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Monthly Maths Innovators in mathematics education www.mei.org.uk I s s u e 9 O c t o b e r 2 0 1 1 The harmony of numbers Maths Club Strategy Games Sprouts Two players start with a few spots drawn on a sheet of paper. Players take turns to draw a line between two spots (or from a spot to itself) and adding a new spot somewhere along the line. The idea is to make it impossible for the other player to draw a line. The line may be straight or curved, but must not touch or cross itself or any other line. The new spot cannot be placed on top of one of the endpoints of the new line. Thus the new spot splits the line into two shorter lines. No spot may have more than three lines leading to or from it. The last person to draw a line is the winner. Maths Careers website has some good background information on the connection between maths and music, including the tale of how Pythagoras came to make the connection. „The story goes that one day, probably deep in thought about the nature of triangles, the Greek philosopher was walking past a blacksmith‟s when he heard something strange. As the blacksmith‟s hammers struck the anvil, some of the hammers would produce harmonious sounds while others sounded discordant. He decided to investigate and found that there was a mathematical relationship between the weight of the hammers and the sounds they produced. ‟ Read more here and view a Science Channel video clip here The magical mathematics of music In an article in Plus magazine, Jeffrey Rosenthal writes: „The astronomer Galileo Galilei observed in 1623 that the entire universe "is written in the language of mathematics", and indeed it is remarkable the extent to which science and society are governed by mathematical ideas. It is perhaps even more surprising that music, with all its passion and emotion, is also based upon mathematical relationships. Such musical notions as octaves, chords, scales, and keys can all be demystified and understood logically using simple mathematics.‟ Read more here and view a short video here Mathematics and Music In the BBC radio programme ‘In our Time‟, Melvyn Bragg and guests including Marcus du Sautoy discuss the mathematical structures that lie within the heart of music. The seventeenth century philosopher Gottfried Leibniz wrote: 'Music is the pleasure the human mind experiences from counting without being aware that it is counting'. Read more and listen to the 45 minute discussion. The Code In his recent BBC documentary The Code: Shapes (part 2), Marcus du Sautoy tells us that „numbers determine how we hear sound‟. He uses an oscilloscope to look at frequencies of notes played an octave apart on a piano and demonstrates that „every combination of notes used in music is defined by simple ratios‟. When complex ratios are used, there is no common pattern, resulting in dissonant, harsh sounds. View a clip here on Differential Dynamics John Whitney first described the idea of spinning shapes producing notes, using the principle of "differential dynamics" in his book “Digital Harmony”, The first digital Whitney Music Box was created by Jim Bumgardner, and is hosted on his website here Inspired by Whitney‟s pre-computer and computer films, Lawrence Ball and Michael Tusch developed a form of mathematics devised for the production of musical, visual and audio/visual art forms, called Harmonic Mathematics. This was applied to graphic visuals, sound timbres and melodic loops that evolved in the 80s. Read more here Useful links Click here to view the Maths Item of the Month “There is geometry in the humming of the strings, there is music in the spacing of the spheres.” Pythagoras BC 580-500 Disclaimer: This newsletter provides links to other Internet sites for the convenience of users. MEI is not responsible for the availability or content of these external sites, nor does MEI endorse or guarantee the products, services, or information described or offered at these other Internet sites. Pythagoras Circa 580 - 500 BC Pythagoras is known as the first pure mathematician and is a very important figure in the development of mathematics. Relatively little is known about his mathematical achievements as he left no writings. Pythagoras was born on the Greek island of Samos around 580 BC. He studied in Egypt and Lebanon. Many of the practices of the society he created later in Italy can be traced to the beliefs of the Egyptian priests who he studied with, such as the codes of secrecy, striving for purity, and refusal to eat beans or to wear animal skins as clothing. He was captured when he tried to return to Greece and continued his studies with Magoi priests in Babylon. He later returned briefly to Samos before settling in Crotona, Southern Italy, where he founded a school of philosophy, mathematics and natural science. The Pythagoreans In the philosophical and religious school where his many followers lived and worked, the main focus of Pythagorean thought was ethics, developed primarily within philosophy, mathematics, music, and gymnastics. In their view everything was composed of number, and the explanation for an object‟s existence could only be found in number. The Pythagoreans followed a code of secrecy, so everything was passed on by word of mouth and written accounts were forbidden. The society studied properties of numbers that are familiar to modern mathematicians, such as even and odd numbers, prime and square numbers. The society also believed in such numerical properties as masculine or feminine, perfect or incomplete, and beautiful or ugly. These opposites, they believed, were found everywhere in nature, and the combination of them brought about the harmony of the world. To the Pythagoreans, each number possessed its own special attributes, e.g. Number Property of the number 1 monad (unity) generator of numbers, the number of reason 2 dyad (diversity, opinion) first true female number 3 triad (harmony = unity + diversity) first true male number 4 (justice, retribution) squaring of accounts 5 (marriage) = first female + first male 6 (creation) = first female + first male + 1 ? 10 (Universe) tetractys The Pythagoreans lived by rules of behaviour, including when they spoke, what they wore and what they ate. Pythagoras was the Master of the society, and the followers, both men and women, who also lived there, were known as mathematikoi. They had no personal possessions and were vegetarians. Although the Pythagorean community was ultraconservative, they accepted women as equals on all terms. When Pythagoras died, his wife Teano (who had been a teacher at the school) ran the school . It is difficult to be certain whether all the theorems attributed to Pythagoras were originally his, or whether they came from the communal society of the Pythagoreans. However, the Pythagoreans gave credit to the Master for the following: The sum of the angles of a triangle is equal to two right angles. The theorem of Pythagoras - for a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. The Babylonians understood this 1000 years earlier, but Pythagoras proved it. Constructing figures of a given area and geometrical algebra. For example they solved various equations by geometrical means. The discovery of irrational numbers is attributed to the Pythagoreans, but seems unlikely to have been the idea of Pythagoras because it does not align with his philosophy that all things are numbers, since number to him meant the ratio of two whole numbers. The five regular solids (tetrahedron, cube, octahedron, icosahedron, dodecahedron). It is believed that Pythagoras knew how to construct the first three but not last two. Pythagoras taught that Earth was a sphere in the centre of the universe, that the planets, stars, and the universe were spherical because the sphere was the most perfect solid figure. He also taught that the paths of the planets were circular. Pythagoras recognized that the morning star was the same as the evening star, Venus. Year 6 Mathematics Resources PowerPoint presentation with notes and worksheet are available to download on the Monthly Maths home page Six dominoes have been placed in this grid and there is only one way of filing the grid with more dominoes. Try it: We can do better than 6 dominoes. The way in which these five dominoes have been placed means that there’s only one way of completing the grid Can we do better still? These four dominoes lead to a unique tiling. Better still! These three dominoes lead to a unique tiling. What is the best you can do with other grids? Number o rows Number of columns 1 2 3 4 5 6 7 8 2 0 1 1 1 2 2 2 3 4 0 1 6 0 2 8 0 3 3 Valhalla Val and Hal take it in turns to place a domino on the grid. Val places dominoes vertically and Hal places dominoes horizontally. You can choose to be either Val or Hal. Your opponent then chooses whether to go first or second. The first player unable to place a domino loses. Can you find a winning strategy?