Euler Characteristic (whooohooooo! Math Math Math!)

Take out a sheet of paper. Pop quiz! (just kidding).

Draw any number of dots on your page. Now connect the dots with lines, subject to the following rules: lines may not cross each other as they move from dot to dot, and every dot on your page must be connected to every other dot through a sequence of lines.

Now count the number dots (D), lines (L), and regions separated by lines (R). (Don't forget to count the outside as a region too.) Compute D-L+R. What do you get?

No matter how you started, the number you will always get is 2!

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In Figure 1, D=9, L=12, R=5, and indeed, D-L+R=2.

If the lines represent fences, and the dots fenceposts, then the regions separated by the fenceposts are the pastures. So, if you are a farmer who wants to fence off 4 pastures together with 55 sections of fence, you can calculate exactly how many fenceposts you need, no matter how you arrange the fences!
(L=55, R=5=4+outside, so D=2+L-R=2+55-5=52 fenceposts.)

Presentation Suggestions:
You may wish to have everyone shout out their answer at the same time... students will be surprised they all get the same answer.

The Math Behind the Fact:
The number D-L+R is called the Euler characteristic of a surface. It is an invariant of a surface, meaning that while it looks like it may depend on the system of fences you draw, it really does not (as long as every pasture, including the outside, is topologically a disk with no holes). Thus the number only depends on the topology of the surface that you are on! For planes and spheres, this number is always 2.

Bike with Square Wheels

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Ride a bike with round wheels and it rolls smoothly on a flat road (i.e., the axle remains level). Can you get a bike with square wheels to roll smoothly on a road of some other shape?

Surprisingly, yes. A road made up of inverted catenaries will do the trick! A catenary is a portion of a cosh curve. Figure 1 shows how such a bike would roll.

Presentation Suggestions:
You can demonstrate this to your class using Mathematica code available in the reference.

The Math Behind the Fact:
You may wonder: what about bikes with other shapes for wheels? What road shape would allow the bike to roll smoothly? The reference by Wagon and Hall discusses the fundamental differential equation that solves this problem.

Brouwer Fixed Point Theorem

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One of the most useful theorems in mathematics is an amazing topological result known as the Brouwer Fixed Point Theorem.

Take two sheets of paper, one lying directly above the other. If you crumple the top sheet, and place it on top of the other sheet, then Brouwer's theorem says that there must be at least one point on the top sheet that is directly above the corresponding point on the bottom sheet! Do you believe that?

In dimension three, Brouwer's theorem says that if you take a cup of coffee, and slosh it around, then after the sloshing there must be some point in the coffee which is in the exact spot that it was before you did the sloshing (though it might have moved around in between). Moreover, if you tried to slosh that point out of its original position, you can't help but slosh another point back into its original position!

More formally the theorem says that a continuous function from an N-ball into an N-ball must have a fixed point. Continuity of the function is essential (if you rip the paper or if you slosh discontinuously, then there may not be fixed point).

Presentation Suggestions:
Bring a coffee cup and 2 sheets of paper with you and demonstrate as you present the fun fact. Draw a grid on the paper, number the gridboxes, then xerox that sheet of paper. After you crumple the paper, you can say that at least one number is on top of the corresponding number on the lower sheet of paper. Alternatively, bring a map of Claremont (or whatever city you are in) to class and drop it on the floor---then there must be some point in the map lying directly over the point that it represents!

The Math Behind the Fact:
Fixed point theorems are some of the most important theorems in all of mathematics. Among other applications, they are used to show the existence of solutions to differential equations, as well as the existence of equilibria in game theory. There are many proofs of the Brouwer fixed point theorem. The advanced student may wish to see if she can show the equivalence of this theorem with Sperner's lemma, which yields a rather nice elementary proof.

Sperner's Lemma

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Divide a triangle T into lots of baby triangles, so that baby triangles only meet at a common edge or a common vertex. Label each main vertex of the whole triangle by 1, 2, or 3; then label vertices on the (12) side by either 1 or 2, on the (23) side by either 2 or 3, and the (13) side by either 1 or 3. Label the points in the interior by any of 1, 2, or 3. For instance, see Figure 1.

Fun Fact: any such labelling must contain a baby (123) triangle! (In fact, there must be an odd number!)

Actually, a version of Sperner's Lemma holds in all dimensions. Can you figure out how it generalizes?

Sperner's Lemma is equivalent to the Brouwer fixed point theorem.

Presentation Suggestions:
Have everyone make their own labelled triangle and see how many (123) triangles they have in their picture.

The Math Behind the Fact:
There are many proofs of this fact. Some short non-constructive proofs rely on parity arguments. Constructive proofs are the key to many fixed point algorithms as well as fair division procedures.

Sphere Eversions

If you take a loop of string in the plane and place an arrow along it pointing clockwise, is it possible to deform the string, keeping it in the plane, so that the arrow points counterclockwise, without causing any kinks in the string?

A moment's reflection seems to indicate that this is impossible. Is it? Can you prove it?

What about a sphere in 3-space? Is it possible to turn the sphere "inside out", allowing self-intersections but not allowing any sharp kinks, creases, or tearing of the surface?

Surprisingly, in 1957, Steve Smale proved that this is in fact possible! Such an operation is called a sphere eversion. And later on, several people constructed explicit methods for doing so, among them William Thurston.

The Math Behind the Fact:
Requiring that loops in the plane have no kinks is equivalent to giving them unit speed parametrizations and requiring that the parametrizations are continuously differentiable, i.e., their rates of change vary continuously. The study of differentiable structures on geometric objects is called differential geometry and the study of smooth deformations of such objects is often called differential topology.

Does Order of Addition Matter?

From your earliest days of math you learned that the order in which you add two numbers doesn't matter: 3+5 and 5+3 give the same result. The same is true for the addition of any finite set of numbers. But what if you are adding an infinite set of numbers?

For instance, in calculus you encounter series, which are infinite sums. Does the order of addition matter?

Surprisingly, yes, but only for some series. A series is said to converge absolutely if the sum of the absolute values of the terms converges. It is a fact that a series which converges absolutely must converge, and no matter how you rearrange the terms the sum will be the same.

Series that do not converge absolutely are wilder; they may or may not converge (so they are called conditionally convergent). But, if such a sequence does converge, it is even stranger still--- by rearranging the terms in the sequence, you can get the series to converge to any value you want!

The Math Behind the Fact:
This fact is due to Riemann, and the basic idea is simple. To obtain a rearrangement that sums to X, just add enough positive terms from the series (in their original order) until the partial sums first exceed X, then add enough negative terms from the series (in their original order) until the partial sums first dip below X, then repeat and continue with the successive positive terms, then the successive negative terms, etc. This will yield a rearrangement of the original series that sums to X.

This fact shows how careful one has to be when switching the order of addition in a conditionally convergent series.

Tesseract

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What does a "cube" look like in higher dimensions?

Well, we can extrapolate by looking at lower dimensions.
A 0-dimensional cube is a point, a vertex.
A 1-dimensional "cube" is a line segment, with 2 vertices at either end. It is obtained from a 0-dimensional cube by thickening it in one dimension.
A 2-dimensional "cube" is square, with 4 vertices, obtained by thickening up the line segment in a second dimension.
A 3-dimensional "cube" is a cube, with 8 vertices, obtained from the square by thickening it in a third dimension.

So, by extrapolation the 4-dimensional "cube", also called a tesseract or hypercube, should have 16 vertices, and is obtained from a cube by thickening it up in a fourth dimension. Since we cannot easily visualize this, there are a number of ways we can understand this object by viewing projections, or "shadows" of it in 3-D.

Regular Solids

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A regular polygon is a polygon whose angles are equal and side lengths are equal. A 3-D polyhedron is said to be regular if all its faces are regular polygons and the same number of faces meet at each corner. How many regular 3-D solid polyhedra are there?

Answer:there are exactly five--- the tetrahedron (pyramid with triangular faces), octahedron (8-sided with triangular faces), dodecahedron (12-sided with pentagonal faces), icosahedron (20-sided with triangular faces), and cube (6-sided with square faces).

These are sometimes called the Platonic Solids, named thus because they were described by Plato in one of his books. Euclid proved that there are only five.

The Math Behind the Fact:
A regular solid with hexagonal faces cannot exist because if it did, the sum of the angles of any 3 hexagonal corners that meet would already equal 360, so such an object would be planar. Thus the only regular solids involve pentagons, squares, and triangular faces and you can further limit the possibilities using the Euler characteristic of the sphere, which is 2. This makes a great exercise.

Inductive Tiling

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Can all but one square of an n by n chessboard be covered by L-shaped trominoes?

In general, it can not necessarily be done, but if n is a power of 2, it can! See the Figure for an 8x8 example.

Is there a method for finding such a tiling? Yes, and we can find it by induction!

The Math Behind the Fact:
A simple proof by induction shows this property. The simplest case is a 2x2 chessboard. Clearly if one square is removed, the remainder can be tiled by one L-shaped tromino.

So, given a chessboard of size 2nx2n, with one square removed, it can be divided into four 2(n-1)x2(n-1) smaller chessboards, one of which contains the removed tile.

In the remaining three 2(n-1)x2(n-1) boards, we'll remove a tile from each by taking the adjacent center corners of the three to give an L-shaped hole.

The inductive hypothesis will show how to cover the four smaller boards, since they each have a square removed, and are size 2(n-1)x2(n-1). The L-shaped hole can also be covered by a tromino. This covers the original board except for the original missing square!

In Figure 1 above, applying the inductive step removes the yellow tiles near the center of the 8x8 board. The 8x8 is divided into 4 quadrants, each with a square missing. Then applying the inductive step removes the green tiles near the center of each 4x4 quadrant. This gives several 2x2 boards with a square removed---the base case!

Mug Trick

Take a mug that has a handle, and fill it with coffee. Hold the mug in your hand. Now, without letting go of the mug (or changing the position of your mug relative to your hand), and without spilling the coffee, see if you can rotate the mug TWO FULL TURNS and return your hand, arm, and cup to their original positions.

If you can do that, can do it the same trick with only ONE full turn? (No!)

Presentation Suggestions:
Do it! Just don't fill the mug with hot coffee! You may wish to point the handle north at the beginning of the trick, so that students can track how many times the mug turns. Now gradually rotate the cup towards your chest, then under your elbow... this will bring the cup through one rotation, but your arm will be twisted. Keeping the cup upright and rotating in the same direction will now force you to rotate the cup over your head. Then bring the cup down and you will have given it two full turns and brought your arm back to its starting position.

Students will enjoy knowing that this trick is evidence of some deep mathematics, even if they don't understand it, so be sure to tell them about the math. Also, even if students only pick up the "buzzwords" (like "fundamental group"), it is still useful, because it all helps to popularize mathematics. (How many of us know what "string theory" really is? But hasn't that given physics a lot of glamor?)

The Math Behind the Fact:
See the Fun Fact on the Fundamental Group. Continuously rotating a mug is equivalent to following a path in "SO(3)", the space of rotations in 3-space, and if you start and end the mug in the same orientation, you have traced a "loop" in SO(3).

The reason this trick works for 2 twists but not 1 twist is because the fundamental group of SO(3) is "Z/2", the cyclic group of order 2. In layman's terms, this means that of all the ways you can twist a mug and return it to its original position, in only half of them will you be able to return your whole arm to its original position, and these are precisely the ones in which the mug rotates an even number of times.

How many Rationals?

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How many rational numbers are there? Yes, infinitely many, I hear you say. But how large is that infinity? Are there just as many rational numbers as integers?

Well, this requires us to be precise about "just as many". A mathematician would say that set A has "just as many" objects as set B if the objects in A and B can be put into one-to-one correspondence with each other. Seem plausible?

Using this definition we can show that lots of infinite sets have the same "size" (or, cardinality). For instance the even integers can be placed in 1-1 correspondence with the odd integers, using N->N+1.

Perhaps surprising is that the set of all integers and the set of even integers have the same cardinality, via N->2N.

The natural numbers {1, 2, 3, ...} and the integers have the same cardinality, because the integers can be "listed" in the order {0, 1, -1, 2, -2, 3, -3, ...} and this ordering gives the correspondence with the natural numbers. Any set with the same cardinality as the natural numbers is called a countable set.

The rationals seem even more densely populated in the real line than the integers, but it is possible show that they are countable! The 1-1 correspondence is given by drawing a double array of rationals (as in the Figure) and then listing them in the order given by snaking diagonally through the array to hit every one.

The Math Behind the Fact:
Mathematicians try to make precise definitions that correspond to our usual intuition, but help to resolve issues when intuition fails. Cantor tried to make precise the notion of the size of an infinite set.

How many Reals? Cantor Diagonalization

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We have seen in the Fun Fact How many Rationals? that the rational numbers are countable, meaning they have the same cardinality as the set of natural numbers.

So are all infinite sets countable?

Cantor shocked the world by showing that the real numbers are not countable... there are "more" of them than the integers! His proof was an ingenious use of a proof by contradiction. In fact, he could show that there exists infinities of many different "sizes"!

Presentation Suggestions:
If you have time show Cantor's diagonalization argument, which goes as follows. If the reals were countable, it can be put in 1-1 correspondence with the natural numbers, so we can list them in the order given by those natural numbers. Now use this list to construct a real number X that differs from every number in our list in at least one decimal place, by letting X differ from the N-th digit in the N-th decimal place. (A little care must be exercised to ensure that X does not contain an infinite string of 9's.) This gives a contradiction, because the list was supposed to contain all real numbers, including X. Therefore, hence a 1-1 correspondence of the reals with the natural numbers must not be possible.

The Math Behind the Fact:
The theory of countable and uncountable sets came as a big surprise to the mathematical community in the late 1800's.

By the way, a similar "diagonalization" argument can be used to show that any set S and the set of all S's subsets (called the power set of S) cannot be placed in one-to-one correspondence. The idea goes like this: if such a correspondence were possible, then every element A of S has a subset K(A) that corresponds to it. Now construct a new subset of S, call it J, such that an element A is in J if and only if A is NOT in K(A). This new set, by construction, can never be K(A) for any A, because it differs from every K(A) in the "A-th element". So K(A) must not run through the entire power set of A, hence the 1-1 correspondence asserted above must not be possible.

This proof shows that there are infinite sets of many different "sizes" (cardinalities) by considering the natural numbers and its successive power sets!

:nosebleed: AdIk BaH!!!

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ang anu? haha! hambal mo pa gani, "Engineer ta ya part!"....... ti amo ni nami da. hahaha!

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