What concepts or facts do you know from math that is mind blowing, awesome, or simply fascinating?
Here are some I would like to share:
- Gödel’s incompleteness theorems: There are some problems in math so difficult that it can never be solved no matter how much time you put into it.
- Halting problem: It is impossible to write a program that can figure out whether or not any input program loops forever or finishes running. (Undecidablity)
The Busy Beaver function
Now this is the mind blowing one. What is the largest non-infinite number you know? Graham’s Number? TREE(3)? TREE(TREE(3))? This one will beat it easily.
- The Busy Beaver function produces the fastest growing number that is theoretically possible. These numbers are so large we don’t even know if you can compute the function to get the value even with an infinitely powerful PC.
- In fact, just the mere act of being able to compute the value would mean solving the hardest problems in mathematics.
- Σ(1) = 1
- Σ(4) = 13
- Σ(6) > 101010101010101010101010101010 (10s are stacked on each other)
- Σ(17) > Graham’s Number
- Σ(27) If you can compute this function the Goldbach conjecture is false.
- Σ(744) If you can compute this function the Riemann hypothesis is false.
Sources:
- YouTube - The Busy Beaver function by Mutual Information
- YouTube - Gödel’s incompleteness Theorem by Veritasium
- YouTube - Halting Problem by Computerphile
- YouTube - Graham’s Number by Numberphile
- YouTube - TREE(3) by Numberphile
- Wikipedia - Gödel’s incompleteness theorems
- Wikipedia - Halting Problem
- Wikipedia - Busy Beaver
- Wikipedia - Riemann hypothesis
- Wikipedia - Goldbach’s conjecture
- Wikipedia - Millennium Prize Problems - $1,000,000 Reward for a solution
Goldbach’s Conjecture: Every even natural number > 2 is a sum of 2 prime numbers. Eg: 8=5+3, 20=13+7.
https://en.m.wikipedia.org/wiki/Goldbach’s_conjecture
Such a simple construct right? Notice the word “conjecture”. The above has been verified till 4x10^18 numbers BUT no one has been able to prove it mathematically till date! It’s one of the best known unsolved problems in mathematics.
Wtf !
For the uninitiated, the monty Hall problem is a good one.
Start with 3 closed doors, and an announcer who knows what’s behind each. The announcer says that behind 2 of the doors is a goat, and behind the third door is
a carstudent debt relief, but doesn’t tell you which door leads to which. They then let you pick a door, and you will get what’s behind the door. Before you open it, they open a different door than your choice and reveal a goat. Then the announcer says you are allowed to change your choice.So should you switch?
The answer turns out to be yes. 2/3rds of the time you are better off switching. But even famous mathematicians didn’t believe it at first.
I know the problem is easier to visualize if you increase the number of doors. Let’s say you start with 1000 doors, you choose one and the announcer opens 998 other doors with goats. In this way is evident you should switch because unless you were incredibly lucky to pick up the initial door with the prize between 1000, the other door will have it.
Imagine a soccer ball. The most traditional design consists of white hexagons and black pentagons. If you count them, you will find that there are 12 pentagons and 20 hexagons.
Now imagine you tried to cover the entire Earth in the same way, using similar size hexagons and pentagons (hopefully the rules are intuitive). How many pentagons would be there? Intuitively, you would think that the number of both shapes would be similar, just like on the soccer ball. So, there would be a lot of hexagons and a lot of pentagons. But actually, along with many hexagons, you would still have exactly 12 pentagons, not one less, not one more. This comes from the Euler’s formula, and there is a nice sketch of the proof here: .
Quickly a game of chess becomes a never ever played game of chess before.
I’m guessing this is more pronounced at lower levels. At high level chess, I often hear commentators comparing the moves to their database of games, and it often takes 20-30 moves before they declare that they have now reached a position which has never been reached in a professional game. The high level players have been grinding openings and their counters and the counters to the counters so deeply that a lot of the initial moves can be pretty common.
Also, high levels means that games are narrowing more towards the “perfect” moves, meaning that repetition from existing games are more likely.
Related: every time you shuffle a deck of cards you get a sequence that has never happened before. The chance of getting a sequence that has occurred is stupidly small.
Borsuk-Ulam is a great one! In essense it says that flattening a sphere into a disk will always make two antipodal points meet. This holds in arbitrary dimensions and leads to statements such as “there are two points along the equator on opposite sides of the earth with the same temperature”. Similarly one knows that there are two points on the opposite sides (antipodal) of the earth that both have the same temperature and pressure.
Also honorable mentions to the hairy ball theorem for giving us the much needed information that there is always a point on the earth where the wind is not blowing.
Seeing I was a bit heavy on the meteorological applications, as a corollary of Borsuk-Ulam there is also the ham sandwich theorem for the aspiring hobby chefs.
Collatz conjecture or sometimes known as the 3x+1 problem.
The question is basically: Does the Collatz sequence eventually reach 1 for all positive integer initial values?
Here’s a Veritasium Video about it: https://youtu.be/094y1Z2wpJg
Basically:
You choose any positive integer, then apply 3x+1 to the number if it’s odd, and divide by 2 if it’s even. The Collatz conjecture says all positive integers eventually becomes a 4 --> 2 --> 1 loop.
So far, no person or machine has found a positive integer that doesn’t eventually results in the 4 --> 2 --> 1 loop. But we may never be able to prove the conjecture, since there could be a very large number that has a collatz sequence that doesn’t end in the 4-2-1 loop.
Godel’s incompleteness theorem is actually even more subtle and mind-blowing than how you describe it. It states that in any mathematical system, there are truths in that system that cannot be proven using just the mathematical rules of that system. It requires adding additional rules to that system to prove those truths. And when you do that, there are new things that are true that cannot be proven using the expanded rules of that mathematical system.
"It’s true, we just can’t prove it’.
The one I bumped into recently: the Coastline Paradox
“The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal curve–like properties of coastlines; i.e., the fact that a coastline typically has a fractal dimension.”
e^(pi i) = -1
like, what?
The Fourier series. Musicians may not know about it, but everything music related, even harmony, boils down to this.
This is a common one, but the cardinality of infinite sets. Some infinities are larger than others.
The natural numbers are countably infinite, and any set that has a one-to-one mapping to the natural numbers is also countably infinite. So that means the set of all even natural numbers is the same size as the natural numbers, because we can map 0 > 0, 1 > 2, 2 > 4, 3 > 6, etc.
But that suggests we can also map a set that seems larger than the natural numbers to the natural numbers, such as the integers: 0 → 0, 1 → 1, 2 → –1, 3 → 2, 4 → –2, etc. In fact, we can even map pairs of integers to natural numbers, and because rational numbers can be represented in terms of pairs of numbers, their cardinality is that of the natural numbers. Even though the cardinality of the rationals is identical to that of the integers, the rationals are still dense, which means that between any two rational numbers we can find another one. The integers do not have this property.
But if we try to do this with real numbers, even a limited subset such as the real numbers between 0 and 1, it is impossible to perform this mapping. If you attempted to enumerate all of the real numbers between 0 and 1 as infinitely long decimals, you could always construct a number that was not present in the original enumeration by going through each element in order and appending a digit that did not match a decimal digit in the referenced element. This is Cantor’s diagonal argument, which implies that the cardinality of the real numbers is strictly greater than that of the rationals.
The best part of this is that it is possible to construct a set that has the same cardinality as the real numbers but is not dense, such as the Cantor set.
The best part of this is that it is possible to construct a set that has the same cardinality as the real numbers but is not dense, such as the Cantor set.
Well that’s not as hard as it sounds, [0,1] isn’t dense in the reals either. It is however dense with respect to itself, in the sense that the closure of [0,1] in the reals is [0,1]. The Cantor set has the special property of being nowhere dense, which is to say that it contains no intervals (taking for granted that it is closed). It’s like a bunch of disjointed, sparse dots that has no length or substance, yet there are uncountably many points.
Integrals. I can have an area function, integrate it, and then have a volume.
And if you look at it from the Rieman sum angle, you are pretty much adding up an infinite amount of tiny volumes (the area * width of slice) to get the full volume.
The four-color theorem is pretty cool.
You can take any map of anything and color it in using only four colors so that no adjacent “countries” are the same color. Often it can be done with three!
Maybe not the most mind blowing but it’s neat.
Thanks for the comment! It is cool and also pretty aesthetically pleasing!
Euler’s identity is pretty amazing:
e^iπ + 1 = 0
To quote the Wikipedia page:
Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:[6]
The number 0, the additive identity.
The number 1, the multiplicative identity.
The number π (π = 3.1415…), the fundamental circle constant.
The number e (e = 2.718…), also known as Euler’s number, which occurs widely in mathematical analysis.
The number i, the imaginary unit of the complex numbers.The fact that an equation like that exists at the heart of maths - feels almost like it was left there deliberately.
Seeing mathematics visually.
I am a huge fan of 3blue1brown and his videos are just amazing. My favorite is linear algebra. It was like an out of body experience. All of a sudden the world made so much more sense.
