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"Math is delicious!"
Examples
- According to a theorem called the Banach-Tarski paradox
 , it's possible to divide a sphere into five "pieces" (subsets), then reassemble them into a sphere with twice the volume using only rotations and translations. (The catch is that the "pieces" are "non-measurable sets" - they're constructed in such a way that they literally don't have a volume that can be calculated.)
- And then there's Whitehead and Russell's Principia Mathematica, a multi-volume opus intended to construct all of mathematics from the most basic of axioms (such as "every statement is either true or false"). After 379 pages of incomprehensibly dense notation, it succeeds in proving that 1+1=2◊. And then Gödel's Incompleteness Theorem undermined the whole thing using a real-life Logic Bomb.
- A googolplex (10^(10^100)) is bigger than the number of elementary particles in the known universe (10^80) and the number of planck times since the big bang (8*10^60)...times each other. Of course, that is nothing compared to...
- Graham's number. A number so large that it could not even be written with conventional scientific notation, and a new form of mathematical notation had to be developed just so it could be expressed. It is often regarded as the largest finite number that pure mathematics actually takes seriously. What's worse? The answer to the problem it was created to solve might actually be twelve.
- Graham's number is so ridiculously mindbogglingly huge that there's aren't enough elementary particles in the whole universe to express it using scientific notation.
- TREE(3). Graham's number, an upper bound is dwarfed by a lower bound!
- Larger than 11, but smaller than a ridiculously large number... has anyone considered 42?
- This troper is known on XKCD's forums for beign a prominent user of A really fast growing function to create a number that dwarves almost any other defined number.
- Möbius strips. It's one thing to have a ridiculous, mind-bending shape like that on paper, but you can make them in real life. It's the weirdest feeling in the world to hold one.
- This troper's mother-in-law has a knitting pattern for a Möbius Scarf. My wife has one. She takes great delight in handing it to people and asking them casually to fold it up.
- If you cut a Möbius strip in half lengthwise it stays in one piece. If you give it not one but three half-twists and cut it in half lengthwise, not only does it stay in one piece, it twists itself into a knot. If you cut a Möbius strip in thirds lengthwise (possibly by making the guide lines on the strip of paper beforehand), you'll finish cutting it in one go, and end up with a regular loop with another Möbius strip attached.
- Möbius strips can be made using legos - if you use the thin treads that have the chain link, give them a twist and connect them. the fun part, is if you run them with a gear, they still have full mobility.
- Klein bottles take it a step further by doing away with those pesky edges. And you can cut them in half into a pair of Möbius strips.
- Hell, it's possible to cut a Klein bottle into a single Möbius strip.
- Here's someone making them in glass.
- And from the same site, Klein Hats.
- In a similar vein, how about some crochet models of hyperbolic space?
- New use for the Mobius strip: the world's smallest-denomination polyhedral die. (OK, monohedral die...is that even a word?)
- 0.999... is exactly equal to 1. Not almost, not "so close there's no practical difference," exactly.
- Well, haven't you always asked about the 1/3 problem, as this troper calls it. 1/3 = .333..., but 3/3 = 1. Think about that.
- So what? 1/3 = .333..., 3/3 = 3 * 1/3 = 3 * .333... = 0.999... = 1.
- 3/3 = 4/4 = 5/5 = 6/6 = 1.
- The more explicit way would be to express 0.999... as a geometric sum of its digits. 9/10 + 9/100 + 9/1000 + ..., which comes out to exactly 1.
- There are the same number of odd numbers (1, 3, 5, ...) as there are natural numbers (1, 2, 3, ...). This troper once spent an entire afternoon trying to explain this to a friend who absolutely would not accept it.
- That's not the half of it. What's even more bizarre is that there are the same number of fractions as there are natural numbers. What's more bizarre even than that is that this is not a trivial fact. There are multiple types of infinity (an infinite number, actually). The set of real numbers (everything that can be plotted on a continuous number line) is a larger type of infinity. (The set of all possible subsets of the real numbers is larger still.) However, most numbers that we care about—"computable numbers"—fit into a countable set.
- As do any numbers recognized by a countable ordinal hypercomputation degree.
- Which brings up that there are numbers that have well defined values and yet are impossible to compute, even approximately.
- All infinities are infinite, but some are more infinite than others. The study of infinite sets is so fantastically weird that when the mathematician Georg Cantor first came up with it, he was accused of challenging God.
- This can be extraordinarily confusing, and tends to prompt discussion when it shows up, likely because it involves things like "all odd numbers" which are widely understood, while the more difficult formal ideas (like cardinality) aren't expressly stated. In short, two sets are defined as having the same cardinality ("size") if you can match up their members in some one-to-one fashion. This is perfectly intuitive for finite sets, but a bit confusing for infinite ones. It is consistent, though, and more sensible than any other definition. Possibly the best example is Hilbert's paradox of the Grand Hotel: in a hotel with an infinite number of rooms, the propositions "every room is occupied" and "no more people can fit" are not equivalent.
- Is there a set with cardinality (size) that is strictly larger than the set of natural numbers but smaller than the set of real numbers? There is no answer (at least in ZFC), this is possible because of Gödel's incompleteness.
- Even trippier? All n-dimentional real spaces have the same cardinality. That is, the number line (x), the Cartesian plane (x,y), three-dimensional space (x,y,z), etc.
- How about a geometrically figure that has an infinite surface area, but a finite volume? Gabriel's Horn is such a figure. And then you get the Koch Snowflake, which has an infinite perimeter but a finite area. Similarly, the Menger Sponge has infinite surface area and finite volume, but unlike Gabriel's Horn, it can fit within a finite bounding box.
- By using the concept of "inner product spaces," you can do absolutely absurd things with anything that can be considered a vector space. Like calculating the angle between two matrices. Or finding the distance between two polynomials.
- That is basically what is going on in modern mathematics, and has been for over a century. Mathematicians have been trying to find the correct generalizations, and see where it gives useful ideas. There is a germ of usefulness in the latter one that you gave for sure.
- Topology has some fun ones, but there is no way to explain why they're fun unless you know a little topology yourself. The simplest one is that there is a notion of "closeness" on the natural numbers so that the sequence 1, 2, 3, ... is close to every natural number. Moreover, this notion of closeness doesn't seem all that unintuitive before you start digging.
- e^(i[pi]) = -1.
- And the graph of that looks like this◊. Makes a nice T-shirt.
- Slightly cooler (it's the generalization) is that e^(i*x)=cos(x)+i*sin(x).
- i^i is a real (not a complex) number. On the other hand, it's not well-defined.
- 10^2+11^2+12^2=13^2+14^2
- If you want proof that Evil Is Cool, play around with the Number Of The Beast. 666 is not only the sum of the squares of the first seven primes, but it's also a palindrome. 1^3+2^3+3^3+4^3+5^3+6^3+5^3+4^3+3^3+2^3+1^3 is another palindrome, which equals 666. And that middle number, 6^3? That's three sixes, of course. Cool, right?
- Want another fun game with 666? You'll discover that it is the sum of all cardinal numbers from 1 to 36. Since 36 = 6^2, that means you can fit it into a Magic Square. Let he who hath understanding reckon the number of the beast.
- Probability can be a very strange thing to behold at times. For example: You can have a situation where the probability of any given event is 0, yet events still happen (thanks to our good old friend "continuum").
- That's defined more precisely as probability density. Instead of probability of specific values, you calculate the probability of landing within selected boundaries by finding a slice of area under the density function (the total area under it being 1 = 100%).
- As a nice example: The probability for a random real number to be rational is zero. That doesn't stop you from getting a rational number once in a while.
- For the entirely useless, but interesting spectrum of math we have the weird number bases. If you know decimal (the normal base) and get your head around binary or hexadecimal, the jump to the the other natural bases is not so great. However, consider a number base of -10. Or 2i. Or even base phi.
- Quiz: What is 1+1 in base phi? 10.01
- There are infinitely many primes
- ...which the above-mentioned Gödel's theorem implicitly relies on, along with integers having unique factorizations
- Mathematicians worry about the strangest things.
- So the rest of the world doesn't have to.
- Want to try and count to infinity? If you start at 1, you've already made a mistake- infinity has no beginning, nor end. Just TRY and picture that.
- That infinity has "no beginning, nor end" doesn't mean that we can't start somewhere; as long you have a countable infinite, for example the set of all counting numbers, you can start at 1 and work your way upwards. Even if your infinity stretches from -∞ to ∞ and passes through every rational number on the way it's still possible to start at 0 and count {0, 1, -1, 2, 1/2, -2, -1/2, 3, 1/3, -3, -1/3, 4, 3/2, 2/3, 1/4, -4, -3/2, -2/3, -1/4,...} and reach infinity that way; there are many forms of infinity; it's all a question on how you reach infinity. If you want to get pedantic, it's the concept of counting to infinity there's something wrong with, not where to start, because infinity isn't a consistent value but rather an expression of a concept; if I start at one and count towards infinity by adding one to the previous term, I am counting to infinity and I've started; no question about it. Even if I were to recite every value in ℵ1, I could still start somewhere.
- New Math. Yes folks, the brilliant folks in charge of our education systems tried to throw matrix ops, set theory, number bases, and abstract algebra at elementary schoolers. Many of the elementary teachers ordered to do the throwing didn't know the material themselves and wouldn't have been able to teach it to high school and college students of appropriate level and background.
- And a truly trivial piece of trivia: you know prime numbers, the numbers which can't be divided by any integer but 1 and themselves? One such number is ... eight six seven five three oh ni-i-ine!
...my brain hurts. I think I'll stick to biology.
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