MATHEMATICSin Wonderland

Dr. Fumiko Futamura, SU Math Club talk, 2/23/12.

Wonderland’s influence on pop culture

Influences on Lewis Carroll?

Alice LiddellBoat trip down the ThamesOxford Natural History Museum

Old Sheep Shop

Firedogs with long necks

Great Hall door

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)

Alice's 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

Mathematical influences on Lewis Carroll?

I'll 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 couldn't 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?' `Yes,' said Alice doubtfully: `it means--to--make--anything-- prettier.' `Well, then,' the Gryphon went on, `if you don't know what to uglify is, you ARE a simpleton.'

The Mock Turtle’s Story

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 up through 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.

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”.

http://www.ma.utexas.edu/users/mks/326K/Negnos.html

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 from another number greater than itself, but to attempt to take it away from a number greater than itself is ridiculous. Yet this is attempted by algebraists, who talk of a number less than nothing, of multiplying a negative number into a negative number and thus producing a positive number, of a number being imaginary.

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.

`That's the reason they're called lessons,' the Gryphon remarked: `because they lessen from day to day.' This was quite a new idea to Alice, and she thought it over a little before she made her next remark. `Then the eleventh day must have been a holiday?' `Of course it was,' said the Mock Turtle. `And how did you manage on the twelfth?' Alice went on eagerly. `That's enough about lessons,' the Gryphon interrupted in a very decided tone: `tell her something about the games now.'

Negative numbers, less than nothing

`Can you do Subtraction? Take nine from eight.'`Nine from eight I can't, you know,' Alice replied very readily: `but -- '`She can't do Subtraction,' said the White Queen.Take a bone from a dog: what remains?‘ Alice considered. `The

bone wouldn't remain, of course, if I took it -- and the dog wouldn't remain; it would come to bite me -- and I'm sure I shouldn't remain!'`Then you think nothing would remain?' said the Red Queen. `I think that's the answer.'`Wrong, as usual,' said the Red Queen: `the dog's temper would remain.'`But I don't see how -- '`Why, look here!' the Red Queen cried. `The dog would lose its temper, wouldn't it?'`Perhaps it would,' Alice replied cautiously. `Then if the dog went away, its temper would remain!' the Queen exclaimed triumphantly.Alice said, as gravely as she could, `They might go different ways.' But she couldn't help thinking to herself, `What dreadful nonsense we are talking!'

Negative numbers, less than nothing

`Take some more tea,' the March Hare said to Alice, very earnestly. `I've had nothing yet,' Alice replied in an offended tone, `so I can't take more.' `You mean you can't take less,' said the Hatter: `it's very easy to take more than nothing.' `Nobody asked your opinion,' said Alice.

Symbolic Algebra1842: George Peacock published Treatise on Algebra, introducing theidea 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 sciences will by no means imply that they possess the same meaning in all their applications. In Arithmetic and Arithmetical Algebra, addition and subtraction are defined or understood in their ordinary sense, and the rules of operation are deduced from the definitions: in Symbolic Algebra, we adopt the rules of operation which are thence derived, extending their application to all values of the symbols…

Symbolic Algebra is not unreal or imaginary, but that it comprehends the representation of large classes of real existences…”3a – 5a = -2a

“This is exclusively a result of Symbolic Algebra.”

'There's glory for you!' 'I don't know what you mean by "glory",' Alice said. Humpty Dumpty smiled contemptuously. 'Of course you don't — till I tell you. I meant "there's a nice knock-down argument for you!"' 'But "glory" doesn't mean "a nice knock-down argument",' Alice objected. 'When I use a word,' Humpty Dumpty said, in rather a scornful tone, 'it means just what I choose it to mean — neither more nor less.' 'The question is,' said Alice, 'whether you can make words mean so many different things.' 'The question is,' said Humpty Dumpty, 'which is to be master — that's all.'

Symbolic Algebra and Humpty Dumpty

Symbolic Algebra and the Caterpillar

Symbolic Algebra and the Caterpillar

`You!' said the Caterpillar contemptuously. `Who are YOU?' Which brought them back again to the beginning of the conversation. Alice felt a little irritated at the Caterpillar's making such VERY short remarks, and she drew herself up and said, very gravely, `I think, you out to tell me who YOU are, first.' `Why?' said the Caterpillar. Here was another puzzling question; and as Alice could not think of any good reason, and as the Caterpillar seemed to be in a VERY unpleasant state of mind, she turned away.

Symbolic Algebra and the Caterpillar

Melanie Bayley, http://www.newscientist.com/article/mg20427391.600-alices-adventures-in-algebra-wonderland-solved.html

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 Alice's 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.”

http://www.newscientist.com/article/mg20427391.600-alices-adventures-in-algebra-wonderland-solved.html

Symbolic Algebra and the Caterpillar

`I HAVE tasted eggs, certainly,' said Alice, who was a very truthful child; `but little girls eat eggs quite as much as serpents do, you know.' `I don't believe it,' said the Pigeon; `but if they do, why then they're a kind of serpent, that's all I can say.' This was such a new idea to Alice, that she was quite silent for a minute or two, which gave the Pigeon the opportunity of adding, `You're looking for eggs, I know THAT well enough; and what does it matter to me whether you're a little girl or a serpent?' `It matters a good deal to ME,' said Alice hastily; `but I'm not looking for eggs, as it happens; and if I was, I shouldn't want YOURS: I don't like them raw.' `Well, be off, then!' said the Pigeon in a sulky tone, as it settled down again into its nest.

`But I'm NOT a serpent, I tell you!' said Alice. `I'm a--I'm a--' `Well! WHAT are you?' said the Pigeon. `I can see you're trying to invent something!' `I--I'm 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. `I've seen a good many little girls in my time, but never ONE with such a neck as that! No, no! You're a serpent; and there's no use denying it. I suppose you'll be telling me next that you never tasted an egg!'

Logic and NonsensePigeon’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 ???

Logic and Nonsense

It was all very well to say "drink me", "but I'll look first," said the wise little Alice, "and see whether the bottle's marked "poison" or not," for Alice had read several nice little stories about children that got burnt, and eaten up by wild beasts, and other unpleasant things, because they would not remember the simple rules their friends had given them, such as, that, if you get into the fire, it will burn you, and that, if you cut your finger very deeply with a knife, it generally bleeds, and she had never forgotten that, if you drink a bottle marked "poison", it is almost certain to disagree with you, sooner or later.

However, this bottle was not marked poison, so Alice tasted it, and finding 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,) she very soon finished it off.

Symbolic Logic and NonsenseLewis Carroll’s Syllogisms

Premises1. 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 crocodile

Conclusion??

Euclidean Geometry

`Come back!' the Caterpillar called after her. `I've something important to say!' This sounded promising, certainly: Alice turned and came back again. `Keep your temper,' said the Caterpillar.

`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 can't explain MYSELF, I'm afraid, sir' said Alice, `because I'm not myself, you see.' `I don't see,' said the Caterpillar. `I'm afraid I can't put it more clearly,' Alice replied very politely, `for I can't understand it myself to begin with; and being so many different sizes in a day is very confusing.' `It isn't,' said the Caterpillar.

Euclidean Geometry“The Caterpillar's warning, at the end of this scene, is perhaps one of the most telling clues to Dodgson's conservative mathematics. "Keep your temper," he announces. Alice presumes he's telling her not to get angry, but although he has been abrupt he has not been particularly irritable at this point, so it's a somewhat puzzling thing to announce.

To intellectuals at the time, though, the word "temper" also retained its original sense of "the proportion in which qualities are mingled", a meaning that lives on today in phrases such as "justice tempered with mercy". So the Caterpillar could well be telling Alice to keep her body in proportion - no matter what her size.”

http://www.newscientist.com/article/mg20427391.600-alices-adventures-in-algebra-wonderland-solved.html

Euclidean Geometry

“This may again reflect Dodgson's love of Euclidean geometry, where absolute magnitude doesn't matter: what's important is the ratio of one length to another when considering the properties of a triangle, for example. To survive in Wonderland, Alice must act like a Euclidean geometer, keeping her ratios constant, even if her size changes. Of course, she doesn't. She swallows a piece of mushroom and her neck grows like a serpent with predictably chaotic results.”http://www.newscientist.com/article/mg20427391.600-alices-adventures-in-algebra-wonderland-solved.html

Projective Geometry

“Keep your temper”

Projective Geometry

Alice caught the baby with some difficulty, as it was a queer- shaped little creature, and held out its arms and legs in all directions, `just like a star-fish,' thought Alice. …Alice was just beginning to think to herself, `Now, what am I to do with this creature when I get it home?' when it grunted again, so violently, that she looked down into its face in some alarm. This time there could be NO mistake about it: it was neither more nor less than a pig, and she felt that it would be quite absurd for her to carry it further.

Imaginary numbers

𝑖=√−1

3a – 5a = - 2a

i-day at SU is Feb 29!!!What is an imaginary number???? Does it exist???? Hmm, does the number 3 exist?????

3i – 5i = - 2i

Imaginary and complex numbers

The set of imaginary numbers forms a group under addition:1. + b = 2. 0 = 0is such that 3.

The set of complex numbers forms a group under addition, and under multiplication (also satisfies other conditions that make it an field):

1. ()+ () = 2. 0 = 0+0is such that (3. 4. )(c)=()+() 5. 1 = 1+0 is such that (6. = = 1

Commutativity of complex numbers

Imaginary 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˚

Commutativity of Rotations

90˚+180˚ = 180˚+90˚2D Rotations are commuative.

What about 3D rotations?

Commutativity of Rotations

NOT COMMUTATIVE!

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 Canal with his wife to preside at a Council meeting of the Royal Irish Academy. Although his wife talked to him now and again Hamilton hardly heard, for the discovery of the quaternions, the first noncommutative algebra to be studied, was taking shape in his mind:- And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples ... An electric circuit seemed to close, and a spark flashed forth. He could not resist the impulse to carve the formulae for the quaternions

i2 = j2 = k2 = i j k = -1. in the stone of Broome Bridge (or Brougham Bridge as he called it) as he and his wife passed it.”

http://www-groups.dcs.st-and.ac.uk/history/Biographies/Hamilton.html

Quaternions and rotations

where represents the axis of rotation, i, j or k. (It's more complicated than this generally, but for the purposes of understanding how it works, this is enough.)

Suppose we have a vector .The way to use algebraic calculations for a 90˚ rotation about an axis is

For example: Suppose we have a vector pointed straight up, 0 , and we want to rotate in the positive direction about the -axis, = .

Quaternions and Mad Tea Party

`Do you mean that you think you can find out the answer to it?' said the March Hare. `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--that's the same thing, you know.' `Not the same thing a bit!' said the Hatter. `You might just as well say that "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 in his sleep, `that "I breathe when I sleep" is the same thing as "I sleep when I breathe"!'

Quaternions and Mad Tea Party

Alice sighed wearily. `I think you might do something better with the time,' she said, `than waste it in asking riddles that have no answers.' `If you knew Time as well as I do,' said the Hatter, `you wouldn't talk about wasting it. It's him.' `I don't know what you mean,' said Alice. `Of course you don't!' the Hatter said, tossing his head contemptuously. `I dare say you never even spoke to Time!' `Perhaps not,' Alice cautiously replied: `but I know I have to beat time when I learn music.' `Ah! that accounts for it,' said the Hatter. `He won't stand beating. Now, if you only kept on good terms with him, he'd do almost anything you liked with the clock. For instance, suppose it were nine o'clock in the morning, just time to begin lessons: you'd only have to whisper a hint to Time, and round goes the clock in a twinkling! Half-past one, time for dinner!'

Quaternions and Mad Tea Party

`And ever since that,' the Hatter went on in a mournful tone, `he won't do a thing I ask! It's always six o'clock now.' A bright idea came into Alice's head. `Is that the reason so many tea-things are put out here?' she asked. `Yes, that's it,' said the Hatter with a sigh: `it's always tea-time, and we've no 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.'

Mathematics in Wonderland? Yes?

3a – 5a = - 2a