. A friend of number curiosities may be delighted by the following property of 153. The sum of the cubes of the digits of this number equals this number: 13 + 53 + 33 = 153. Numbers in Every Day Life and Culture Numbers in Every Day Life and Culture • C M Y K : This combination of letters is omnipresent and yet not obvious. We meet this notation indirectly when we read the newspaper in the morning, look through the mail during the day or thumb through the TV guide in the evening. Many color prints are based on the primary colors Cyan, Magenta and Yellow. A rich red results from the color composition C: 0%, M: 100%, and Y: 100%. All three basic colors together make black. Because this combination does not produce a rich black for printing reasons, black is also used as Keycolor. • No. 5 is the first perfume launched by French fashion designer Gabrielle „Coco“ Chanel (1883 – 1971). The chemical formula for the fragrance was compounded by the French-Russian chemist Ernest Beaux (1881 – 1961). Chanel No. 5 is perhaps the world’s most famous perfume. • 4711 is a traditional German Eau de Cologne (“water from Cologne“). It has been produced in Cologne since at least 1799. The first small factory was in Glockengasse 4. Additionally the houses at Cologne got numbers, and the fragrance factory received the number 4711, which became a world - famous brand name. • The term 08/15 refers to a German Army’s standard machine gun. During World War I it was manufactured in such large quantities that it became the army slang for anything that was standard issue. Nowadays this term is also used in normal life. • The fractions 3/4, 6/8, and 4/4 denote time signatures in music. E.g. the waltz is a dance in three-four time. Cavalry march music is often written in six-eight time that means we have six eight notes in one beat. Four-four time is a tango rhythm. Since 2009 this partner dance (it takes two to tango) has been included in the UNESCO Intangible Cultural Heritage Lists. • Sumer in southern Mesopotamia is one of the earliest known civilizations in the world (the late 6th millennium to the early 2nd millennium BC). Around the middle of the 4th millennium cuneiform script was invented. Cuneiform means “wedge-shaped”, due to the triangular tip of the stylus used for impressing signs on wet clay. Two characters were enough to represent all numbers: • The Mesopotamians used a sexagesimal (base 60) numeral system. This is the source of our 60-minute or 3600-second hours and 24-hour days. There are 12 months in a year. The Sumerian calendar also measured weeks of 7 days each. • Chess is played on a square board of eight rows (denoted with numbers 1 to 8) and eight columns (denoted with letters a to h). e2 – e4 means to move a chess piece from position e2 to position e4.
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