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Home , Angle trisection, Platonic solid, Spiral of Theodorus

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 ”, “Good Luck”, “Blue ”, 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 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 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 . 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 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 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 are the triple (3,4,5), the corresponding square numbers, and the generalization with semicircles on the sides of a right .

• 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 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 formula V + F – E = 2 shows the relation between the numbers of vertices (V), faces (F), and edges (E) of simple .

• An example for a simple polyhedron is a . Next to the polyhedron formula we see the 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 . This hexagram can be viewed as a compound of two equilateral .

• 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 “-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.

• The two principal properties of the electric power supply, voltage and frequency, differ between regions. A voltage of 220 V (volts) and a frequency of 50 Hz (hertz) are used in Europe and many other parts of the world.

• A postal code or ZIP code is a series of letters or digits or both included in a postal address for the purpose of sorting mail. The great majority of the 190 member countries of the Universal Postal Union have a postal code system. 3604 is the ZIP code of Eugen Jost’s Swiss hometown.

Prime Time

Prime Time

• A dot pattern runs through the painting like a winding snake. The individual dots symbolize the natural numbers (positive integers) in ascending order. The prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, ... are especially marked by white spots.

• A or prime is an integer greater than 1 that can be divided only by itself and 1.

• By the way, the largest known prime 277 232 917- 1 (effective May 2018) has 23 249 425 digits. Of course, the number snake in the painting is “a bit” shorter!! Such a prime number that is one less than a power of two is called a Mersenne prime, named after the Minorites friar Marin Mersenne (1588 – 1648).

• Mersenne stated that the numbers 2p – 1 were prime for the primes 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257. He claimed to have tested the formula for all primes less than or equal to 257. A really challenging task at his time without calculating machines or even computers. It took until 1947 to completely check Mersenne’s list. He was not entirely accurate, because he excluded 61, 89, 107 and he included incorrectly 67 and 257. It is not known, whether the number of Mersenne primes is finite or infinite.

• Nowadays we would characterize Mersenne as an excellent networker. He travelled extensively through Europe and met some of the greatest people in mathematics of his time (like Descartes, Desargues, Fermat, Pascal, Galileo, Huygens) on a regular basis. Thus he was able to communicate mathematical knowledge. In a time with no scientific journals, with no regular mail, with no telephone and with no internet an extremely important activity to disseminate scientific results.

• Every can be composed of primes. This representation is unique except to the order of the factors. For example, the number 28 can be expressed in exactly one way as a product of prime numbers: 28 = 2*2*7. Therefore, primes are called the building blocks of natural numbers.

• More than 2300 years ago it was already known that there is an infinite number of primes. A proof can be found in the famous textbook of (c. 300 BC).

• Just one conjecture can make you immortal. This happened to Christian Goldbach (1690 – 1764) who exchanged letters with Leonhard Euler (1707 -1783), the outstanding mathematician of the time. By trial and error, Goldbach made an astounding observation: Every even number greater than 4 is the sum of two odd primes (sometimes in more than one way). Some examples: 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7 = 5 + 5, …

• Goldbach failed to prove his conjecture and therefore wrote to Euler. After Euler’s death the letter was discovered among his enormous legacy. It is not known whether Euler tried to solve this problem. Despite numerous attempts the conjecture is still unproved.

• Also unproved up to this day is the conjecture that there is an infinite number of twin primes. A twin prime is a pair of primes of the form p, p+2: (3, 5), (5, 7), (11, 13), (17, 19), …

• There is only one triplet prime p, p+2, p+4, namely 3, 5, 7.

• There can be arbitrarily large gaps between successive primes.

• As a student in Budapest in 1931, Paul Erdös (1913 – 1996) found an elegant elementary proof that there is always at least one prime between an integer n greater than 1 and its double number 2n. Erdös was one of the most prolific mathematicians of the 20th century. He was known both for his social practice of mathematics and for his eccentric lifestyle.

• The Sieve of is a most efficient way of finding all small primes (e.g. those less than 1 000 000). Eratosthenes of Cyrene (c. 276 BC – c. 195 BC) was a Greek scientist with many fields of interest. He is regarded as founder of the discipline of geography and also was the principal of the .

• There are also prime numbers with very special properties. A right-truncatable prime is a prime which remains prime when the last (“right”) digit is successively removed. The largest is the 8-digit number 73939133.

• Even cicadas know at least one prime number as we learn from Thomas Jefferson (1743 – 1826), the principal author of the Declaration of Independence and 3rd President of the United States. In 1775 he recorded in his "Garden Book" the 17-year periodicity of a special locust, the Magicicada septendecim. In that year these cicadas had again emerged from the ground at his home Monticello.

• Consider a rectangular grid. We start at the central point at 41 and arrange the successive positive integers in a fashion (anticlockwise). In the diagonal we find the primes generated by the n2 + n + 41. This polynomial provides an uninterrupted sequence of prime numbers for n = 0 to 39. This kind of graphical depiction of prime numbers was devised by Stanislaw Ulam (1909 – 1984).

• There are two facts about the distribution of prime numbers: One is that, despite their simple definition and role as building blocks of natural numbers, prime numbers are among the most arbitrary, unruly objects that mathematicians study. They grow like weeds among natural numbers, seemingly subject to no law other than chance, and no man can predict where again one will sprout, nor see a number whether it is prime or not. The other fact is much more astonishing, because it says just the opposite, that the prime numbers show the most monstrous regularity, that they are indeed subject to and obey them with almost embarrassing accuracy. (Don Zagier, inaugural lecture, Bonn 1975).

Hidden Pythagoras I

Hidden Pythagoras

• If you are a bit familiar with elementary geometry and you think of the of Pythagoras you normally have the following in mind:

In a the sum of the squares of the lengths of the legs equals the square of the length of the .

1

(In a right triangle, the side opposite the right is called the hypotenuse. The other two sides are called legs.)

• The painting of Eugen Jost looks completely different in comparison to the above . We see a square that is dissected in eight congruent isosceles right-angled triangles.

Where is “Pythagoras”? We try to find out the relation of this figure to the . It’s kind of a puzzle. Two or four of these triangles form squares; the first one has half the area of the second one.

Their side lengths correspond to the lengths of a leg and of the hypotenuse of the right triangle, respectively. Therefore our puzzle shows a special case of the Theorem of Pythagoras and simultaneously provides a descriptive proof.

2

• Now we consider the general case. The Pythagorean Theorem states (written algebraically):

In any right triangle with legs of lengths a, b and hypotenuse of length c holds: a2 + b2 = c2.

We arrange four identical right triangles of side lengths a, b, and c in a large square of side length a + b, leaving two square spaces of sides a and b, respectively (left figure below). The four triangles can also be arranged in the large square of side length a + b to leave a central quadrangle space with equal sides of length c (right figure below). Explain why this quadrangle is a square!

In both cases the contained squares equal the large square minus four times the triangle. Therefore the sum of the two smaller squares a2 + b2 equals the larger square c2.

• In his Elements Euclid generalizes the Pythagorean Theorem. We draw any geometric figure with its base equal to one of the sides of a right triangle. Then we draw similar figures with bases equal to the other two sides. It is still true that the figure on the hypotenuse is equal in area to the sum of the areas of the other two figures.

3

• The next painting once more shows a special case of the Pythagorean Theorem. Again we consider an isosceles right-angled triangle. The squares on the sides of the triangle are dissected in congruent smaller squares. We can determine the areas of the three squares by counting the small squares inside. For the squares on the legs we get 25 small squares each, and for the square on the hypotenuse we get 49.

4

25 + 25 = 49

Something strange seems to go on. Following the Pythagorean Theorem we get the equation

52 + 52 = 72 or 25 + 25 = 49.

Do we see an optical illusion? How can we explain this mysterious result?

5

Hidden Pythagoras II (Le Quattro stagioni)

Le Quattro stagioni

• Four seasons, four colors, four congruent triangles, again a Pythagorean puzzle.

6

• A right triangle has legs of

lengths a and b (a>b) and a hypotenuse of length c. The sum of the measures of the in every triangle is 180°. If one angle equals 90°, then the other two complement one another to a too.

By arranging two congruent right triangles such that different non right angles meet at a vertex, we get a right angle at this vertex. This property is used in the following proof.

• The proof is based on the diagram from Jost’s painting.

small (inner) square + 4 * triangle = big square (a – b)2 + 4 * ab = c2 a2 - 2ab + b2 + 2ab = c2 a2 + b2 = c2

• The converse of the Theorem of Pythagoras is also true. If for a triangle of side lengths a, b, c holds a2 + b2 = c2, then it is right-angled.

7 Positive integers that satisfy the equality a2 + b2 = c2 are known as Pythagorean triples. A well known construction of a right angle from a loop of string with 3 + 4 + 5 = 12 equally spaced knots is based on the Pythagorean triple (3, 4, 5).

A Babylonian clay tablet lists integers corresponding to Pythagorean triples, which suggests that this theorem or at least its converse may have been known long before Pythagoras.

8 Dynamic Pythagoras

Dynamic Pythagoras I

• The picture sequence indicates a dynamic process where an additional square is generated in the interior of a fixed square. The interior square becomes smaller and smaller and finally disappears, but then it grows again until it is finally congruent with the fixed square.

• Interesting questions arise: How do we get the interior squares? What connections to the Pythagorean theorem do exist?

• Pythagorean Theorem:

9 In a right triangle the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse.

• The Greek philosopher Pythagoras of Samos (c. 570 – c. 500 BC) was the founder of a brotherhood that, although religious in nature, formulated principles that influenced the thought of (428 – 348 BC) and (384 – 322 BC) and contributed to the development of Western rational philosophy. “Everything is number” was the credo of the Pythagoreans. For certainty the above theorem was known long before Pythagoras.

• Elisha Scott Loomis (1852 - 1940) collected and classified approximately 370 proofs of the theorem in his book The Pythagorean Proposition. But there are still more proofs. James A. Garfield (1831 - 1881), the 20th President of the United States, published an interesting proof in 1876.

• We take a square from Eugen Jost’s picture. This cutout gives an idea for a proof. We only need two of the three quadrilaterals in this figure.

But let us begin with the beginning. We arrange four congruent right triangles with side lengths a,b,c to a square, as shown below.

Our final goal is to show that a2 + b2 equals c2. The area of the entire square is (a + b)2. The area of each triangle is ab , so the sum of the areas of the four triangles is

10 2ab. Therefore the area of the quadrilateral in the center is (a + b)2 – 2ab = (a2 + 2ab + b2) - 2ab = a2 + b2. Now we have to show that the quadrilateral in the center is a square with area c2. We know that all sides have length c, but we need to show that the angles are right angles. It holds: � + � = 90°, because the third angle in each right triangle is 90°. Therefore the angle at E is 90° as well (AB is a straight ). The same logic applies to the three other angles of the quadrilateral, so it is a square with area c2.

• Also the following picture sequence contains a proof of the Pythagorean theorem.

Dynamic Pythagoras II

11 First of all we consider the dynamic process. We start with the big yellow square. The vertices of a – in the initial phase - hidden square consistently glide on the sides of the big square. The area of the gliding square decreases and reaches its minimum (half the area of the yellow big square) when the vertices are the midpoints of the big square. Then the process goes into reverse and ends up with the big blue square.

The center square of Jost’s picture shows a special case of the Pythagorean theorem. We recognize isosceles right triangles, all in all eight congruent triangles.

Any two make the square on a leg of the right triangle and four make the square on the hypotenuse. Thus we have got a triangle puzzle that provides an intuitively clear proof.

For a proof of the general case we choose another square of the picture.

The legs of the congruent right triangles have side lengths a, b and the hypotenuse length c. Therefore the large square has area (a + b)2. Now it is easy to verify the Pythagorean theorem.

12

the Wolf

13

Playing with Pythagoras

14 Blue Stars

Blue Stars

• In a square we connect two neighboring vertices with the midpoint of the opposite side. We repeat this procedure for each side of the square.

• The “octagon-star” in the center of the square is the basic figure in the painting Blue Stars. Nine of these stars form the underlying structure of the complete construction. This special octagon-star has got a name of its own, Knauth’s figure. Johannes Knauth (1864 - 1924) was a master builder of cathedrals in Strasbourg. With a brand new technical invention he saved the tower of Strasbourg Cathedral from collapsing and he also was involved in finishing Cologne Cathedral.

• Regular are polygons that are both equilateral and equiangular. What do you think: Is the octagon-star of the Knauth figure regular?

• Now we trisect the sides of a square. The marked points are the vertices of an octagon. What do you think: Have we got a regular octagon?

• On the four sides of a square we draw equilateral triangles looking into the interior. The vertices of the triangles that do not coincide with the vertices of the fixed square are the vertices of a new square (red lines in the figure). Now we consider the midpoints of the sides of the red square together with the eight midpoints of the triangle sides in the interior. These points are the vertices of a regular 12-gon (grey shaded in the figure below).

• Studying ground plans of Romanesque churches we can recognize that often a square was used as unit of measurement for components of the building. Inside the square we can find a triangle that has the same basic line and height as the square.

• Why especially a square? We cannot help speculating. A square contains the number four that symbolizes a part of the Creation: the four seasons, the four points of the , the four elements. Additionally we have the inscribed triangle. Three times four equals twelve. The year has twelve months, we differentiate between twelve hours daytime and nighttime, we know twelve signs of the zodiac.

• Twelve is a so-called sublime number. What does that mean? The factors of twelve are 1, 2, 3, 4, 6, 12. Altogether we have 6 factors. They add up to 1 + 2 + 3 + 4 + 6 + 12 = 28. Both 6 and 28 are perfect numbers. In , a sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number. This is a very rare property. Up to now there are only two sublime numbers known; besides 12, a number consisting of 76 decimal digits.

Playing within Knauth’s Grid

5 x 20%

0% - 20% - 40%- 60% - 80% - 100%

Golden Ratio

Golden Ratio

“Geometry has two great treasures: one is the Theorem of Pythagoras, the other one the division of a line into extreme and mean ratio. The first may be compared to a measure of gold, the second may be named a precious jewel.” (1571 - 1630)

• Nowadays the “extreme and mean ratio” is better known by the name “golden section” or “golden ratio”. Here is the definition: The golden ratio is a special number found by dividing a into two parts so that the whole length divided by the longer part is equal to the longer part divided by the smaller part.

In other words: The whole is to the section as the section is to the remainder.

To determine the value of the golden ratio, we “translate” the above definition into mathematical language (the whole = a + b, section = a, remainder = b, where a > b). Therefore we have to solve the equation = : 2 2 = = 1 + = 1 + ↔ ( ) = + 1 ↔ ( ) - - 1 = 0 → = .

• The value of the golden ratio is an and is generally denoted by the Greek letter �. The number surrounding the painting shows the beginning of the decimal representation of � that never ends or repeats. � = 1,61803398874989484820458683436563811772030917980576286213544….

• The symbol � (phi) for the golden ratio was established at the beginning of the 20th century. It forges a link to Phidias (c. 480 – c. 430 BC), the outstanding sculptor and architect, by taking his initial.

• Most probably the golden ratio was discovered by the ancient Greeks, and its documented history begins with the Elements of Euclid of Alexandria, written around 300 BC. This famous treatise is composed of 13 books (=chapters). The golden ratio appears in book II in relation to areas. Book VI contains a clearer definition: “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.” By the way the first English translation of Euclid’s work was made in 1570.

• Euclid’s objectives in writing his Elements were twofold. On the one hand, he wanted to compile all the basic mathematical findings of his era and compose a sort of encyclopedia that would be a textbook used for teaching. On the other hand, he introduced a certain methodology for demonstrating proofs and he created a new mathematical theory based on axioms (self-established facts) and the laws of deduction.

• Construction of point P that divides the given segment [AB] in the golden ratio:

- [BC] ⊥ [AB] and |BC| = ½|AB| - k(C; r =|BC|) ∩ [AC] = D - k(A; r =|AD|) ∩ [AB] = P

• The pentagram or star was a distinctive badge of the Pythagorean society. This five-pointed star can be made by drawing all the diagonals of a regular pentagon.

In this figure we can find the golden ratio twice: o The diagonals (starting at different vertices) divide themselves in this ratio, || || e.g. = = �. || || o The ratio of the length of a diagonal to the length of a side equals �.

• A regular pentagon with a strip of paper:

There is a simple way of “constructing” a regular pentagon as long as we don’t take the result too serious. All we have to do is to take a strip of paper and tie an ordinary knot. Then we carefully tighten it and press it flat.

We recognize one entire diagonal, a bit of a second diagonal, and their point of intersection. This point divides the diagonals in the golden ratio.

• In a golden rectangle the ratio of length to width is the golden ratio. Ancient Greek architects and artists were aware of its harmonious impact. The Parthenon is an example of the early architectural use of the golden rectangle. The golden rectangle also appears in arts as paintings of Leonardo da Vinci, Albrecht Dürer, Piet Mondrian, and Salvador Dali show.

• The words Girasole and Parthenon in Jost’s painting indicate that there are links between the Fibonacci sequence and the golden ratio. Girasole is the Italian word for sunflower. When we look at the head of such a flower we notice both clockwise and counterclockwise spiral patterns formed by the florets. The numbers of theses depend on the size of the sunflower, but they are adjacent Fibonacci numbers. As previously mentioned we can find several golden rectangles in the world-famous facade of the Parthenon.

• We consider three congruent golden rectangles. They are perpendicular to each other and intersect each other symmetrically. The corners of these rectangles coincide with the vertices of an whose center is the common point of the rectangles.

• � is the solution of the equation x2 – x – 1 = 0. Therefore it holds �2 = � + 1. By multiplying the two sides several times by � we get:

� = �2 + � = � + 1 + � = 2� + 1 � = �3 + � = (2� + 1) + (� + 1) = 3� + 2 � = �4 + � = (3� + 2) + (2� + 1) = 5� + 3 � = �5 + � = 8� + 5 � = �6 + � = 13� + 8 … .. We see that to obtain any power of � it is sufficient to sum up the results of the two preceding powers. Do you recognize the pattern? Do you recognize the Fibonacci numbers?

• The quotients of Fibonacci numbers fn divided by their predecessors fn-1 approximate towards �: lim = �. →

• How can we determine the value x of the unusual expression of square roots of 1?

x = 1 + 1 + 1 + 1 + ⋯

We square both sides of the above equation. Squaring the right-hand side simply removes the outermost and we obtain

x2 = 1 + 1 + 1 + 1 + 1 + ⋯

As the second expression on the right-hand side goes on to , it is actually equal to x. We therefore get the quadratic equation x2 = x + 1. But this is precisely the equation that defines the golden ratio. Therefore our endless expression x is equal to �.

• Some amount of research has been devoted just to the simple question of the origin of the name golden section. Although the ratio itself has been known since antiquity, this name was first used around the 1830s by the German mathematician Martin Ohm (1792 - 1872), brother of the famous physicist Georg Simon Ohm (1789 - 1854). He writes in his book The Pure Elementary Mathematics: “Diese Zertheilung einer beliebigen Linie in 2 solche Teile nennt man wohl auch den goldenen Schnitt. (One also customarily calls this division of an arbitrary line in two such parts the golden section.”) Ohm surely did not invent the term, it seems likely that he used a commonly accepted name especially popular in nonmathematical . Following Ohm’s book, the term “golden section” started to appear frequently in the German mathematical and art history literature and is nowadays used all over the world. • Eugen Jost’s sketch indicates a construction of the golden section. The figure next to it explains this construction. We consider the ABC with circumcircle. X and P are the midpoints of sides [AC] and [BC] respectively. The line XP intersects the circumcircle in points Y and Z. Then P divides |XY| in the golden section.

Fibonacci meets Pythagoras

Fibonacci meets Pythagoras

• Certainly the name Pythagoras is well known from school days, the name Fibonacci is probably less familiar. About 1700 years lie between these two celebrities. How can an encounter occur nevertheless?

• Pythagoras (c 570 BC – c 500 BC) came from the island of Samos and founded a brotherhood dedicated to mathematics in Lower Italy (at that time part of Magna Graecia). The name Pythagoras is strongly connected with a special relationship between the lengths of the sides in a right triangle. The Theorem of Pythagoras states: In a right triangle the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. • Fibonacci (c 1170 – c 1240), actually called Leonardo da Pisa, was a merchant and mathematician who mostly lived in the of Pisa. Nowadays he is mainly remembered for a little problem on the number of offsprings of rabbits he posed in his famous “Book of Calculation (Liber Abaci)”. This book has been very influential because here Leonardo advocated the Arabic numeral system and explained its application, especially for merchants.

• The little problem mentioned above leads to the so-called Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, … How is this number sequence generated?

• Which Fibonacci number follows 987?

• Is 12 586 269 025 a Fibonacci number?

• The scenario behind the Fibonacci numbers is the following problem:

How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair produces a new pair that becomes productive from the second month on?

• What about the meeting of Fibonacci and Pythagoras? To be precise we consider Fibonacci squares in a Pythagorean figure:

Two squares with side length 1 (and thus area 1) are laid together in such a way that they touch each other in a vertex and the adjacent sides are perpendicular to each other. These sides are the legs of a right triangle. The square over the corresponding hypotenuse has area 2. By appropriate positioning of one of the leg squares to a free vertex of the hypotenuse square we get a new right triangle whose hypotenuse square has area 3. Now the larger leg square is positioned at the hypotenuse square and the procedure is continued. In this way a Pythagoras spiral gradually develops.

The areas of the two starting squares together with the areas of the hypotenuse squares represent the elements of the Fibonacci sequence.

• The meaning of the term spiral in mathematics differs from the use of the word in everyday language. When we hear the word spiral, we often think of circular staircases, snail shells, helices or coil springs, i.e. spatial structures. In mathematics, only plane figures are called spirals.

Turn, Turn, Turn

Turn, Turn, Turn

• A spiral is a that emanates from a point, moving further away as it turns around the point.

• The request “turn, turn, turn” can be regarded as a rough guideline for the behavior of a spiral. A much more detailed description is provided by the inscription on Jakob Bernoulli’s (1654 – 1705) tombstone at the cloisters of the Basel Münster: “Eadem mutata resurgo – though changed, I shall arise the same.” These words summarize the many features of a spiral: stretch it, rotate it, or invert it, it always stays the same.

• The colorful shows rectangles arranged in form of a spiral. This is best visible when we connect the intersection points of the diagonals of the individual rectangles with circular arcs. We start at the black rectangle and proceed to the white one, yellow one, green one, …

• All these rectangles have one feature in common: Their side lengths have the ratio 2 : 1.

• A sheet of paper normally has a rectangular shape. The international paper size standard ISO 216 is based on the aspect ratio 2 : 1, too. With every sheet of paper we figuratively hold the irrational number 2 in our hand. Thus we often touch irrationality and most of us do not even realize it!

• Why does especially 2 occur in connection with a paper size standard? For answering this question we consider the definition of paper sizes in the series A1, A2, A3, A4, A5, … - Successive paper sizes are defined by halving the preceding paper size across the larger side. This also halves the area of each sheet effectively. - All paper sizes are similar, that means the ratio is always equal.

• We translate this definition in mathematical notation: On a sheet of paper let a be the long side and let b be the short side. Then we fold it in half widthwise. Since the aspect ratio remains the same, we get:

2 2 = and a = 2b and = 2 and = 2. That means the ratio of the long side to the short side of a sheet of paper in the A series always is 2. • The most frequently used paper size A4 measures 210 x 297 millimeters. The base A0 size of paper is defined as having an area of 1 m2.

• 2 is an irrational number. Therefore we won’t find any pattern in its decimal fraction. But what a surprise when we look at the continued fraction of 2 (to the right of the rectangular tessellation)! Here we have got a repeating (periodic) pattern. By the way, this property holds for every square root. In contrast the continued fraction of a is finite.

• The . ( was a Libyan Greek who lived during the 5th century BC.) We can construct the square root of the positive integer n if we have previously constructed the square root of the positive integer n-1. This recursive process is based on the Pythagorean theorem. We start with an isosceles right triangle with each leg having unit length. The hypotenuse has the length 2. Another right triangle is formed with one leg being the hypotenuse of the prior triangle (with length 2) and the other leg having the length of 1. The length of the hypotenuse of this second right triangle is 3. This process then recurs. The nth triangle in the sequence is a right triangle with leg lengths � and 1, and with hypotenuse length � + 1.

• A short history of the radical sign Nikomachos (c 160 AD) used the Greek word rhiza that originally means root of a plant, but also basis and origin in a metaphorical sense. (c 480 AD – c 524 AD) translated it literally with radix into Latin. Radix was abbreviated as R or R with a slash through the right leg of the R. Our familiar symbol √ was first used in a German algebra textbook in 1525. If one had a long term under the radical sign, the term was put in parentheses, and later there was drawn a line over it.

SQRT 2

On Squaring the

On

• The squaring or quadrature of a circle means to construct a square that has the same area as a given circle (area r2 �). • Greek philosophers viewed the straight line and the circle as perfect . Therefore only ruler (unmarked) and compass were allowed as construction tools.

• Lunes are plane regions bounded by arcs of two different circles. The word lune originates from the Latin word lunar that means moon shaped.

1 • In his three long narrow plates Eugen Jost demonstrates how figures bounded by arcs of different circles are turned into squares.

• The lune and the corresponding equilateral right triangle have the same area. Therefore it is possible to transform four congruent lunes in a square of equal area.

• (c 460 – c 380 BC) succeeded in squaring lunes. He found and proved: The sum of the areas of two lunes constructed on the two legs of a right triangle equals the area of that triangle.

• Hippocrates probably believed that his result would lead to a solution of squaring the circle itself. But his efforts were unsuccessful. In the subsequent period all attempts failed as well.

• Originally Hippocrates was a merchant, but not a successful one. Aristotle tells us that he was cheated by Byzantine customhouse officials. According to other sources his trading ship was captured by pirates. Therefore he went to Athens to try to save his property in the law court. We do not know how his complaint turned out. But during his time in Athens he studied geometry and published a lot of contributions in plane geometry.

• We must not confuse “our” Hippocrates of Chios with the famous physician Hippocrates of Kos, the author of the Hippocratic oath.

• Squaring a circle belongs to the three famous problems of the ancient Greeks. The other two are trisecting an angle (dividing an angle into three congruent angles) and duplicating a cube (constructing a cube with twice the of a given cube).

2 • These problems were solved by the Greeks, but with methods that required the use of curves other than the straight line and the circle. This breach of the rules was not accepted.

• It was not until the 19th century that it was shown that the three constructions are impossible with ruler and compass. The proofs used algebraic ideas that were not known to the Greeks.

• In 1837 Pierre Laurent Wantzel (1814 – 1884) proved the impossibility of trisecting a 60° angle. Nearly 50 years later (1882) Ferdinand Lindemann (1852 – 1939) showed that � is a transcendental number. This proved that it is not possible to construct a segment with length � or �. Therefore we cannot square a circle.

• Squaring the circle is a phrase that is used in everyday life for a particularly difficult problem to be solved. Strictly speaking, this phrase only makes sense for an insoluble problem.

3 3.14159....

3.14159...

• It is really amazing: We measure the circumference and diameter of a circle, calculate the ratio of these two quantities and always get the same result for any circle. This ratio is called ’ constant �.

• � is an irrational number, which means that it cannot be expressed as a fraction , where m and n are integers. Consequently its decimal representation never ends or repeats itself. A first impression gives the start of the digit sequence of � in the painting.

4 • In the following sentence the number of letters in each word represents the successive digits of �: How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.

• Beyond being irrational, � is a transcendental number.

• Real numbers that are not algebraic are called transcendental. These numbers are all irrational, but the converse is not true. For instance, √2 is irrational, but not transcendental, being a solution to the equation x2 – 2 = 0.

• The name transcendental has nothing mysterious about it. It simply implies that such numbers transcend the realm of algebraic (polynomial) equations.

• Operations performed with ruler and compass have corresponding algebraic descriptions. Any length that can be constructed with these classical tools nowadays is called algebraic. That is, it can be obtained as the solution of a polynomial equation of the form

n n-1 2 anx + an-1x + … + a2x + a1x + a0 = 0, where an, … , a0 are all integers.

• The first workable algorithm for approximating � seems to have been carried out by Archimedes of Syracuse (c 287 BC – 212 BC). His idea was to inscribe regular polygons of 6, 12, 24, … sides inside a circle, find the perimeter of each , and divide it by the diameter of the circle. Then he repeated the process with circumscribing polygons. From the 96-sided inscribed and circumscribing polygons Archimedes obtained the approximation:

3 < � < 3

• � is also used in the Bible. But without taking decimal places into account, as a look in the Book of Kings (I Kings 7,23) shows: Also, he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about.

• Every year we can celebrate two different � - days: March 14th and July 22th. Can you imagine why?

5

Three missing Points

6 Good Luck

Good Luck

• Games of chance have been the start of probability theory. We only need to throw a coin or one or several . But first we are going to explain what probability means. The probability of a particular outcome is the ratio of the number of desired outcomes to the number of possible outcomes.

• For centuries rolls have been used to teach probability. If a single fair dice is thrown, any one of its six faces is equally likely to turn up. Thus, for example, we have one chance in six of throwing a 4; therefore its probability is . This does not guarantee that a 4 will turn up exactly once if we throw the dice six times. But the Law of Large Numbers says that if we throw the dice 100 or 1,000 or 1,000,000 times, the

7 number of times that 4 appears will tend to get closer and closer to of the number of total throws.

• The probability to get a prime number when you throw a dice is . Can you explain why?

• The probability of throwing a particular sequence of n numbers is . For example, the chance of throwing a 2 followed by a 5 is = .

• If we use a pair of dices we get a slightly different situation. Because each dice can come up in six different ways, there are 36 (= 6*6) combinations possible. We can get the total 4, for example, with a 1 and a 3, or a 3 and a 1, or a 2 and a 2. So the probability of getting the total 4 is = . The probability of throwing a 7 is = , the probability of throwing 12 is only .

• Let us summarize: Probability is the measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur.

• The artist Eugen Jost turns out to be a gambler because he is going to throw dices while generating an arts work. He starts with a 21x21 grid in which the upper left square signals the beginning, and the center square that includes Jost’s signature marks the end. The remaining 439 squares are painted in six different colors. To color these squares Jost uses a dice whose six faces have exactly these colors. Starting in the first row following the entrance arrow Jost throws the dice 439 times to decide which color to fill each square with. In a spiral he continues to the center. Then he does the same by throwing a standard dice 439 times to determine a number between 1 and 6 to be written as figure or Roman numeral or depicted with dots or line patterns in each square. Actually the creation of this painting is kind of a stochastic process. Only the six different colors for the squares and the look of the numbers are the artist’s decision.

• Around 1654 Antoine Gombaud (1607 - 1684), better known by the name of Chevalier de Méré, proposed a gaming problem to the mathematician Blaise Pascal (1623 - 1662). The problem concerns a game of chance with two players who have equal chances of winning each round. The players contribute equally to a prize pot, and agree in advance that the first player to have won a certain number of rounds will collect the entire prize. Now suppose that the game is interrupted by external

8 circumstances before either player has achieved victory. How is the pot divided up fairly?

• Pascal discussed the problem in his ongoing correspondence with Pierre de Fermat (1607 - 1665). In the course of this discussion Pascal and Fermat not only came up with a convincing, self-consistent solution to the division of the prize pot, but also developed concepts that have continued to be fundamental in probability to this day.

• The starting insight for Pascal and Fermat was that the division was not to depend so much on the history of the part of the interrupted game that actually had taken place, but more on the possible ways in which the game might have continued, if it had not been interrupted. In a series of letters both mathematicians laid the foundation for the modern theory of probability.

• In Europe, the subject of probability was first formally developed in the 16th century by the work of Gerolamo Cardano (1501 - 1576), whose interest in the branch of mathematics was largely due to his habit of gambling. In 1525 he published a book titled Liber de ludo aleae (Book on Games of Chance).

9 Morley’s Miracle

Morley’s Miracle

• Not all knowledge about triangles can be found as far back as in the writings of antiquity. Thus in 1899 the mathematician Frank Morley (1860 - 1937) discovered an astonishing result that had apparently been overlooked by the ancient geometers. His colleagues were so enthusiastic that they even spoke of Morley's Miracle.

• What does this miracle look like? Take any triangle ABC and draw two lines through each vertex so that each interior angle is divided into three equal parts (angle trisectors). Now consider the two angle trisectors adjacent to side [AB]; they meet at a point. Similarly, the trisectors adjacent to side [BC] meet at a point and those adjacent

10 to side [CA], too. These three intersection points form an equilateral triangle (Morley triangle). C

A

B

Morley’s Theorem:

11 In any triangle the angle trisectors lying near the sides intersect in an equilateral triangle.

• Using dynamic geometry software provides a first impression that the theorem holds true for all triangles. The proof itself is a bit challenging. But the beauty of Morley’s theorem lies in the simplicity of the statement.

• Starting in 1900 Frank Morley was professor at Johns-Hopkins-University in Baltimore (Maryland). He was an excellent chess player and even once defeated the world champion Emanuel Lasker (1868 - 1941) who was a mathematician too. After completing his doctorate at the University of Erlangen, Lasker intended to pursue an academic career. But he was not successful in Europe or the USA and eventually became a professional chess player.

• With compass and ruler a given segment can be bisected, trisected, quartered, etc. Now we consider angles, and the situation is different. Any angle can be bisected with compass and ruler, but only certain angles can be trisected. For example, the 90° angle is easy to trisect. This may be a possible explanation why the ancient geometers did not discover Morley’s theorem. Because they were unable to trisect an arbitrary angle, they neglected problems involving !

• Instead of the interior angles, you can consider the exterior angles of a triangle and Morley’s theorem holds true as well. If you trisect the exterior angles of a triangle, then the pairs of exterior trisectors adjacent to sides [AB], [BC], and [CA] meet at a point, respectively.

12 Starry Starry Night

Starry Starry Night

• It's a starry night. We are looking at a shining star in the deep blue sky. How big might its surface be? We notice that our special star is embedded in a lattice with its peaks and notches lying exactly on grid points. Counting squares doesn’t work in this case, because many squares lie only partly inside the star. But there is a special method to determine the area only by counting grid points.

• That sounds pretty amazing, but it works – not only for star-shaped figures. Let’s take a closer look at this method. To do this we draw a simple polygon in a square grid such that all the vertices of the polygon are grid points. The grid points inside the figure are important as well as all the grid points on the boundary, not only the vertices. The area of the polygon is calculated according to the following rule:

• You count the number of points i in the interior of the figure. Add half the number of points b on the boundary. Then subtract 1 from this sum.

This is the so-called Pick’s formula: Area = i + – 1 Take the polygon below as an example. In this polygon i = 16 (blue points) and b = 8 (red points). Applying Pick’s formula we get: Area = 16 + - 1 = 19

• Pick’s formula is useful for calculating or approximating the area of an irregularly shaped region. First you have to put the shape on a grid and then use the formula. It gives an exact area only of a polygon whose vertices are at grid points. Otherwise the formula approximates the area of a polygon.

13 • The Austrian mathematician Georg Alexander Pick (1859 – 1942) published this formula in 1899. He spent most of his academic life in Prague as a professor at the German University there. With his friend Albert Einstein (1879 – 1955) he shared the passion for science and music. The political atmosphere of the late 1930s overshadowed Pick’s last years. As a Jew he was arrested and deported to the concentration camp Theresienstadt, where he died in 1942.

• Mathematical exercises: - Determine the area of the star in Jost’s painting. - Use Pick's formula to show that you cannot draw an equilateral triangle on a lattice so that each vertex is on a grid point.

14 Dunking Cube

Dunking Cube I

• What patterns can be produced on the surface of a mug filled with black coffee, if we dip in a solid cube? Of course the pattern depends on how the cube hits the smooth surface. We differentiate between three cases: The surface is first hit either by a or a vertex or an of the cube. The following picture sequence shows some “screenshots” of varying dunking depths of the three possibilities. The starting picture is kind of a combination of several of these screenshots.

15

Dunking Cube II

16

• If we unfold a cube we get a so-called net consisting of six joined identical squares. We get different nets depending on the edges of the cube we cut along to unfold it. Here are two examples:

Two nets are different, if it is not possible to transform one into the other by reflection or . Altogether we can find 11 different nets for a cube. Now two tasks are left: (1) Draw all these nets. (2) Show that there exist only these 11 nets.

• The cube is the best-known . These solids are convex three-dimensional multifaceted objects whose faces are all identical regular polygons (sides of equal lengths and angles of equal degrees). Additionally the same number of faces meets at each vertex. The ancient Greeks recognized and proved that there exist only five Platonic solids:

17

• Plato (c. 428 - c. 348 BC) was overwhelmed by their beauty and . According to ancient philosophers these five regular solids are the basis for an early atomic theory of matter. They argued that the four basic elements (, , water, ) were all aggregates of tiny solids (atoms) and assigned a regular solid to each of them. The remaining represents the entire .

• The German astronomer Johannes Kepler (1571 - 1630) constructed models of Platonic solids nested within one another in an attempt to describe the orbits of around the sun. Although Kepler’s theories were wrong, he was one of the first scientists to insist on a geometrical explanation for celestial phenomena.

• Duplication of the cube (or the Delian problem) is one of the three classical problems of antiquity. This problem is not solved by doubling the edges of the cube, because this procedure would increase the volume eightfold. Our task is: Given an edge of a cube of length a, we have to produce a new edge of length x such that x3 = 2a3. Of course it’s only allowed to use the Platonic tools (ruler) and compass for the construction. Some two thousands years mathematicians were unable to solve this problem, and in 1837 Pierre Laurent Wantzel (1814 - 1848) proved it to be impossible. Using algebraic methods that were not available to the Greeks, he set conditions under which a geometric construction, translated into a set of equations, could be achieved with Platonic tools. Duplication of the cube leads to the equation x = 2a, and this is not a .

18 Symmetry

Symmetry

• In everyday language symmetry (from Greek ���������) refers to a sense of harmonious and beautiful proportion. In mathematics symmetry has a more precise definition: A symmetric object is invariant to any of various transformations, including reflection, rotation etc.

• A geometric object has symmetry if there is a transformation that maps the figure or object onto itself. There are several kinds of symmetry such as reflectional (mirror), rotational, translational symmetry, , and glide reflection.

19

• A palindrome (from the Greek words ����� and ������) is a nice example for a special kind of symmetry. It is a word, number, or other sequence of characters that read equally backwards as forwards, such as otto or madam or racecar or 4884.

• Longer palindromes are possible when we neglect capital letters, punctuation, and word dividers. Some well-known English examples are, ”Madam, I’m Adam”, “Never odd or even”, “A man, a plan, a canal, - Panama”.

• The following palindrome has been attributed to emperor Napoleon, though the likelihood of his knowing enough English for this composition is questionable: ABLE I WAS ERE I SAW ELBA

• The outstanding mathematician Peter Hilton (1923 - 2010) composed one of the world’s longest palindromes (51 letters): DOC, NOTE: I DISSENT. A FAST NEVER PREVENTS A FATNESS. I DIET ON COD.

• Take any number and add the related mirror number (digits in reverse order). Add to this sum the related mirror number. Repeat this procedure until you get a palindrome. •

Up to now it is not known whether this procedure always leads to a palindrome. The number 196, for example, is still a mystery.

• (220, 284) is the smallest pair of amicable numbers. In what circumstances may numbers become friends? Two numbers are called amicable if the sum of the proper divisors of each is equal to the other number. (A proper divisor of a number is a positive factor of that number other than the number itself.) 220 and 284 are amicable because the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, of which the sum is 284. The proper divisors of 284 are 1, 2, 4, 71, and 142, of which the sum is 220.

Other examples of amicable numbers are (1184, 1210), (2620, 2924), (5020, 5564).

20

Amicable numbers were already known to the Pythagoreans. A first general formula by which some of these pairs could be derived was published c. 850 by the Iraqi mathematician Thabit ibn Qurra (826 - 911).

• The number triple (8, 6, 12) contains the key figures of a cube. It consists of 8 vertices, 6 faces and 12 edges. According to the theory of Empedocles (c. 490 - c. 430 BC) the cube is allocated to the element earth because of its solid structure. What Platonic solids correspond to the number triples (4, 4, 6), (12, 20, 30), (6, 8, 12), and (20, 12, 30)? Each of the following graphs represents a Platonic solid that has been projected down onto the plane (so-called Schlegel diagram).

ƒ* • The equation V – E + F = 2 is known as Euler’s polyhedron formula. All the above number triples satisfy this equation, where V is the number of vertices, E the number of edges, and F the number of faces. This formula was rediscovered and proven by the famous Swiss mathematician Leonhard Euler (1707 - 1783). It had already been known to the French philosopher and mathematician René Descartes (1596 - 1650). Some historians even link this formula to Archimedes (c. 287 - 212 BC).

21 Infinity I

Infinity I

“Infinity is a square without vertices.“ (Chinese saying)

• This saying describes the underlying structure of the painting. The midpoints of the sides of the first large square are the vertices of a new square. This means we cut off the vertices of the first square. Now we repeat this procedure with the second square, and so on. The squares getting smaller and smaller, the procedure comes never to an end.

• The Ionian philosopher and astronomer (c. 500 – 428 BC) puts infinity into the following words:

22 “There is no smallest among the small and no largest among the large; but always something still smaller and something still larger.”

This imagination of infinity is visualized in Jost’s painting. It is possible to continue the squares in both directions, getting smaller or getting larger in size.

• Now we study the areas of the squares. The area is halved while going from one square to the next smaller one. The following sketches show the transition:

• We assume that the area of the largest square is 1. What is the sum of the areas of all the squares? We get the following sum

1 + + + + + + …

One can show that the value of this sum equals 2. At first sight it is astonishing that an infinite sum can have a finite value.

• To get a better understanding we take a look at the following dissection:

2 = 1 + 1 = 1 + + = 1 + + + = 1 + + + + = 1 + + + + + = ….

Since 4 = 22, 8 = 23, 16 = 24, … we get the infinite series of the reciprocals of the power of two - step by step:

1 + + + + + + …

By the way, such a series, in which each term has a fixed ratio to its predecessor, is called a geometric series. As mentioned above, this special series has the value 2.

23 • By exchanging the base and the exponent in the denominator of the above infinite series we get another infinite series with a finite value:

1 + + + + + …

• The determination of this sum had puzzled mathematicians especially in the first half of the 18th century. In 1736 Leonhard Euler (1707 – 1783) proved that the value of this sum of the reciprocals of the squares of positive integers equals .

• A famous example in this connection is a paradox of the Greek philosopher (c. 490 – c. 430 BC). The story is about a race between the athletic hero Achilles and a tortoise. The tortoise gets a lead of 100 meters over Achilles. We suppose that each racer starts running at some constant speed, one very fast and one quite slow. After some finite time, Achilles will have run 100 meters, and reaches the tortoise’s starting point. During this time the tortoise has run a much shorter distance, say, 10 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther, and so on. Thus whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can reach the tortoise. Actually he does not succeed in passing the tortoise. Doesn’t that sound strange?

24

Infinity II

Infinity II

• In the painting we can see a continued of a square of area 1. This procedure descriptively shows that a sum of an infinite number of positive integers can have a finite value.

We start with a square of area 1 and bisect the area. We get two isosceles right triangles, each of the area . We bisect the white triangle and get two smaller isosceles right triangles, each of the area . Again we bisect the new white triangle, and so on. This

25 procedure shows that it is possible to dissect a finite area in infinite pieces. In our case the area is halved step by step. Thus it becomes obvious that an infinite sum of positive elements can have a finite value.

1 = + = + + = + + + = + + + + = ….

1 = + = + + = + + + = + + + + = ….

• Has the number 2048 a special meaning in respect of the painting? The answer is yes. 2048 is a power of 2, it holds 2048 = 2*2*2*2*2*2*2*2*2*2*2 = 211. Of course Eugen Jost couldn’t draw an infinite number of . If we count the halved triangles in the painting we reach number 11.

• Not every infinite sum has a finite value, even if the elements in the sum get smaller and smaller. We regard the sum of the reciprocals of the positive integers

1 + + + + + + + + + + + ……

To demonstrate that this sum approaches infinity we do a kind of regrouping:

1 + + ( + ) + ( + + + ) + ( + + + ……

Now we replace every term by a number less or equal to it, e.g.

+ > + = and + + + > + + + = , etc.

The new sum is less than the one we started with, and we notice that our substitutes add up to infinity:

1 + + + + + + + + + ….

That means that the sum of the reciprocals of the positive integers is also infinite.

• In mathematics the above infinite series

1 + + + + + + + + + + + ……

26

is called a harmonic series. The name derives from the concept of overtones, or harmonics in music: The wavelengths of the overtones of a vibrating string , , , … (represented by sinusoidal waves) are derived from the fundamental wavelength of the string. Every term of the series (i.e. the higher harmonics) after the first one is the "harmonic mean" of the neighboring terms.

27 It’s a Kind of Magic

It’s a Kind of Magic

• The artist likes to play with square grids. His painting with the blue square is based on the idea of a magic square. The order of such a square is defined by the number of its rows or its columns. For example the blue magic square has order 5 because it has 5 rows.

• A magic square of order n is normally composed of the positive integers 1, 2, 3, … , n2. The magic of a magic square arises from the special arrangement of these n2 numbers in the square grid: The sum of the n numbers in each row, each column, and each diagonal is the same. This sum is called the magic constant.

28 • How do we compute the magic constant of an nth - order magic square? We have to add the numbers 1, 2, 3, … , n2. Because in each of the n rows the sum has to be equal () we must divide our result by n. As magic constant we finally get .

• In the blue magic square the artist has replaced nearly all the numbers by dot patterns and symbols. The magic constant 65 of a (5x5)-magic square helps to find out the “hidden” numbers.

• The numbers of different (nxn)-magic squares is still an unsolved problem in mathematics. We know the result only for n =3, 4, 5. There are 1 square of order 3, 880 of order 4, and 275 305 224 of order 5 (not counting rotations and reflections). For n = 6 there exists an estimation, a number greater than 1019.

• Magic squares have intrigued people for centuries. In fact, the earliest record of the appearance of a magic square is about 2200 BC in China. It was called lo-shu. Legend has it that this 3 by 3 magic square was first seen by Emperor Yu on the back of a sacred turtle on the bank of the Yellow River. Here we see the lo-shu square rendered in our familiar Arabic numerals:

• In 1531 Agrippa of Nettesheim (1486 – 1535) published De Occulta Philosophia, a manual of occult beliefs and practices. Here he expounded the magical virtues of seven magic squares of orders 3 to 9, each associated with one of the planets. E.g. the 5x5 square was dedicated to .

• The Martian square helps to mitigate bilious complaints. This is claimed by the well- known contemporary of Agrippa, the medical doctor Paracelsus (1493 – 1541). He also assigns therapeutic properties to other magic squares.

29 • The 4th order magic square is associated with the and was used as a talisman to drive away melancholy. This may be the reason why Albrecht Dürer (1471 – 1528) selected an order 4 magic square for his famous engraving . And it is really an exceptional magic square, probably the first one seen in European art.

The magic constant 34 can be found in the rows, columns, diagonals, each of the quadrants, the center four squares, the corner squares, etc. Try to find all these possibilities! The two numbers in the middle of the bottom row give the date of the engraving: 1514. The numbers 4 and 1 at either side of the date correspond to the letters D and A respectively, which are the initials of the artist.

• We finish our excursion to magic squares with a visit to a special playground. The artist invites us to describe the patterns and symbols in the following squares of order 5. He hopes that we take delight in discovering all the hidden numbers.

Magic Squares

30

Lo Shu

31

Magic Square

32

V.G.Z.: Past – Present – Future

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100 Squares

34 Eugen Jost was born and raised in Zurich. For him, playing with numbers, signs, and patterns has top priority. That’s why he calls his canvases playgrounds. He is interested in mathematics, geometry and its history. Euclid wrote: A point is that which has no part; a line is a length without width. Notwithstanding that claim, Archimedes drew his broad-lined circles in the sand of Syracuse with his finger. Nowadays it is much easier to meet Euclid’s demands: with a few clicks of the mouse you can reduce the width of a line to almost. In the last decade Jost has created most of his pictures and illustrations with special software on his computer. All pictures in this exposition, however, are acrylic on canvas. Jost’s pictures and illustrations can be found in a lot of books, among them: • Alles ist Zahl (together with Peter Baptist) • Alles ist Zahl - Mathematik andersARTig (together with Peter Baptist and Carsten Miller) • Beautiful Geometry (together with Eli Maor) If you have questions regarding pictures or exhibitions, don’t hesitate to contact Eugen Jost [email protected]

Currently Peter Baptist is the chairman of the Center for Mobile Learning with Digital Technology at the University of Bayreuth (Germany). There he held the chair of mathematics and mathematics education until 2013. Before he went to Bayreuth he had professorships at the Universities of Dresden and Erlangen. His academic career commenced at the University of Würzburg. His collaboration with Eugen Jost started in 2007 on the occasion of the year of mathematics in Germany in 2008.

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