DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARA Donald in Mathmagic Land MATHEMATICS 1º ESO IES SIERRA DE SANTA BÁRBARA 2012-13BACKGROUND INFORMATION Rectangles are found in classical architecture The Pythagoreans exactly equal line #3. This partition is a and art. Pythagoreans were students of the Golden Section. And lines #2 and #3 exactly equal line #4. This partition is also a Golden mathematical, philosophical, and religious Section. The ratio of the lengths of the two The Magic Spiral school started by Pythagoras (c. 580 B.C. – c. Golden Sections is (square root of 5 + 1)/2, A magic spiral isn’t magic at all. It is a spiral 500 B.C.). Some historians think that approximately 1.618. When this ratio is used that repeats the proportions of the Golden Pythagorean students were expected to to create the length and width of a rectangle, Sections of a Golden Rectangle into infinity. listen but not contribute during their first five the result is called a Golden Rectangle. Magic spirals can be seen in many of nature’s years at school, and that they were to credit spirals, such as the shell of the sea snail. any mathematical discovery to the school or to Pythagoras. The Golden rectangle A Golden Rectangle is a rectangle whose The Golden Section ratio of length to width is approximately 8 to The lines of a pentagram can be divided into 5, or 1.618. These proportions were greatly four different lengths. Lines #1 and #2 admired by the Greeks, and Golden
DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARA | 2 Porch of the Caryatids The Caryatids are six female statues supporting the south porch roof of the Erechtheum temple. The temple, located on the Acropolis in Athens, Greece, was built between 420 B.C. to 406 B.C. The Caryatids that adorn the temple today are copies, but four of the six originals are housed in the Acropolis Museum. Venus de Milo This famous ancient Greek sculpture depicts the goddess Venus. The sculpture was named for the Greek island of Melos where it was discovered in 1820 as it was about to be United Nations Secretariat crushed into mortar. The sculpture was Building restored, but its broken arms were lost and This 39-story building is one of several United never replaced. The sculpture is now housed Nations (U.N.) buildings located on the 18- in the Louvre Museum in Paris, France. acre U.N. complex in New York City. John D. Rockefeller Jr. donated the land and design began in 1947. The Secretariat building was completed in 1950. The U.N. Secretary General’s offices are located on the 38th floor. Mona Lisa This portrait of a Florentine woman was painted between the years 1503 and 1506 by Leonardo da Vinci (1452-1519). The painting was stolen from France’s Louvre Museum in 1911, but was found in a Florence hotel room two years later and returned to the Louvre. Cathedral of Notre Dame of Paris The Cathedral of Notre Dame in Paris, France, is regarded as the greatest masterpiece of Gothic architecture. It was constructed between 1163 and 1250 and was dedicated to Mary, the Mother of Jesus (“Notre Dame” means “Our Lady” in French). It was restored after the French Revolution ended in 1799.
DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARA | 3POSTVIEWING QUESTIONS 1. Pythagoras was a Greek mathematician. What are some of the mathematical contributions he made? a. Parthenon. b. Billiards. c. Music. 2. What is weird about the trees Donald sees in Mathmagic Land? a. The trees are black. b. Their roots are square. c. Their fruits are numbers. 3. Where is Donald going? a. Europe. b. Egypt. c. Greece. 4. Where is Maths found in nature?. Give three examples. a. b. c. 5. What does Donald mean when he says: “There’s a lot more to mathematics than two times two”? 6. We have seen in the video that there are many games that are developed in geometric spacesIn what games you can see maths? Give three examples. a. b. c.
DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARA | 4 7. Answer the following questions: a. What is the only game in which there are circles? i. Baseball ii. Basketball iii. Billiards b. How many squares form the chess board? i. 100 ii. 48 iii. 64 8. In the game of billiards mathematical calculations are essential. Say if the following statements are true or false. a. To calculate the position where I shoot you only need a sum. i. True ii. False b. The position of the diamond is marked with integers. i. True ii. False 9. List 10 geometry terms and shapes that you have seen in the film. 10. Write down three places where you can find the pentagon in nature. a. b. 11. Indicate whether the following statements are true or false: a. The voice that speaks with Donald is "the spirit of mathematics" b. Donald does not like mathematics.
DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARA | 5 12. Fill the gaps: Pythagoras is the father of and . He invented the musical scale using a . He tightened the rope and it in two equal parts 13. Look at the pictures and fill in the gaps: a. The objects represented in these images are obtained from the section of a b. In this case the objects are obtained from a c. Finally, these are obtained from a 14. Try to imagine your world without numbers. How would you telephone your friends? How would you change your TV channel? How would you know what size you are?
DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARA | 6 15. What instruments appear in the video? 20. What do you get when you spin a circle? a. Drums b. Recorder c. Piano 21. What do you get when you spin a triangle? 16. How do you get the pythagorean star from a pentagon? 22. What examples does the spirit give to explain a. By joining consecutive vertices. that the circle has been the basis of many human inventions? b. By joining every two vertices. c. The pythagorean star and a pentagon are not related. 17. Where can we find the Golden rectangle? 23. What’s the meaning of the closed doors of the film? a. The Parthenon (Athens) a. The past. b. The sculptures. b. The present. c. The cathedral of Burgos. c. The future. d. Some skyscrapers. e. Donald duck. 24. What is the key? 18. The strange creature which recites the digits of a. Imagination. the PI number is made of: b. Luck. a. A circle. c. Mathematics. b. A triangle. c. A square. 25. Who said “Mathematics is the alphabet with which God has written the universe” d. A rectangle. a. Pythagoras. b. Galileo Galilee. 19. In which scenes of the film do numbers appear? c. Donald. a. Animals 26. In the video it is said that Pithagoras claimed: “ b. Footprints Everything is ruled by numbers and c. Trees mathematical shapes”. Do you agree with Pithagoras? Why? d. The river
DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARA | 7APPENDIX A: THE GOLDEN RATIO THE GOLDEN RATIO IN ART AND ARCHITECTURE The appearance of this ratio in music, in patterns of human behaviour, even in the proportion of the human body, points to its universality as a principle of good structure and design. Parthenon (Athens) The Last Supper (Leonardo da Vinci)
DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARA | 8 Taj Mahal (Agra) Mona Lisa (Leonardo da Vinci) CN Tower (Toronto) Notre Dame (Paris) Status of Athena Eiffel Tower (Paris)
DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARA | 9 VISUAL POINTS OF INTEREST INSIDE A GOLDEN RECTANGLE Any square or rectangle (but especially those based on the golden ratio) contains areas inside it that appeal to us visually as well. Here’s how you find those points: 1. Draw a straight line from each bottom corner to its opposite top corner on either side. They will cross in the exact center of the format. 2. From the center to each corner, locate the midway point to each opposing corner. These points—represented by the green dots in the diagram above—are called the “eyes of the rectangle.” One strategy often used by artists is to locate focal points or areas of emphasis around and within these eyes, creating a strong visual path in their compositions. Let’s see some examples: Edward Hopper’s composition, below, sets the sailboat right on the lower right eye (with the tip of the sails extending nearly to the upper right eye).
DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARA | 10 J.M.W. Turner uses the angle of his waves to create an arch that circles through the lower right and lower left eyes. In this painting, Carolyn Anderson places her subject’s hands around that spot too.
DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARAAPPENDIX B: CONSTRUCT A GOLDENRECTANGLE
DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARA | 12APPENDIX C: GLOSSARY TERM ANDDEFINITION PENTAGRAM. A pentagram is a five- PI (π) Pi represents the ratio of the pointed star. Formed by five straight lines, circumference of a circle to its diameter a pentagram connects the vertices of a (approximately 3.14). pentagon and encloses another pentagon in the complete figure. INFINITY. Infinity is an immeasurably PENTAGON. A pentagon is a polygon large amount that increases indefinitely with five sides. and has no limits. CONE. A cone is a pyramid-like object ANGLE. An angle consists of two rays with a circle-shaped base. with a common end point. SPHERE. A sphere is the set of all GOLDEN SECTION. The Golden Section refers to a ratio between two points in space at a given distance from a dimensions of a plane figure, observed given point called the center. especially in art. RATIO. A ratio is the comparison between two numbers.
DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARA | 13BIBLIOGRAPHY “Phi and the Golden Section in Architecture” [online]. Phi 1.618 The Golden Number. May 11, 2012. http://www.goldennumber.net/architecture “LAB: The golden rectangle” [online]. Math Bits. http://mathbits.com/MathBits/MathMovies/LABGolden%20Rectangle.pdf MIZE, Dianne. “A guide to the Golden Ratio for Artists” [online]. http://emptyeasel.com/2009/01/20/a-guide-to-the-golden-ratio-aka-golden-section-or-golden-mean-for-artists/ FREITAG, Mark. "Phi: That Golden Number." 15 June, 2005. [online] http://jwilson.coe.uga.edu/emt669/Student.Folders/Frietag.Mark/Homepage/Goldenratio/goldenratio.html ELLIOTT, Ruth. “The golden rectangle (and the golden spiral)”, 2008. [online] http://www.edudesigns.org STEWART, Ian. The magical Maze: Seeing the World Through Mathematical Eyes. Wiley & Sons, Inc., 1997 PERDIGUERO, Eva María. “Donald en el país de las matemágicas” [online]. http://edu.jccm.es/proyectos/cuadernia-descartes/eXe/donald-mates/index.html
DONALD IN MATHMAGIC LAND IES SIERRA DE STA. BÁRBARA | 14 Teresa Martín Gómez J. César Bárcena Sánchez