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Math in wonderland


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Math in wonderland

  1. 1. in WonderlandDr. Fumiko Futamura, SU Math Club talk, 2/23/12.
  2. 2. Wonderland’s influence on pop culture
  3. 3. Oxford Natural History MuseumBoat trip down the Thames Alice Liddell Influences on Lewis Carroll? Old Sheep Shop Great Hall door Firedogs with long necks
  4. 4. Lewis Carroll: a.k.a. Charles Ludwidge Dodgson, Mathematician• Obtained first-class honors in Mathematics at Oxford• Taught mathematics at Christ Church, Oxford for 26 years• Wrote a number of mathematics books, including • A Syllabus of Plane Algebraic Geometry (1860) • The Fifth Book of Euclid Treated Algebraically (1858 and 1868) Alices Adventures in Wonderland (1865) •An Elementary Treatise on Determinants, With Their Application to Simultaneous Linear Equations and Algebraic Equations (1867) Through the Looking Glass (1872) • Euclid and his Modern Rivals (1879) • Symbolic Logic Part I (1896) • Symbolic Logic Part II (published posthumously) • The Game of Logic
  5. 5. Mathematical influences on Lewis Carroll? Ill try if I know all the things I used to know. Let me see: four times five is twelve, and four times six is thirteen, and four times seven is--oh dear! I shall never get to twenty at that rate!The Pool of Tears `I couldnt afford to learn it. said the Mock Turtle with a sigh. `I only took the regular course. `What was that? inquired Alice. `Reeling and Writhing, of course, to begin with, the Mock Turtle replied; `and then the different branches of Arithmetic-- Ambition, Distraction, Uglification, and Derision. `I never heard of "Uglification," Alice ventured to say. `What is it? The Gryphon lifted up both its paws in surprise. `What! Never heard of uglifying! it exclaimed. `You know what to beautify is, I suppose?The Mock Turtle’s `Yes, said Alice doubtfully: `it means--to--make--anything-- prettier. `Well, then, the Gryphon went on, `if you dont know what to uglify Story is, you ARE a simpleton.
  6. 6. Negative numbers, less than nothing Negative numbers "... darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple". Francis Maseres, British Mathematician, 1758Negative numbers were controversial in Europe all the way upthrough the Victorian era.• Although negative numbers have been used since 200 BC in China, 600 AD in India, and 800 AD in the Middle East, negative numbers did appear in Europe until the 1400s.• The Greeks dealt mostly with Geometry, which used positive lengths, areas and volumes.• Arithmetic and later, algebra came from the Arabs, Al – Khwarizmi in particular, first brought over to Europe by Leonardo of Pisa, aka Fibonacci in the 1200s. His book, Liber Abaci, didn‘t contain any negative numbers, despite dealing with money.
  7. 7. Negative numbers, less than nothing Mathematician Gerolamo Cardano (1501-1576) called positive numbers ―numeri ueri‖ (real) negative numbers ―numeri ficti‖ (fictitious) Michael Stifel (1486-1567) referred to negative numbers as ―absurd‖ and ―fictitious below zero‖.Blaise Pascal (1623-1662) regarded the subtraction of 4 from 0 as utter nonsense.The principles of algebra: By William Frend and Francis Maseres (1796)You may make a mark before one, which it will obey: it submits to be taken away fromanother number greater than itself, but to attempt to take it away from a number greaterthan itself is ridiculous. Yet this is attempted by algebraists, who talk of a number lessthan nothing, of multiplying a negative number into a negative number and thusproducing a positive number, of a number being imaginary.
  8. 8. Negative numbers, less than nothing `And how many hours a day did you do lessons? said Alice, in a hurry to change the subject. `Ten hours the first day, said the Mock Turtle: `nine the next, and so on. `What a curious plan! exclaimed Alice.`Thats the reason theyre called lessons, the Gryphon remarked:`because they lessen from day to day. This was quite a new idea toAlice, and she thought it over a little before she made her nextremark. `Then the eleventh day must have been a holiday?`Of course it was, said the Mock Turtle. `And how did youmanage on the twelfth? Alice went on eagerly. `Thats enoughabout lessons, the Gryphon interrupted in a very decided tone:`tell her something about the games now.
  9. 9. Negative numbers, less than nothing `Can you do Subtraction? Take nine from eight. `Nine from eight I cant, you know, Alice replied very readily: `but -- `She cant do Subtraction, said the White Queen.Take a bone from a dog: what remains?‗ Alice considered. `The bone wouldntremain, of course, if I took it -- and the dog wouldnt remain; it would come to bite me -- and Im sure I shouldnt remain!`Then you think nothing would remain? said the Red Queen. `I think thats the answer.`Wrong, as usual, said the Red Queen: `the dogs temper would remain.`But I dont see how -- `Why, look here! the Red Queen cried. `The dog would lose its temper, wouldnt it?`Perhaps it would, Alice replied cautiously. `Then if the dog went away, its temperwould remain! the Queen exclaimed triumphantly.Alice said, as gravely as she could, `They might go different ways. But she couldnt helpthinking to herself, `What dreadful nonsense we are talking!
  10. 10. Negative numbers, less than nothing`Take some more tea, the March Hare said toAlice, very earnestly.`Ive had nothing yet, Alice replied in an offendedtone, `so I cant take more.`You mean you cant take less, said the Hatter: `its veryeasy to take more than nothing.`Nobody asked your opinion, said Alice.
  11. 11. Symbolic Algebra 1842: George Peacock published Treatise on Algebra, introducing the idea of symbolic algebra. 1849: Augustus DeMorgan published Trigonometry and Double Algebra. 1854: George Boole published An Investigation of the Laws of Thought.―The use however, of the same terms (addition and subtraction) in these two scienceswill by no means imply that they possess the same meaning in all their applications. InArithmetic and Arithmetical Algebra, addition and subtraction are defined or understoodin their ordinary sense, and the rules of operation are deduced from the definitions: inSymbolic Algebra, we adopt the rules of operation which are thencederived, extending their application to all values of the symbols…Symbolic Algebra is not unreal or imaginary, but that it comprehends the representationof large classes of real existences…‖ 3a – 5a = -2a ―This is exclusively a result of Symbolic Algebra.‖
  12. 12. Symbolic Algebra and Humpty DumptyTheres glory for you!I dont know what you mean by "glory", Alice said.Humpty Dumpty smiled contemptuously. Of course you dont — till I tell you.I meant "theres a nice knock-down argument for you!"But "glory" doesnt mean "a nice knock-down argument", Alice objected.When I use a word, Humpty Dumpty said, in rather a scornful tone, itmeans just what I choose it to mean — neither more nor less.The question is, said Alice, whether you can make words mean so manydifferent things.The question is, said Humpty Dumpty, which is to be master — thats all.
  13. 13. Symbolic Algebra and the Caterpillar
  14. 14. Symbolic Algebra and the Caterpillar`You! said the Caterpillar contemptuously. `Who are YOU?Which brought them back again to the beginning of the conversation. Alicefelt a little irritated at the Caterpillars making such VERY shortremarks, and she drew herself up and said, very gravely, `I think, you out totell me who YOU are, first.`Why? said the Caterpillar.Here was another puzzling question; and as Alice could not think of anygood reason, and as the Caterpillar seemed to be in a VERY unpleasant stateof mind, she turned away.
  15. 15. Symbolic Algebra and the CaterpillarMelanie Bayley,
  16. 16. Symbolic Algebra and the Caterpillar ―The first clue may be in the pipe itself: the word "hookah" is, after all, of Arabic origin, like "algebra", and it is perhaps striking that Augustus De Morgan, the first British mathematician to lay out a consistent set of rules for symbolic algebra, uses the original Arabic translation in Trigonometry and Double Algebra, which was published in 1849. He calls it "al jebr e al mokabala" or "restoration and reduction" - which almost exactly describes Alices experience. Restoration was what brought Alice to the mushroom: she was looking for something to eat or drink to "grow to my right size again", and reduction was what actually happened when she ate some: she shrank so rapidly that her chin hit her foot.‖
  17. 17. Symbolic Algebra and the Caterpillar `But Im NOT a serpent, I tell you! said Alice. `Im a--Im a-- `Well! WHAT are you? said the Pigeon. `I can see youre trying to invent something! `I--Im a little girl, said Alice, rather doubtfully, as she remembered the number of changes she had gone through that day. `A likely story indeed! said the Pigeon in a tone of the deepest contempt. `Ive seen a good many little girls in my time, but never ONE with such a neck as that! No, no! Youre a serpent; and theres no use denying it. I suppose youll be telling me next that you never tasted an egg!`I HAVE tasted eggs, certainly, said Alice, who was a very truthful child; `but little girlseat eggs quite as much as serpents do, you know.`I dont believe it, said the Pigeon; `but if they do, why then theyre a kind ofserpent, thats all I can say.This was such a new idea to Alice, that she was quite silent for a minute or two, which gavethe Pigeon the opportunity of adding, `Youre looking for eggs, I know THAT wellenough; and what does it matter to me whether youre a little girl or a serpent?`It matters a good deal to ME, said Alice hastily; `but Im not looking for eggs, as ithappens; and if I was, I shouldnt want YOURS: I dont like them raw.`Well, be off, then! said the Pigeon in a sulky tone, as it settled down again into its nest.
  18. 18. Logic and Nonsense Pigeon‘s assertion: Alice is a serpent, not a girl. Stated facts: 1. Girls don‘t have long necks. 2. Alice has a long neck. 3. Serpents have long necks.Girls don‘t have long necks. Alice has a long neck. Therefore, Alice is not a girl.Serpents have long necks. Alice has a long neck. Therefore, Alice is a serpent. Q.E.D.?Let A = Girl, B = long neck, C = Alice, D = serpent.A → not B. C → B. Since B → not A, we can conclude that C → B → not A.D → B. C → B. Since B → D, we can conclude that C → B → D ???
  19. 19. Logic and NonsenseIt was all very well to say "drink me", "but Ill look first," said the wiselittle Alice, "and see whether the bottles marked "poison" or not," forAlice had read several nice little stories about children that got burnt, andeaten up by wild beasts, and other unpleasant things, because they wouldnot remember the simple rules their friends had given them, suchas, that, if you get into the fire, it will burn you, and that, if you cut yourfinger very deeply with a knife, it generally bleeds, and she had neverforgotten that, if you drink a bottle marked "poison", it is almostcertain to disagree with you, sooner or later.However, this bottle was not marked poison, so Alice tasted it, andfinding it very nice, (it had, in fact, a sort of mixed flavour of cherry-tart, custard, pine-apple, roast turkey, toffy, and hot buttered toast,) shevery soon finished it off.
  20. 20. Symbolic Logic and Nonsense Lewis Carroll’s Syllogisms Premises 1. All babies are illogical. 2. Nobody is despised who can manage a crocodile. 3. Illogical persons are despised. A = babies, B = illogical, C = despised, D = manage a crocodileConclusion??
  21. 21. Euclidean Geometry `Who are YOU? said the Caterpillar. This was not an encouraging opening for a conversation. Alice replied, rather shyly, `I--I hardly know, sir, just at present-- at least I know who I WAS when I got up this morning, but I think I must have been changed several times since then. `What do you mean by that? said the Caterpillar sternly. `Explain yourself!`I cant explain MYSELF, Im afraid, sir said Alice, `because Im not myself, you see.`I dont see, said the Caterpillar.`Im afraid I cant put it more clearly, Alice replied very politely, `for I cant understand itmyself to begin with; and being so many different sizes in a day is very confusing.`It isnt, said the Caterpillar.`Come back! the Caterpillar called after her. `Ive something important to say! Thissounded promising, certainly: Alice turned and came back again. `Keep your temper,said the Caterpillar.
  22. 22. Euclidean Geometry ―The Caterpillars warning, at the end of this scene, is perhaps one of the most telling clues to Dodgsons conservative mathematics. "Keep your temper," he announces. Alice presumes hes telling her not to get angry, but although he has been abrupt he has not been particularly irritable at this point, so its a somewhat puzzling thing to announce.To intellectuals at the time, though, the word "temper"also retained its original sense of "the proportion inwhich qualities are mingled", a meaning that lives ontoday in phrases such as "justice tempered with mercy". Sothe Caterpillar could well be telling Alice to keep her bodyin proportion - no matter what her size.‖
  23. 23. Euclidean Geometry―This may again reflect Dodgsons love of Euclidean geometry, whereabsolute magnitude doesnt matter: whats important is the ratio of onelength to another when considering the properties of a triangle, forexample. To survive in Wonderland, Alice must act like a Euclideangeometer, keeping her ratios constant, even if her size changes. Ofcourse, she doesnt. She swallows a piece of mushroom and her neckgrows like a serpent with predictably chaotic results.‖
  24. 24. Projective Geometry ―Keep your temper‖
  25. 25. Projective GeometryAlice caught the baby with some difficulty, as it was a queer- shaped littlecreature, and held out its arms and legs in all directions, `just like a star-fish, thoughtAlice.…Alice was just beginning to think to herself, `Now, what am I to do with this creaturewhen I get it home? when it grunted again, so violently, that she looked down into itsface in some alarm. This time there could be NO mistake about it: it was neithermore nor less than a pig, and she felt that it would be quite absurd for her to carry itfurther.
  26. 26. Imaginary numbers i-day at SU is Feb 29!!!What is an imaginary number???? Does it exist???? Hmm, does the number 3 exist????? 3a – 5a = - 2a 3i – 5i = - 2i
  27. 27. Imaginary and complex numbers
  28. 28. Commutativity of complex numbersImaginary numbers and complex numbers are COMMUTATIVE: A◦B=B◦A 2i+3i=3i+2i (2+5i)(3-7i)= (3-7i)(2+5i) 90˚+180˚ = 180˚+90˚
  29. 29. Commutativity of Rotations 90˚+180˚ = 180˚+90˚2D Rotations are commuative. What about 3D rotations?
  30. 30. Commutativity of RotationsNOT COMMUTATIVE!
  31. 31. Quaternions and rotations Sir William Rowan Hamilton (1805-1865) Struggled to find an algebraic way of describing rotations in 3D, but all known algebraic systems were commutative.―On 16 October 1843 (a Monday) Hamilton was walking in along the Royal Canalwith his wife to preside at a Council meeting of the Royal Irish Academy. Althoughhis wife talked to him now and again Hamilton hardly heard, for the discovery of thequaternions, the first noncommutative algebra to be studied, was taking shape in hismind:-And here there dawned on me the notion that we must admit, in some sense, a fourth dimension ofspace for the purpose of calculating with triples ... An electric circuit seemed to close, and a sparkflashed forth.He could not resist the impulse to carve the formulae for the quaternions i2 = j2 = k2 = i j k = the stone of Broome Bridge (or Brougham Bridge as he called it) as he and hiswife passed it.‖
  32. 32. Quaternions and rotations
  33. 33. Quaternions and Mad Tea Party`Do you mean that you think you can find out the answer to it? said the MarchHare. `Exactly so, said Alice.`Then you should say what you mean, the March Hare went on.`I do, Alice hastily replied; `at least--at least I mean what I say--thats the samething, you know.`Not the same thing a bit! said the Hatter. `You might just as well saythat "I see what I eat" is the same thing as "I eat what I see"!`You might just as well say, added the March Hare,`that "I like what I get" is the same thing as "I get what I like"!`You might just as well say, added the Dormouse, who seemed to be talking inhis sleep,`that "I breathe when I sleep" is the same thing as "I sleep when I breathe"!
  34. 34. Quaternions and Mad Tea PartyAlice sighed wearily. `I think you might do something better with the time, shesaid, `than waste it in asking riddles that have no answers.`If you knew Time as well as I do, said the Hatter, `you wouldnt talk about wasting it.Its him.`I dont know what you mean, said Alice.`Of course you dont! the Hatter said, tossing his head contemptuously. `I dare say younever even spoke to Time!`Perhaps not, Alice cautiously replied: `but I know I have to beat time when I learnmusic.`Ah! that accounts for it, said the Hatter. `He wont stand beating. Now, if you onlykept on good terms with him, hed do almost anything you liked with the clock. Forinstance, suppose it were nine oclock in the morning, just time to begin lessons: youdonly have to whisper a hint to Time, and round goes the clock in a twinkling!Half-past one, time for dinner!
  35. 35. Quaternions and Mad Tea Party`And ever since that, the Hatter went on in a mournful tone, `he wontdo a thing I ask! Its always six oclock now.A bright idea came into Alices head. `Is that the reason so many tea-things are put out here? she asked.`Yes, thats it, said the Hatter with a sigh: `its always tea-time, and weveno time to wash the things between whiles.`Then you keep moving round, I suppose? said Alice.`Exactly so, said the Hatter: `as the things get used up.
  36. 36. Mathematics in Wonderland? Yes? 3a – 5a = - 2a