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)
“Mom, how long would it take a supercomputer running at 10^15 additions per second to calculate the 1000th Fibonacci number?”
Simon has learned this problem from the new course he is following on Brilliant.org: Computer Science Algorithms. Simon worked it out on an A3 sketch book sheet and got the answer correct: it would take longer than the age of the Universe!
Simon has already finished the Computer Science Fundamentals course! It has been Simon’s idea to take up the courses on Brilliant.org again and he has been working independently, driven entirely by his intrinsic motivation.
The course has also inspired Simon to work on a very large scale project: record a series of tutorials where he explains all the best known sorting algorithms and comes up with the Python code for them on his RaspberryPi!
Here is a fun math trick! Simon and Neva have made a 8 x 8 cm square (with an area of 64 cm²) and cut it into four pieces, turning the square into a puzzle. Using the same four pieces, they built a 5 x 13 cm rectangle. But wait a minute! 5 x 13 equals 65, so the area of the rectangle is one cm² larger than that of the square!
They also made a similar puzzle using bigger pieces. A 13 x 13 = 169 cm² square turned into a 8 x 21 = 168 cm² rectangle! So now the area of the rectangle is one cm² smaller than that of the square! What’s gong on?
You have probably recognized the numbers in this trick: 5, 8, 13, 21… Those are Fibonacci numbers! Simon explains, that with Fibonacci numbers, the effect of the rectangle area being greater or smaller than the square area is alternating. Fibonacci have a converging ration to φ (Phi), but not φ. The pieces only look like they are golden ratio bigger/ smaller. In reality, there is a little gap between the pieces in the first rectangle and a little overlap in the second.
Simon has been inspired by Mathologer to build this.
Simon explains: “Induction is a mathematical term, type of mathematical proof, if you have a couple of base cases (n base cases), then the inductive hypothesis implies that for the previous n values the statement holds. It proves that if the inductive hypothesis is true, the next value will also hold”.
Below, Simon used induction to prove that “any Lucas number and Lucas number after that divided by 5 equals the Fibonacci number between the Lucas numbers”:
“If I get the next two digits right, I’ll be ecstatic!” Simon says, as he hurries on with a φ (Phi) approximation algorithm using Fibonacci numbers. He keeps dividing every following Fibonacci number by the previous one and eventually gets quite a good Phi approximation a precision of 6 digits! This experiment is inspired by Mathologer, who applied this algorithm to approximate both Pi and Phi and show how “wildly less irrational Pi is than Phi”, Simon says. Simon calculated more terms for Phi though.
Simon saw this proof on the Numberphile channel.
This has probably been proven before, but Simon likes to come up with his own proof. Here he uses proof by induction, that is a proof that proves that some property holds for all natural numbers.
Simon came up with this trick today and had Neva solve his riddle: any Fibonacci number is equal to the sum of its surrounding Lucas numbers divided by 5. And a Lucas number is Phi to the n, rounded to the next integer:
Simon made a game out of this (the purple ones are the Lucas numbers and the red ones are Fibonacci numbers):
This is a fun number guessing trick, based on powers of 2 and the Fibonacci sequence, that even little kids can enjoy. You don’t have to know anything about the powers of 2 or Fibonacci to play this game, just basic addition up to 30. Yet, if you are more advanced, it is very interesting to see what lies underneath and even apply binary numbers to your guessing technique. Simon learned this trick from the Numberphile video on Brown’s Criterion.
Simon also made his own version of the game, based on prime numbers:
In this second part of the cool number guessing trick session, Simon shows his own version of the game, based on prime numbers. He discovered that it’s impossible to create this game for all numbers between 1 and 30 because some numbers (4 and 6) cannot be expressed as a sum of two different primes and was very upset about it. Yet he did manage to make the game and it works for all numbers except 4 and 6. To play the game, one player thinks of a number and the other player tries to guess it by asking whether the number is present on different sheets of paper. The answer is the sum of the numbers located in the top left corners of all the yes-sheets.
And please check out Part 3, where Simon actually programmed this game in Java (Processing):
Now it’s the computer guessing the number! The game is available on Simon’s GitHub to download at: https://github.com/simon-tiger/browns-criterion
Simon explained the rules in the GitHub README (because he “has a different writing style than Mom”, he said): https://github.com/simon-tiger/browns-criterion/blob/master/README.md