Combinatorics. Combinatorics began as a formalized treatment of efficient ways of counting certain collections of objects which arise relatively often. Nowadays the word ‘combinatorics’ can be used to refer to pretty much all of finite mathematics, and the original field is more specifically called “enumerative combinatorics”.

The things in parentheses are not fractions (they don’t have a fraction bar). These are calledbinomial coefficientsand (n [over] k) represents the number of ways to choose k balls from a set of n balls. Pascal’s triangle is an arrangement of these numbers where (n [over] k) is the k-th number in the n-th row; it is very famous to be a useful way of visualizing many of the properties of the binomial coefficients.I’m experimenting a little with how to display the theorem statements. Not really sure what I like yet.

Pulling up to Liz and Gary’s, the first thing you notice is the garage - or more specifically, the garage doors. Two of them, each containing 16 raised panels arranged in a 4 x 4 grid. Standard enough, except when you consider that each of the 32 panels is painted in one of four colors - purple, turquoise, indigo, and sky - and that, within the confines of either singular door, each color appears exactly once in any given column or row. If you’re thinking this sounds a lot like Sudoku or the more recently-popularized KenKen, then you’re absolutely right. Sudoku, KenKen, and Liz and Gary’s garage doors are all examples of the combinatorial

Latin square.For the sakes of clarity and concision, here are a few definitions:

Definition.Ann x narray containingndistinct symbols, so that each distinct symbol occurs exactly once in each row and each column is called a.Latin square of order-n

Definition.Two order-nLatin squares are said to beif, when you superimpose one square on top of the other, each of theorthogonaln^{2}ordered pairs arising occurs exactly once. An integral collection ofk> 2 order-nLatin squares are said to beif any arbitrary pair of squares selected from the collection ofmutuallyorthogonalkis orthogonal.It may be of interest to note that if we are considering collections of Latin squares of order-

n, the collection cannot be mutually orthogonal unless it containsn-1or fewer squares. That is

Theorem.A maximal set of mutually orthogonal order-nLatin squares containsn-1elements.

Proof.Suppose we have some mutually orthogonal collection of order-nLatin squares, which we denoteA. Notice that if we permute the symbols of one square in the collection and call our new resultant collectionA’,A’is still a mutually orthogonal collection. Moreover notice that this is true no matter how many squares fromAwe choose to permute.Since we can permute every member of

Aand still have a mutually orthogonal collection, suppose that we permute each and every square comprisingAso that it’s top row is an increasing index [for example, if the “symbols” concerned are the natural numbers, each Latin square would be permuted until its top row read “1 2 3 … n”]. Consider next, for any two elements ofA, the coordinate at the intersection of the second row and first column - we refer to this as the (2,1) coordinate. Clearly, no element inAcan have the first symbol in the increasing index in its (2,1) position, as this would violate the “exactly once” condition required of a Latin square. Moreover, orthogonality necessitates that for any two distinct elements ofA, the (2,1) coordinates of these elements are too distinct, so that we have at most n-1 distinct possible entries for the (2,1) coordinate, and thus a maximal set (if it exists) of cardinality n-1, as desired.

We get more compassionate as we evolve. More humble. More subtle. More aware of how little we know. We don’t get superior. We don’t form cults of personality. We don’t think we have it all worked out. If we imagine ourselves ‘all that’, then we have actually devolved. I trust the ones who know a little something but don’t know a whole lot, more than the ones who ‘know it all’. I trust the ones who realize how far they have yet to travel. We have so much more to learn. All of us.

Jeff Brown (via parkstepp)

Probably the coolest part of intro to analysis. Working out the expansion of the series and then watching pieces of it cancel out along the way is supremely satisfying.

It is not easy for the lay mind to realise the importance of symbolism in discussing the foundations of mathematics, and the explanation may perhaps seem strangely paradoxical. The fact is that symbolism is useful because it makes things difficult. (This is not true of the advanced parts of mathematics, but only of the beginnings.) What we wish to know is, what can be deduced from what. Now, in the beginnings, everything is self-evident; and it is very hard to see whether one self-evident proposition follows from another or not. Obviousness is always the enemy to correctness. Hence we invent some new and difficult symbolism, in which nothing seems obvious. Then we set up certain rules for operating on the symbols, and the whole thing becomes mechanical. In this way we find out what must be taken as premiss and what can be demonstrated or defined. For instance, the whole of Arithmetic and Algebra has been shown to require three indefinable notions and five indemonstrable propositions. But without a symbolism it would have been very hard to find this out. It is so obvious that two and two are four, that we can hardly make ourselves sufficiently sceptical to doubt whether it can be proved. And the same holds in other cases where self-evident things are to be proved.

Bertrand Russell, *Mathematics And The Metaphysicians*

Happiness, like every other emotional state, has blindness and insensibility to opposing facts given it as its instinctive weapon for self-protection against disturbance. When happiness is actually in possession, the thought of evil can no more acquire the feeling of reality than the thought of good can gain reality when melancholy rules. To the man actively happy, from whatever cause, evil simply cannot then and there be believed in. He must ignore it; and to the bystander he may then seem perversely to shut his eyes to it and hush it up.

William James

No company is preferable to bad, because we are more apt to catch the vices of others than their virtues, as disease is far more contagious than health.

C. C. Colton

One of the answers to the topic:

Visually stunning math concepts which are easy to explainatMathematics Stack Exchange.I think if you look at this animation and think about it long enough, you’ll understand:

- Why circles and right-angle triangles and angles are all related
- Why sine is opposite over hypotenuse and so on
- Why cosine is simply sine but offset by pi/2 radians