II. Work in pairs. Decide if the sentences 1–7 are T (true) or F (false).
1. The golden ratio is usually represented by the Latin letter ‘q’.
2. German mathematician Martin Ohm must have introduced the popular name ‘the golden ratio or the golden number’ in 1835.
3. The golden number isn’t considered to be associated with elegance of proportion in art and architecture, and there is much evidence for these claims.
4. The golden number is related to the logarithmic spiral.
5. The spiral of the nautilus has the ratio ϕ.
6. Fibonacci numbers are very common in kingdoms of plants and animals.
7. Fibonacci numbers occur in the seed heads of sunflowers and daisies.
III. Many aspects of the natural world display strong numerical patterns and strong connection with Mathematics. Think of a fractal or any other phenomenon. Find some information on the Internet and share it with other students.
Text 11
Reading and Speaking
A STRONG MATHEMATICAL COMPONENT
Maurits Cornelis Escher was born on June 17, 1898 in Leeuwarden (Friesland), the Netherlands. His father George Arnold Escher, who was a civil engineer, and his mother Sarah Gleichman Escher, had three sons of which Maurits was the youngest. The Escher family lived in Leeuwarden for the first 5 years of Maurits Cornelius life in a large house called ‘Princessehof’. 1.___________________________________________________ In 1903 the Escher family moved to Arnhem where Maurits took carpentry and piano lessons until he was thirteen. From 1912 until 1918, Maurits Cornelius Escher attended secondary school. 2._______________
___________________________________________ He never succeeded in his final exam, so M.C. has never officially graduated. Later he started studying at the Haarlem School of Architecture and Decorative Arts. After three years, Maurits decided he had gained enough experience in drawing and woodcutting and left the school.
Escher traveled to Italy regularly in the following years, and it was in Italy that he first met Jetta Umiker, the woman who would become his wife in 1924. 3._____________________________ When the political climate under Mussolini became unbearable, the family moved to Chateau-d’Oex , Switzerland, where they stayed for two years. 4.__________________________________________So two years later, in 1937, the family moved again, this time to Ukkel, a small town near Brussels, Belgium. World War 2 forced them to move a last time in January 1941, this time to Baarn, the Netherlands, where Escher lived until 1970. Most of Eschers better known pictures date from this period. 5. ______________________________. Only in 1962, when he had to undergo surgery, there was a time when no new images were created. Escher moved to the Rosa-Spier house in Laren in the northern Netherlands in 1970, a retirement home for artists where he could have a studio of his own. He died there on March 27, 1972.
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Escher and Umiker had three sons.
Well known example of his work include Drawing Hands, a work in which two hands are shown drawing each other, Sky and Water, in which plays on light and shadow convert fish in water into birds in the sky, and Ascending and Descending, in which lines of people ascend and descend stairs in an infinite loop, on a construction which is impossible to build and possible to draw only by taking advantage of quirks of perception and perspective.
Esher’s works has a strong mathematical component, and many of the worlds which he drew are built around impossible objects such as the Necker cube and the Penrose triangle.
6._________________________________________________________________
For example, in Gravity, multi-colored turtles poke their heads out of a stellated dodecahedron. His work has been referenced in the 1980 Pulitzer Prize-winning
book Godel, Escher, Bach by Douglas Hofstadter.
I. I. Match the words (1–4) with their definitions/explanations (a–d):
| 1 | ascend | a | twist out of the useful shape |
| 2 | distort | b | shape produced by a curve |
| 3 | loop | c | go or come up |
| 4 | carpentry | d | work of a workman who makes and repairs |
II. Choose from (a–g) the one which best fits each of (1–6). There is one choice you do not need to use.
a. Escher’s artwork is well-liked by scientists, especially mathematicians who enjoy his use of polyhedra and geometric distortions.
b. Escher, however, who had been very fond of and inspired by the landscape in Italy, was decidedly unhappy in Switzerland.
c. This house would later be turned into a museum to host work from M.C. Escher.
d. Though he excelled at drawing, his grades were generally poor.
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e. The young couple settled down in Rome after marriage and stayed there until 1935
f. Sometimes the cloudy, cold wet weather of the Netherlands allowed him to focus entirely on his works.
g. He was not interested in Mathematics.
III. In pairs, find and then say what events the following years refer to:
| 1898 | 1903 | 1912 | 1924 | 1937 | 1941 | 1962 | 1970 |
IV. Do you know an artist (a writer) having a strong mathematical component in his/her creative work? Search for information on the Internet and give a presentation on the subject.
Text 12
Reading and Speaking
FRACTAL
A fractal is ‘a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,’ a property called self-similarity. 1 __________. There are three types of self-similarity found in fractals: Exact self-similarity – This is the strongest type of self-similarity; the fractal appears identical at different t scales. Fractals defined by iterated function systems often display exact self-similarity. For example, the Sierpinski triangle and Koch snowflake exhibit exact self-similarity.
Quasi-self-similarity – This is a looser form of self-similarity; the fractal appears approximately (but not exactly) identical at different scales. Quasi-self-similar fractals contain small copies of the entire fractal in distorted and degenerate forms. Fractals defined by recurrence relations are usually quasi-self-similar. The Mandelbrot set is quasi-self-similar, as the satellites are approximations of the entire set, but not exact copies.
Statistical self-similarity – This is the weakest type of self-similarity; the fractal has numerical or statistical measures which are preserved across scales. Most reasonable definitions of ‘fractal’ trivially imply some form of statistical self-similarity. (Fractal dimension itself is a numerical measure which is preserved across scales.) Random fractals are examples of fractals which are statistically self-similar. The coastline of Britain is another example; one cannot expect to find microscopic Britains (even distorted ones) by looking at a small section of the coast with a magnifying glass.
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Possessing self-similarity is not the sole criterion for an object to be termed a fractal. 2 _____________. These do not qualify, since they have the same Hausdorff dimension as topological dimension. Roots of the idea of fractals go back to the 17th century, while mathematically rigorous treatment of fractals can be traced back to functions studied by Karl Weierstrass, Georg Cantor and Felix Hausdorff a century later in studying functions that were continuous but not differentiable 3_____________. A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion. There are several examples of fractals, which are defined as portraying exact self-similarity, quasi self-similarity, or statistical self-similarity. While fractals are a mathematical construct, they are easily found in nature. Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. 4 _____________. Even coastlines may be loosely considered fractal in nature. Trees and ferns are fractal in nature and can be modeled on a computer by using a recursive algorithm. The connection between fractals and leaves is currently being used to determine how much carbon is contained in trees. Fractal patterns have been found in the paintings of American artist Jackson Pollock. While Pollock’s paintings appear to be composed of chaotic dripping and splattering, computer analysis has found fractal patterns in his work. Decalcomania, a technique used by artists such as Max Ernst, can produce fractal-like patterns. 5 ____________. Cyberneticist Ron Eglash has suggested that fractal-like structures are prevalent in African art and architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles.
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