Simon explains what Gaussian formula is to check a shape’s curvature and shows how to make a triangle with three 90° angles. Or is it a square, since it’s a shape with all sides equal and all angles at 90°? He also says a few words about the curvature of the Universe we live in.
I have composed a piece of music based on the Fibonacci sequence, using modular arithmetic (I assigned numbers from 0-6, the remainders after ÷ by 7, to notes C-B, i.e. 1-C, 2-D, 3-E, 4-F, 5-G, 6-A, 0-B. Then I added harmonies to the left hand). I noticed that after 16 notes, the sequence comes back to where it started!
But what really amazed me, is:
> I tried the same with Lucas #s, and Double fibonacci #s, and it always came back to where it started! Not only that, but always with the same length of period as well! It’s amazing!!!!
So, when you see something like this, you try to go over to a whiteboard and prove it, right? This is exactly what I did. In the vid below, you can see my proof of why this happens. I also analyze it a bit more, by seeing what is special of the Fibonacci #s, and also try ÷ by different numbers, instead of 7.
Disclaimer: Numberphile has already done a musical piece based on the Fibonacci numbers and a discussion of Pesano periods. What’s specific to my video:
* Trying different fibonacci-style sequences
* What’s then special about the Fibonacci #s
* Making a table of different divisors
* (And, mathematics-aside, doing my composition in a more mathematical way, by being more strict about the melody)
This is a project that Simon started a few weeks ago but never finished, so I think it’s time I archive it here. It’s based upon this wonderful Numberphile video, in which Ben Sparks shows a curious math problem – a game of cat and mouse – in a computer simulation he’d built. The setting is that the mouse is swimming in a round pond and is trying to escape from a cat that is running around the pond. What is the strategy that the mouse should apply to escape, considering that it swims at a quarter of the speed the cat runs?
Simon came up with his own code to recreate the simulation from the Numberphile video. In the four fragments I recorded, he showcases what he has built. Please ignore my silly questions, at the time of the recording I hadn’t viewed the Numberphile video yet and had no idea what the problem entailed.
Simon explains that the Van Eck Sequence is and shows the patterns he has discovered in the sequence by programming it in Python and plotting it in Wolfram Mathematica. Simon’s project in Wolfram is online at: https://www.wolframcloud.com/objects/4066d93a-893b-4a99-9fdc-54e265d27888
He also shows Neil Sloane’s proof of why the sequence is not periodic and adds an extra bit to make the proof more complete.
This video is inspired by the Numberphile video about the Van Eck sequence.
This project is a simulation of how many people can stem from the same ancestor, something Simon has learned from James Grime’s “Every Baby is a Royal Baby” video on Numberphile. In this simplified version, there’re only 6 people per generation. Simon was throwing two dice to determine who the two parents were for every person (in the case when both dice came out to be the same number, this was considered “virgin birth” or simply that the father had come from outside the limited sample Simon was working with).
Mesmerised by the 3D printed gears on Numberphile: “If you move two of these, the third one appears to be hovering in mid-air!”, Simon made a similar construction of his own – 6 straws forming 3 gears.
Simon has tried Matt Parker’s multiplicative persistence challenge on Numberphile: by multiplying all the digits in a large number, looking for the number of steps it takes to bring that large number to a single digit. Are there numbers that require 12 steps (have the multiplicative persistence of 12)?
Simon has worked on this for two days, creating an interface in Wolfram Mathematica. He wrote the code to make the beautiful floral shapes above, they are actually graphs of how many steps three digit numbers take to get to single digit numbers (each ”flower” has the end result at its center).
What about the numbers with many more digits than three? Simon has tried writing code to look for the multiplicative persistence of really large numbers and also came up with some efficiencies, i.e. shortcuts in the search process. He did manage to find the persistence for 2^233 (the persistence was 2):
However after he applied one of his efficiencies to the code to be able to search through many numbers at once, the code didn’t run anymore. You can read Simon’s page about this project and see his code here:
277777788888899 is the smallest number with a persistence of 11.
The largest known is 77777733332222222222222222222:
This code works with a few efficiencies:
1. They’ve already checked up to 10^233, so we don’t have to check those again.
2. We can rearrange the digits, and the multiplication will be the same. So we don’t have to check any of the rearrangements of any of the numbers we’ve already checked.
3a. We should never put in a 0 (a digit of the number). Because then you would be multiplying by 0, which would result in 0 in 1 step!
3b. We should also never put in a 5 and an even number. Because, in the next step, the number would be divisible both by 5 and by 2, so it’s also divisible by 10. That would put a 0 in the answer, which we saw we should never do!
3c. With similar reasoning (assuming we want to find the smallest number of the type we want), we’ll see we should never put in:
– Two 5s
– A 5 and a 7
– When we put in a… (- means anything, the order doesn’t matter):
1,- , remove the 1
2,2, put 4 instead
2,3, put 6 instead
2,4, put 8 instead
3,3, put 9 instead
So, we can reduce the search space and time collossaly, with just some logic!
Simon explains strong and weak tree sequences and reveals the greatest finite number used in mathematics: TREE(3), a lot larger than Graham’s number. The TREE sequence is a fast-growing function arising out of graph theory.
Simon comments: “What is you make TREE(TREE(3))?”
Simon is enchanted by Gödel’s incompleteness theorem (that he has learned about from Numberphile) and keeps talking about it:
“There’re problems that we just can’t solve. But if we prove that we can’t prove them, then we prove them! We can’t prove that we can’t prove that we can’t prove, and so on… Quirky! Standard math doesn’t really accept that because the statement goes on forever: you’ll just never get to what we can’t prove. What follows from Gödel’s incompleteness theorem is that that statement is actually true!”
The same evening, Simon is also bothered about the lies pupils are told in school. He repeatedly quotes James Grime that it’s a big lie that mathematics is about numbers. — “What is mathematics about? Mathematics is actually about proving! But there’s one more lie that even professional mathematicians don’t know. It’s that it’s about logic. Actually, mathematicians are a lot more creative!”