Monday, November 28, 2005

Easy solution

I liked this solution to the eternal "Find x"-exercise a lot...

(I don't know the source - it just fell into my mailbox...)

Saturday, November 26, 2005

Diary of a Black Mathematician

I'm spending a little time seeing a little of what is out there in the mathematical blog sphere. One of the interesting ones are Diary of a Black Mathematician, with several interesting postings. (Among others, the comic strips added at the end of this posting.)

He mentions that he has kept a mathematical diary for years. That reminds me of the usefulness of keeping a diary for other purposes as well. For more than a year some years ago, I kept a diary of what happened at work. All good ideas of what to change in the way we taught and so on were recorded there. At the end of each month, I made a summary. And at the end of the academic year, I was able to write a detailed evaluation of the course - based on the diary. (After writing it, I let the students read it and react to it - they agreed that I had caught the main points.) Maybe I should try to get the time to take up this habit again...

(Okay, I know they turn out quite small. Right-click and choose "Show image" (or whatever is the way of doing it in your browser) to see a bigger version.)

Pascal's triangle (in Chinese)

I like to give my students the following illustration and ask them for the pattern:

The illustration "dates from a book of 1303 CE written by Chu Shï-kié. The earliest known use of the pattern was by Yang Hui, whose books date from 1261 & 1275 CE. Chu Shï-kié refers to the triangle as already being old. Jamshid Al-Kashi, who died around 1436 CE, was an astronomer at the court of Ulugh Beg in Samarkand in the 15th Century. Al-Kashi was the first known Arabic author to consider 'Pascal's' Triangle.", to quote the BBC.

They quickly come up with the pattern (every number is the sum of the two above), and thereby can also agree on the interpretation of the chinese numerals. Then they are ready to start exploring some of the wonderful "hidden" patterns in the triangle. These are some of the ones they may find - an additional exercise is of course to explain the pattern:

- adding the numbers in a row gives a power of 2.
- start from the top row, and move downwards - every step either down to the left or down to the right. In how many ways can you get to the number 20 (in the middle of the seventh row)? (20)
- coloring the even numbers gives us some triangles. Why? And what about coloring the numbers divisible by three? Or the ones that give a remainder of 1 when divided by 3?
- the rows with prime numbers in have a special property - which?
- when ordering a pizza, you get a choice of 8 toppings. In how many ways can you choose two toppings? Three? Four? Why are these numbers found in one and the same row of Pascal's triangle?
- choose any number in the triangle (though not on the edge). Color the six numbers circling your chosen number red-blue-red-blue-red-blue. Multiply the red numbers. Then multiply the blue numbers. What do you see? Why is this so?
- calculate and so on (if you have the patience). Do you see a connection with Pascal's triangle?
- n people meet and everybody wants to shake hands with everybody else. How many handshakes does that make? Where do you find those numbers in the triangle?
- choose a row. Let the number to the far right be ones, the next tens, the next hundreds and so on. Factorize the combined number. What do you see?
- draw a circle and place five dots on the periphery of the circle. How many lines can you make between the dots? How many triangles can you make with corners in the dots? How many quadrilaterals? And so on...

These small exercises show some connections in mathematics between geometry, algebra, combinatorics and numbers - and most of them could easily be used in primary or lower secondary school. Given in the right way, they may also work as a "landscape of investigation" (Skovsmose).

Please tell me if you know of more patterns in the triangle, by the way. (I know that the Fibonacci numbers are there, but I always find the "diagonals" where they occur too odd to point them out to my students...)

Friday, November 25, 2005

Foundations for fluxions

I just got the idea that I should use this blog to promote (shamelessly) my Master Thesis. The title of it is "Foundations for fluxions", and the topic is Newton's several attempts to put his theory of fluxions on a sound footing, and some English (/Scottish) attempts to do the same in the period after his death (including Colin MacLaurin and Roger Paman).

The thesis was written in LaTeX, but the web version is only in ascii - sorry about that...

How to catch a lion in the Sahara desert

Luckily, mathematics har good practical applications. Here are a few legendary ones - on how to catch a lion in the Sahara desert:

The method of inverse geometry: We place a spherical cage in the desert and enter it. We then perform an inverse operation with respect to the cage. The lion is then inside the cage and we are outside.

The set theoretic method: We observe that the desert is a separable space. It therefore contains an enumerable dense set of points from which can be extracted a sequence having the lion as the limit. We then approach the lion stealthily along this sequence bearing with us suitable equipment.

The Dirac method: We observe that wild lions are ipso facto not observable in the Sahara desert. Consequently if there are any lions in the Sahara, they are tame. The capture of a tame lion is left as an exercise for the reader.

The thermodynamic method: We construct a semi-permeable membrane which is permeable to everything except lions and sweep it across the Sahara.

The Kalra method: Make a list of the lion's whereabouts. Classify them into different fuzzy sets. The lion will get confused and fall into your trap.

Topological method: We observe that the lion has at least the connectivity of the torus. We transport the desert into four-space. It is then possible to carry out such a transformation that the lion can be returned to 3-space in a knotted condition. He is then helpless.

The Schrodinger method: At any given moment there is a positive probability that there is a lion in the cage. Sit down and wait.

The Heisenberg method: You will disturb the lion when you observe it before capturing. So keep your eyes closed.

The Einstein method: Run in the direction opposite to that of the lion. The relative velocity makes the lion run faster and hence he feels heavier and gets tired.

The Newtonian method: Let the lion catch you (let's assume you remain alive here). For every action there is an equal and opposite reaction. Therefore, you would have captured the lion.

The cartesian method: Take the origin as close as possible to the lion. Then perform rotation operation again and again. Initially, the lion will feel dizzy. Finally it will fall down.

The software method: Make a linked list of all objects in the desert. Then delete the pointers on either side of the lion. (Make sure you are not AFTER the lion.)

The automata method: Use a Non-Deterministic Finite Automaton with epsilon moves from all states to the final state, and no moves from the final state. The lion will soon enter the final state and be trapped.

The time-cop method: Use a time-machine and take the entire Sahara back a few years in time. The lion is just a cub now, and all you need is a mouse-trap.

The integro-differential method: Integrate the Sahara over its entire surface. The lion is now somewhere in the result. Differentiate the result w.r.t. the earth's rotation. The resulting value is zero, and the lion is no more.

The Shakespeare method: Hold the lion still for a moment (I don't care how you do it), and recite Shakespeare`s Hamlet to it. The lion will change from 'To be to Not-to-be'.

The Unix method: Put the lion on (the hard) disk. Do : ls -l | grep -i "lion" | cut $9 > trapfile. The “-i” option ensures that size of the lion does not matter. Now the lion is caught in a trapfile, and can be safely tar-ed or backed up into tape !

The Quantum Measurement Method: We assume that the sex of the lion is _ab initio_ indeterminate. The wave function for the lion is hence a superposition of the gender eigenstate for a lion and that for a lioness. We lay these eigenstates out flat on the ground and orthogonal to each other. Since the (male) lion has a distinctive mane, the measurement of sex can safely be made from a distance, using binoculars. The lion then collapses into one of the eigenstates, which is rolled up and placed inside the cage.

The nuclear physics method: Insert a tame lion into the cage and apply a Majorana exchange operator on it and a wild lion. As a variant let us assume that we would like to catch (for argument's sake) a male lion. We insert a tame female lion into the cage and apply the Heisenberg exchange operator, exchanging spins.

The Newton method (II): Cage and lion attract each other with the gravitation force. We neglect the friction. This way the lion will arrive sooner or later in the cage.

The Special relativistic method: One moves over the desert with light velocity. The relativistic length contraction makes the lion flat as paper. One takes it, rolls it up and puts a rubber band around the lion.

The general relativistic method: All over the desert we distribute lion bait containing large amounts of the companion star of Sirius. After enough of the bait has been eaten we send a beam of light through the desert. This will curl around the lion so it gets all confused and can be approached without danger.

The software method: Make a linked list of all objects in the desert. Then delete the pointers on either side of the lion. (Make sure you are not AFTER the lion.)

The Heisenberg method (II): Position and Velocity from a moving lion can not be measure at the same time. As moving lions have no physical meaningfull position in the desert, one can not catch them. The lion hunt can therefore be limited to resting lions. The catching of a resting, not moving lion is left as an exercise for the reader.

The atomic fission method: We irradiate the desert with slow neutrons. The lion becomes radioactive and starts to disintegrate. Once the disintegration process is progressed far enough the lion will be unable to resist.

The magneto-optical method: We plant a large, lense shaped field with cat mint (nepeta cataria) such that its axis is parallel to the direction of the horizontal component of the earth's magnetic field. We put the cage in one of the field's foci. Throughout the desert we distribute large amounts of magnetized spinach (spinacia oleracea) which has, as everybody knows, a high iron content. The spinach is eaten by vegetarian desert inhabitants which in turn are eaten by the lions. Afterwards the lions are oriented parallel to the earth's magnetic field and the resulting lion beam is focussed on the cage by the cat mint lense.

The Hilbert (axiomatic) method: We place a locked cage onto a given point in the desert. After that we introduce the following logical system: Axiom 1: The set of lions in the Sahara is not empty. Axiom 2: If there exists a lion in the Sahara, then there exists a lion in the cage. Procedure: If P is a theorem, and if the following is holds: "P implies Q", then Q is a theorem. Theorem 1: There exists a lion in the cage.

The projective geometry method: Without loss of generality, we can view the desert as a plane surface. We project the surface onto a line and afterwards the line onto an interiour point of the cage. Thereby the lion is mapped onto that same point.

The Bolzano-Weierstrass method: Divide the desert by a line running from north to south. The lion is then either in the eastern or in the western part. Let's assume it is in the eastern part. Divide this part by a line running from east to west. The lion is either in the northern or in the southern part. Let's assume it is in the northern part. We can continue this process arbitrarily and thereby constructing with each step an increasingly narrow fence around the selected area. The diameter of the chosen partitions converges to zero so that the lion is caged into a fence of arbitrarily small diameter.

The Peano method: In the usual way construct a curve containing every point in the desert. It has been proven [1] that such a curve can be traversed in arbitrarily short time. Now we traverse the curve, carrying a spear, in a time less than what it takes the lion to move a distance equal to its own length. ([1]: After Hilbert, cf. E. W. Hobson, "The Theory of Functions of a Real Variable and the Theory of Fourier's Series" (1927), vol. 1, pp 456-457)

The search method: We assume that the lion is most likely to be found in the direction to the north of the point where we are standing. Therefore the REAL problem we have is that of speed, since we are only using a PC to solve the problem.

The parallel search method: By using parallelism we will be able to search in the direction to the north much faster than earlier.

The Monte-Carlo method: We pick a random number indexing the space we search. By excluding neighboring points in the search, we can drastically reduce the number of points we need to consider. The lion will according to probability appear sooner or later.

The practical approach: We see a rabbit very close to us. Since it is already dead, it is particularly easy to catch. We therefore catch it and call it a lion.

The common language approach: If only everyone used ADA/Common Lisp/Prolog, this problem would be trivial to solve.

The standard approach: We know what a Lion is from ISO 4711/X.123. Since CCITT have specified a Lion to be a particular option of a cat we will have to wait for a harmonized standard to appear. $20,000,000 have been funded for initial investigations into this standard development.

Linear search: Stand in the top left hand corner of the Sahara Desert. Take one step east. Repeat until you have found the lion, or you reach the right hand edge. If you reach the right hand edge, take one step southwards, and proceed towards the left hand edge. When you finally reach the lion, put it the cage. If the lion should happen to eat you before you manage to get it in the cage, press the reset button, and try again.

The Dijkstra approach: The way the problem reached me was: catch a wild lion in the Sahara Desert. Another way of stating the problem is: Axiom 1: Sahara elem deserts. Axiom 2: Lion elem Sahara. Axiom 3: NOT(Lion elem cage).We observe the following invariant: P1: C(L) v Not(C(L)) where C(L) means: the value of "L" is in the cage. Establishing C initially is trivially accomplished with the statement ;cage := {}Note 0: This is easily implemented by opening the door to the cage and shaking out any lions that happen to be there initially. The obvious program structure is then: ;cage:={} ;do NOT (C(L)) -> ;"approach lion under invariance of P1" ;if P(L) -> ;"insert lion in cage" [ ] not P(L) ->;skip ;fi ;od where P(L) means: the value of L is within arm's reach. Note 1: Axiom 2 ensures that the loop terminates. Exercise 0: Refine the step "Approach lion under invariance of P1". Note 2: The program is robust in the sense that it will lead to abortion if the value of L is "lioness". Remark 0: This may be a new sense of the word "robust" for you. Note 3: From observation we can see that the above program leads to the desired goal. It goes without saying that we therefore do not have to run it.

Mathematicians hunt lions by going to Africa, throwing out everything that is not a lion, and catching one of whatever is left. Professors of mathematics prove the existence of at least one lion and leave the capture of an actual lion as an exercise for one of their graduate students.

Computer scientists hunt lions using algorithm A: 1. Go to Africa, 2. Start at the Cape of Good Hope, 3. Work northward in an orderly manner, traversing the continent alternately East and West. 4. During each traverse a. Catch each animal seen, b. Compare each animal caught to a known lion, c. Stop when a match is detected. Experienced computer programmers modify Algorithm A by placing a known lion in Cairo to ensure that the algorithm
will terminate.

Engineers hunt lions by going to Africa, catching yellow animals at random, and stopping when any one of them weighs within plus or minus 15 percent of any previously observed lion.

Economists don't hunt lions, but they believe that if lions are paid enough they will hunt themselves.

Statisticians hunt the first animal they see N times and call it an lion.

Consultants don't hunt lions, but they can be hired by the hour to advise those who do. Operations research consultants can measure the correlation of hat size and bullet color to the efficiency of lion hunting strategies, if someone else will identify the lions.

Politicians don't hunt lions, but they will share the lions you catch with the people who voted for them.

Lawyers don't hunt lions, but they do follow the herds around arguing about who owns the droppings. Software lawyers will claim that they own an entire herd based on the look and feel of one dropping.

When the Vice President of R&D tries to hunt lions, his staff will try to ensure that all lions are completely prehunted before he sees them. If the VP sees a nonprehunted lion, the staff will (1) Compliment the vice president's keen eyesight and (2) enlarge itself to prevent any recurrence.

Senior managers set broad lion hunting policy based on the assumption that lions are just like field mice, but with deeper voices.

Quality assurance inspectors ignore the lions and look for mistakes the other hunters made when they were packing the jeep.

Salespeople don't hunt lions but spend their time selling lions they haven't caught, for delivery two days before the season opens. Software salespeople ship the first thing they catch and write up an invoice for an lion. Hardware salespeople catch rabbits, paint them yellow and sell them as "desktop lions."

Good method: Let Q be the operator that encloses a word in quotation marks. Its square Q2 encloses a word in double quotes. The operator clearly satisfies the law of indices, Qm*Qn=Qm+n. Write down the word 'lion', without quotation marks. Apply to it the operator Q-1. Then a lion will appear on the page. It is advisable to enclose the page in a cage before applying the operator.

Roselius method: Let L denote the category whose objects are lions, with 'ancestor' as the only nontrivial morphism. Let l be the category of caged lions. The subcategory l is clearly complete, is nonempty (by inspection), and has both generator and cogenerator. Let F:l->L be the forgetful functor, which forgets the cage. By the Adjoint Functor Theorem, the functor F has a coadjoint C:L->l, which reflects each lion into a cage. We remark that this method is obviously superior to the Good method, which only guarantees the capture of one lion, and which requires an application of the Weierkäfig Preparation Theorem.

Surgical method: A lion may be regarded as an orientable three-manifold with a nonempty boundary. It is known that by means of a sequence of surgical operations (known as 'spherical modifications' in medical parlance) the lion can be rendered contractible. He may then be signed to a contract with Barnum and Bailey.

Logical method: A lion is a continuum. According to Cohen's theorem he is undecidable (especially when he must make choices). Let two men approach him simultaneously. The lion, unable to decide upon which man to attack, is then easily captured.

Functorial method: A lion is not dangerous unless he is somewhat gory. Thus the lion is a category. If he is a small category then he is a kittygory and certainly not to be feared. Thus we may assume, without loss of generality, that he is a proper class. But then he is not a member of the universe and is certainly not of any concern to us.

Method of differential topology: The lion is a three-manifold embedded in Euclidean 3-space. This implies that he is a handlebody. However, a lion which can be handled is tame and will enter the cage upon request.

Sheaf theoretic method: The lion is a cross-section of the sheaf of germs of lions on the Sahara Desert. Merely alter the topology of the Sahara, making it discrete. The stalks of the sheaf will then fall apart releasing the germs which attack the lion and kill it.

Method of transformation groups: Regard the lion as a surface. Represent each point of the lion as a coset of the group of homeomorphisms of the lion modulo the isotropy group of the nose (considered as a point). This represents the lion as a homogeneous space. That is, this
representation homogenizes the lion. A homogenized lion is in no shape to put up a fight.

Postlikov method: A male lion is quite hairy and may be regarded as being made up of fibres. Thus we may regard the lion as a fibre space. We may then construct a Postlikov decomposition of the lion. This being done, the lion, being decomposed, is dead and in bad need of burial.

Steenrod algebra method: Consider the mod p cohomology ring of the lion. We may regard this as a module over the mod p Steenrod algebra. Doing this requires the use of the table of Steenrod cohomology operations. Every element must be killed by some of these operations. Thus the lion will die on the operating table.

Homotopy method: The lion has the homotopy type of a one-dimensional complex and hence he is a K(Pi, 1) space. If Pi is noncommutative then the lion is not a member of the international commutist conspiracy and hence he must be friendly. If Pi is commutative then the lion has the homotopy type of the space of loops on a K(Pi, 2) space. We hire a stunt pilot to loop the loops, thereby hopelessly entangling the lion and rendering him helpless.

Covering space method: Cover the lion by his simply connected covering space. In effect this decks the lion. Grab him while he is down.

Game theoretic method: A lion is big game. Thus, a fortiori, he is a game. Therefore there exists an optimal strategy. Follow it.

Group theoretic method: If there are an even number of lions in the Sahara Desert we add a tame lion. Thus we may assume that the group of Sahara lions is of odd order. This renders the situation capable of solution according to the work of Thompson and Feit.

Biological method: Obtain a number of planarians and subject them to repeated recorded statements saying: 'You are a planarian'. The worms should shortly learn this fact since they must have some suspicions to this effect to start with. Now feed the worms to the lion in question. The knowledge of the planarians is then transferred to the lion. The lion, now thinking that he is a planarian, will proceed to subdivide. This process, while natural for the planarian, is disastrous to the lion.

Moore-Smith method: Letting A = Sahara Desert, one can construct a net in A converging to any point in the closure of A. Now lions are unable to resist tuna fish, on account of the charge atoms found therein (see Galileo Galilei, Dialogues Concerning Tuna's Ionses). Place a tuna fish in a tavern, thus attracting a lion. As noted above, one can construct a net converging to any point in a bar; in this net enmesh the lion.

Method of analytical mechanics: Since the lion has nonzero mass it has moments of inertia. Grab it during one of them.

Mittag-Leffler method: The number of lions in the Sahara Desert is finite, so the collection of such lions has no cluster point. Use Mittag-Leffler's theorem to construct a meromorphic function with a pole at each lion. Being a tropical animal a lion will freeze if placed at a pole, and may then be easily taken.

Method of natural functions: The lion, having spent his life under the Sahara sun, will surely have a tan. Induce him to lie on his back; he can then, by virtue of his reciprocal tan, be cot.

Boundary value method: As Dr. Morphy has pointed out, Brouwer's theorem on the invariance of domain makes the location of the hunt irrelevant. The present method is designed for use in North America. Assemble the requisite equipment in Kentucky, and await inclement weather. Catching the lion then readily becomes a Storm-Louisville problem.

Method of moral philosophy: Construct a corral in the Sahara and wait until autumn. At that time the corral will contain a large number of lions, for it is well known that a pride cometh before the fall.

The Ziplockian method: Assuming that the lion is wandering through the desert, do 1 of three things: 1. Follow closely behind the lion, stepping on it's heels, chanting loud mantras in it's ears, feeding it Italian Soda, giving it the finger, and farthing, and this attack on the senses will inevitably cause the lion to have a breakdown resulting in its inability to avoid capture. 2. Do not acknowledge it's authority. Place it, in your mind, at an equal or lower level of importance to yourself and it will appear less dangerous to ones self. By removing all fear, this will enable you to capture it. 3. Shoot it in the head.

The psychiatric method: Prescribe a drug such as Prozac to it. Upon takin this drug, it will have no emotional stimulus on which to react, thus causing it to return for further counselling.

The inductive method: Initial Condition: If you center a large cage on any one grain of sand, and a lion is on or close to the grain of sand. then he will be trapped by the cage. By close we mean within epsilon grains of sand. Given a cage the size of 2 * (size of lion * epsilon) it works. First Hypothesis: Given the first grain of sand in the desert, if the lion is standing on it you will trap him. Proof: Given by the initial condition. Induction Hypothesis: Assume that a lion is on a grain of sand n, and is trappable. Now, for grain n+1 (assume all grains of sand are ordered, inorder) n+1 is close to n, hence n is close to n+1. If the lion is on grain n, and is trappable; then he is close to n+1, and by the above condition, trappable. Hence, no matter where the lions are if you drop a cage centered on a piece of sand you will catch a lion.

The divide and conquer method (by recursion): Divide the desert in half. Repeat the process until you have the lion, a grain of sand, or some other object that cannot be divided without blood shed. You have the lion. The order of this method = O(insane). (Where sanity is anything reasonable.)

The square method: Square Sahara. Somewhere you will get a square lion. If the lion was negative, it will now be positive, which makes catching it a whole lot easier. (The actual catch is left as an exercise for the reader). Put lion in cage. Draw square root. If you prefer to have a negative lion, change signs. Warning: Do not draw square root of lions not already squared and put in a cage. If the lion was not square, it will become irrational. And if the lion was negative, you will start seeing imaginary lions.

The Darwin method: Put amoeba in a glass of water in a cage. Sooner or later, the amoeba will evolve into a lion.

The Bill Gates method: Convince Bill Gates that lions are important. Then he will put a lion in your next Windows. Therefore you will have a lion, even already framed.

The George W Bush method: Bomb (… somewhere). Then nobody will remember that you were supposed to catch a lion.

My own imagination

Thursday, November 24, 2005

Mathematics music

A student asked me this week (in an attempt to change the subject away from mathematics) what music I liked. I answered that at the moment I'm listening to for instance:

- Kate Bush: Pi
(Oh he love, he love, he love/He does love his numbers/And they run, they run, they run him/In a great big circle/In a circle of infinity/3.1415926535 897932 3846 264 338 3279)

- Tom Lehrer: New maths
(Hooray for New Math,/New-hoo-hoo Math!/It won't do you a bit of good to review math./It's so simple,/So very simple,/That only a child can do it!) - probably the only song in the world which includes the calculation of 342 minus 173, both in base 10 and in base 8...

- deLillos: Geometri
(Then you must tell yourself that you are in the centre, friend/it's only the circle around you that changes/and you can live friendly and reasonably in the middle of it/on the periphery of the others' (my translation))

- Tre små kinesere: Pi err i andre
(We all know the circle is round/Pi times r squared/The circle is eternal, it never breaks/That can't be changed (my translation))

There's lots of good music out there...

Edwin A. Abbott: Flatland: A romance of many dimensions

This is one of many sites which has the full text of Flatland, Edwin A. Abbott's classic "romance of many dimensions" from 1884, wherein the "people" are geometrical shapes and live in two dimensions. The book is an interesting attempt at showing us how foolish we are when we assume three dimensions are all there is, and at the same time it is a satire of society in 1884.

I haven't yet found a good way to use this text with students, I'm afraid, but good ideas are always appreciated...

Geometrical look at tangens

As mathematics is an important part of my life (I'm a teacher educator in mathematics), I thought it would be a good idea to start a blog where interesting small mathematical details could be posted...

Just a few weeks ago, I had a little revelation in trigonometry. I've been teaching trigonometry for years, and have given the definition of sin, cos and tan according to the triangle and the unit circle. I have never seen tangens clearly in the unit circle however - not before I came across this application. While cos u is the x-coordinate of the intersection of the one leg of the angle and the circle, and sin u is the y-coordinate of the same intersection, I have never before seen that tan u is the intersection of the leg and the vertical line x=1. A nice fact...

(And yes, I know I should have an illustration here, but for now the link to the Java-thing will do...)