Phi and the TRIBONACCI monster

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Today's video is about explaining a lot of the miracles associated with the golden ratio phi, the Fibonacci sequence and the closely related tribonacci constant and sequence.

Featuring the truely monstrous monster formula for the nth tribonacci number, the best golden ratio t-shirt in the universe, rabbits, mutant rabbits, Kepler's wonderful Fibonacci-Phi link, Binet's formula, the Lucas numbers, golden rectangles, icosahedra, snub cubes, Marty, a very happy Mathologer, etc.

Special thanks to my friend Marty Ross for some good-humoured heckling while we were recording the video and Danil Dimitriev for his ongoing Russian support of this channel.

Also check out my other videos featuring the golden ratio and the Fibonacci numbers.

Enjoy!

[nitsanbenhanoch]

"you're allowed to do this?"
"who's gonna stop me"
<evil laugh>

[AttilaAsztalos]

Please, never stop making these vids! Each one feels like it's Christmas... :)

[a-wx]

'Wait, you're allowed to do that?'
'Who's gonna stop me?'
I love this.

[The-cyber-imbiber]

As a graphics artist, whenever I am trying to make components in a scene fit together in an aesthetically pleasing way, I will often adjust their scales and proportions by phi. It feels extremely satisfying to actually know a bit of the maths behind this beautiful constant!!! Thanks for the awesome video.

[robertdibenedetto]

At around 5:40 you mention "cheating a little bit" with the rounding for the first number. Instead of rounding to the nearest whole number, you can "round down, round up" successively, and you'll reach the correct integer every time.

[Antediluvian]

This was brilliant - love how it was laid out. A lot of information at first and then carefully breaking down how an equation like this could be discovered.
Marvelous work put into these videos, much thanks!

[eshel]

3 fastest things in the universe:
1. The speed of light
2. The expansion of the universe
3. Me clicking on a new mathologer video

[daiconk]

The relationship between the icosahedron and three golden rectangles is my favorite detail from any of your videos, and that's saying a lot. What a beautiful geometric connection. Thank you.

[Mathologer]

Got distracted again and just had to make this video about my tribonacci friends. Will be interesting to see whether anybody comes up with a nice answer to the puzzle about the mutant tribonacci rabbit population at the end. Anyway, enjoy :)

[whygreen]

I think I have a solution...


You have three stages in the life cycle of the Tribonacci bunnies:
1. The adolescent bunnies have no babys
2. The mature bunnies have 1 baby
3. The extra-mature bunnies have twins!

[alexhancu]

See? This is why I love maths. People might think it's all just adding and subtracting and all messy stuff, but if you get deep enough, you start finding beautiful patterns that amazingly link with each other in many ways, the Fibonacci numbers and the Luka numbers being one of many examples

[Iridiumalchemist]

Of all the mathematics popularizers I follow on the internet, I think you do the best job at combining rigorous mathematics and sound explanations. Keep up the great work! (I'm doing my PhD in number theory at UF right now- so it's enjoyable to learn things far afield from what I'm doing.)

[expchrist]

Anybody who does not agree that this mathologer video is wonderful does not have a soul!!!

[blue_tetris]

Is there a general case for the n-onacci sequence? That is, a sequence of integers where you start with 0 ... 01, then add up the most recent n digits to get the next in the sequence?

[yxlxfxf]

15:15 it's actually quite easy to see why the formula always spits out integers. Just by expanding both terms, we can notice all even powers of sqrt(5) from the 1st term will cancel with the evens from the second one. The remaining odd powers will all be some power of 5 multiplied by a factor of sqrt(5) which cancels out and the remaining numerators will be even (because adding two equal numerators), being divisible by 2.

[buckleysangel]

You’re not cheating by rounding. You’re adding or subtracting one over phi to the same power as phi! So beautiful

[kellsierliosan]

Hey, I've got to say that your channel is the closest thing to mathematical disneyland in youtube, if not in the internet as a whole. Amazing work, seriously! :)

[Fassislau]

Omg this was SO GOOD !!!! Hands down best mathematics video in the internet. Thank you so much for this. I absolutely loved it.

[voltrevo]

16:17 In general, you can derive formulas for any sequence of the form s_n = A*s_(n-1) + B*s_(n-2) using a similar technique. Step 1: suppose that s_n = X^n. Step 2: Derive a quadratic for X to obtain Xa and Xb. Step 3: Observe that s_n = C*Xa^n + D*Xb^n satisfies the definition of the sequence for any values of C and D. Step 5: Substitute two known values (probably the first two values) of the sequence to obtain two equations in the two unknowns (C and D). Step 6: Solve for C and D. Step 7: Now that Xa, Xb, C, and D are all known, s_n = C*Xa^n + D*Xb^n.

[JCOpUntukIndonesia]

I have ever thought about this variant of Fibonacci Sequence, also, how the ratio would be, but I didn't know that it already existed. Your explanation is on spot prof. It is easy to understand, and thank you for it.