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Major Facilities for Mathematical Thinking and Understanding. (1) Language. We would like to expound some examples on how language can help us in mathematical thinking. Our discussion will eventually lead to organized descriptions of some plants and trees. Gao Xingjian Nobel Prize in Literature for 2000 “Language is the ultimate crystallization of human civilization. It is intricate, incisive and difficult to grasp ....” “ It is a writer’s insights in grasping truth that determine the quality of a work…” Languages are wonderful creations of human being. Our linguistic facility is an important tool for thinking. For examples: ‘Ex’ equals minus ‘bee’ plus or minus the square root of ‘bee’ squared minus four ‘ay’ ‘see’ all over two ‘ay’ Aleph Bet/Ve Gim t el א ב ג Lame d Mem ל מ נ ם ן Dalet He ד ה Vav Zayi K n ו ז Nun Samek Ayi Pe/F Tsad K h n e i ס ע פ צ ף ץ b b 4ac x . 2a 2 ‘Ay’ square plus ‘bee’ square equals ‘see’ square. a b c 2 2 2 -Pythagoras’ theorem. Besides language, mathematicians also derive symbols. This may dramatically simplify the thinking process. For examples, A B x A f (x) 100 n n 1 : A is a subset of B : x is an element in A : a function of x 2 : summation : angle PQ : statements P and Q P Q : statements P or Q negation of the ~ P : statement P The use of language is also reflected in computer programming. After one month, the pair of rabbits is mature and mated. One more month later, a pair of rabbits are born. Let’s use the letters ‘b’ to denoted a pair of new born rabbits, and ‘a’ a mature pair. The situation can be represented by the rewriting rules a ab b a. Starting with ‘b’ we proceed b a ab aba abaab abaababa abaababaabaab abaababaabaababaababa : Let Cn be the number of letters in the n-th step. E.g. Co 0, C1 1, C2 1, C3 2, C4 3, C5 5, C6 8, C7 13, C8 21, ... We conjecture that Cn Cn 1 Cn 2 . (Think about a proof in the tutorial.) The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, ... (add the last two numbers to get the next number). There are 13 notes in the span of any note through its octave, with 8 white keys and 5 black keys.. The first few Fibonacci numbers appear in this plant. 1, 1, 2, 3, 5, 8, 13, 21, ... The sequence of ratios 1 2 3 5 8 13 1, , , , , , ,... 2 3 5 8 13 21 ‘converges’ to the golden ratio (number) 0·6180339887... Many books on oil painting and photo taking in your local library will point out that it is better to position objects not in the centre of the picture but to one side or "about one-third" of the way across. 1 x x 1 x 1 x x 1 x x 1 0 2 1 5 2 x x 0.618 . 2 3 Position the objects about 1/3 of the way across. The position of baby Christ is described by horizontal and vertical lines using the Golden ratio number. The exquisite Parthenon temple in Athens, which has height : breadth : ~ 0.618 (recall the golden section number 0·6180339887…). Pantheon in Rome (~120 AD). Classicism – clarity, symmetry and proportion. http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html Woodcut from the Divina Proportione by Luca Pacioli (1509) depicting the golden proportion as it applies to the human face. http://www.beautyanalysis.com (Extra slide for interest) http://zhurnal.lib.ru/m/muratow_s_w/violin_design.shtml (Extra slide for interest) Curtate cycloid A violin scroll usually approximates a Logarithmic Spiral. (Extra slide for interest) (Extra slide for interest) (Extra slide for interest) (Extra slide for interest) tan 31.6 ~ 0.615