A New Way to Look at Fibonacci Numbers
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- Опубликовано: 16 июл 2024
- A look at how Pisano periods and the modulo function can turn the Fibonacci sequence into strange and fun visual designs.
More links:
Interactive circle designs by towerofnix: www.khanacademy.org/computer-...
On-Line Encyclopedia of Integer Sequences: oeis.org
Fibonacci Sequence (A000045): oeis.org/A000045
List of Pisano Periods (A001175): oeis.org/A001175
PICTURE CREDITS
Lewitt, Sol. "Wall Drawing #33." 15 bytes. Artists of Utah, Kiki Karahalios, 27 Feb. 2019, artistsofutah.org/15Bytes/index.php/43029/.
Lewitt, Sol. "Wall Drawing #51." MASS MoCA. n.d., massmoca.org/event/walldrawing51/.
Lewitt, Sol. "Wall Drawing # 1115." Rice University News and Media Relations. Katharine Shilcutt, Estate of Sol Lewitt / ARS, 1 Nov. 2019, news.rice.edu/2019/11/01/rices-glasscock-school-to-display-sol-lewitt-drawings/.
u/julekca. "[OC] Blade Runner 2049..... represented by 1600 captures of the movie. Each of these is resized to 1px wide and extracted with the same time interval." r/dataisbeautiful. Reddit, c. Oct. 2019, www.reddit.com/r/dataisbeautiful/comments/d7nw9p/oc_blade_runner_2049_represented_by_1600_captures/.
VIDEO CREDITS
"Fibonacci Mystery." RUclips, uploaded by Numberphile, 18 Sep. 2013, • Fibonacci Mystery - Nu... .
"The Golden Ratio (why it is so irrational)." RUclips, uploaded by Numberphile, 8 May 2018, • The Golden Ratio (why ... .
"Times Tables, Mandelbrot and the Heart of Mathematics." RUclips, uploaded by Mathologer, 6 Nov. 2015, • Times Tables, Mandelbr... .
MUSIC
Used with permission:
Yombai - Karmic Tea
Licensed under Attribution 3.0 Unported (CC BY 3.0) creativecommons.org/licenses/...
- Louigi Verona -
Window in the Sky (parts 3 & 5, edited)
Mirrors
Patterns on a Silk Cloth Lit by the Morning Sun that Reflects off of a Newly Printed Map of Australia
Silk Companion
Licensed under Creative Commons CC-BY 4.0 creativecommons.org/licenses/...
- Seclorance -
Xinachtli
Autumn (Still Life) 
Наука
Some notes and responses to common questions:
- The video was made using Adobe Illustrator and After Effects. I would not recommend doing a similar video this way, as it requires laying out every shot perfectly beforehand and animating every line more or less individually, rather than relying on a coding background (I have basically none) and a program that could simply generate the animations instead. A drawback of the way I did it is stuff like the missing line in the decagon that people have pointed out at 0:50.
- Despite looking similar, I assure you there's no connection between the 10 graph and the Brilliant logo :D. (Also, what do we call these images? Designs? Graphs? Patterns? Symbols? Let me know what you think)
- Could this be done in 3D? I'm not exactly sure. You could pick a point on the sphere to start, but how do you go about distributing the rest of the points on the sphere, in a regular pattern? It's easy to do it with a circle because you just go around the circle. But with a sphere, you have to choose between two axes of movement.
- Thanks to everyone who reassured me that the mod operation can apply to fractions as well as integers!
last reply
The golden spiral and the fibonacci sequence are time scale from the floor plan of the temple mount in old Jerusalem. Newton.
Peace and Ahev
When you mentioned the problem with 3D I immediately started to think about the 360 HSL color range and how to do art with it. Thank you!
there's a python library for mathematical animations called Manim, 3Blue1Brown himself was the developer
They are patterns, so let me put it like this.
All designs are patterns, but not all patterns are designs. There is a difference between making a pattern and discovering them.
I don’t know why I’m getting this as a suggested a year later but I ain’t complaining
@Hakan hasşerbetçi yes
Hm
Same here.
Neither am I.
I'm here a year after your comment, I'm not complaining either 👍✌️🌍☮️
This is a hidden gem.
Secrets shhh
@Amanda Lane No secrets here. Its the Mandela Effect.
@@melvo_ke hei. what do you mean ? how is it related to mandela effect?
Hidden geom*
6:18 This mod 10 design was brought to you by, brilliant
I bet this video will inspire a lot of tattoos in some math enthusiasts around the globe
Will look forward to put something like this on my skin. Thank you for the help!
I will never understain tattoos
@@ranua9327 your telling me that i can have a design I made, that makes me happy when I look at it, on my body? becoming the art rather than just being the artist??? yeah sounds sick af sign me up
I’m an artist and have been focused on rotating objects and the visualization of mathematical patterns for my entire life. The information in this video absolutely provides the most inspiring information I’ve ever come across, thank you!
do U knoW About 369🕉🕉🕉
EyE "ITs''... A seCRet SeerEt 🕉🕉🕉
Can thou hintest me of THE program to use for making animations, geometrical animations?
Wait until you find out about quaternions! :)
Hi. I discovered these exact patterns a few years back and it feels strangely validating to have someone else discover them too. I would like to be the first to have discovered these things but that's highly unlikely since there's nothing new under the sun. Let me recommend that you stop limiting the periods to bouncing within a circle and give them the angles of a triangle or a pentagon or a hexagon. Whatever polygon you like. You will see some very beautiful and awesome line designs, there's one that even looks like a profile of a brain. It's fascinating. Also, I used a different kind of modulo that does not allow zeroes to be produced, I guess you could say it is an 'inclusive modulo' since it produces the dividend if the divisor fits exactly. Be careful though, you may lose many many hours watching the designs produced:D
@Ben Romero Wow I really wish I fully knew what you were talking about because it sounds fascinating
Could you share your findings in a video please 😊
^
@@Rudxain I'm very happy to hear that! Please let me know when your programming is running. I unfortunately don't have the skills needed to build a visualization tool.
@@Rudxain Hey! if it's just a passion project, I am looking for open source projects to contribute to. I have experience in mathematics up to group theory and have built apps/websites in may languages, let me know if you want to work together!
Any way add a third dimension? It would be interesting to see some of the irregular designs in a sphere.
Have a look at some of Simon holmedals work. He uses alot of vector math based equations in houdini to create some insane 3D stuff.
Maybe use the third dimension if anytime a number visited multiple times?
Maybe something complex?
That's not possible and we don't talk about that here
@@clementello That's right...because the earth is flat :)
5:30 When you just want to do mathematics but accidentally start summoning a demon.
Here's a prof who accidentally summons one with an equation (Twilight Zone) ruclips.net/video/BoQ6ZC8EUQ0/видео.html
Exactly... But even deeper than that, is there a force in nature that involves that part of the spiral of the sequence to form that pattern in our brain or neural fibers upon receiving certain electrical signals or frequencies?
@@antiprismatic Creating tricks in our brains people are gonna start to say we are getting crazy or using crack.😂
Hi Jacob. I found some interesting ones.
Just woow:
For mod = 675
and every [fib+fib] * 947
with a fib start position of 6,7
Butt/Mushroom:
For mod = 2529
and every [fib+fib] * 2
with a fib start position of 0,1
eye:
For mod = 2529
and every [fib+fib] * 2
with a fib start position of 2:2
Infinity mandala:
For mod = 376
and every [fib+fib] * 2
with a fib start position of 2:2
Regular mandana:
For mod = 688
and every [fib+fib] * 662
with a fib start position of 2:8
I also made an online demo where everyone can experiment with values I tried linking it before but it didn't work, will now try in the reactions of this comment.
rutgerklamer.nl/maths/fibonacci_modulus/
Nice. My brother coded a program for me to do this 30 years ago when we were in high school, sonifications too! Your GUI is much better, though. ^_^
If you still have that online demo (and can't link it here) could you link it somewhere on your channel page? I really want to find it but I'm struggling to find it on Google.
Link is gone.... :(
Can you say the name of the site without using a link?
9:42 It contains 1, 3, 7, and 9 *because* the chosen mod is 10. Except for two and five, all of these numbers are coprime with ten-because primes are necessarily coprime with every number that isn't a multiple of themselves. Two and five are the *only* exceptions because they are the factors of ten.
The more lay explanation is the straightforward realisation that numbers ending in 5 are divisible by 5, and even numbers are divisible by 2, and therefore neither type (excepting 2 and 5 themselves) are prime candidates.
All primes , after the single digit primes, end in 1, 3, 7, or 9 so a multiple of 10 will always have one of those 3 numbers as the remainder.
This is one of the most interesting, math related videos I've seen in a while. I love these types of math visualizing videos, so I hope you continue making them!
10:14 I actually used this framework a couple years ago to solve an interesting puzzle I came across at a conference: “arrange the digits 1 through 16 so that every pair of digits sums to a perfect square.” I used this visitation method to find other sequences of digits, 1 to n, for which this is possible, and their respective solutions. Turns out they’re connected to Pythagorean triples, and the visitation of all possible sequence of digits makes nice parallel lines.
Don't rearrange the counting numbers, take them in sequence. Sum the first number (1). You get 1 or 3 to the zeroth power. Sum the next 3 numbers. You get 9 or 3 squared. Sum the next 9 numbers. You get 81 or 3 to the 4th. Sum the next 27 numbers. You get 729 or 3 to the 6th. Sum the next 81 numbers. You get 6561 or 3 to the 8th. So the number of numbers you sum is the next power of 3 and the result is the next even power of 3. Lots of patterns to find.
@@jayspenceranderson I feel like I know the basics of how this might work (for any succesion of 3 numbers, adding them together will always be divisible by 3 because their mods will be 0, 1, and 2. You add 0, 1, and 2 together and it's divisible by 3), but I have no idea why the rest of it would work. Like, why the nth power specifically? I'm sure there's a perfectly reasonable reason which could be shown in a formula, but I don't get it lol.
@@trickytreyperfected1482 It works because the median (and so the mean) of each succession of numbers is 3^k. And since each succession is length 3^k, its sum is (3^k)^2. This pattern holds for any odd base ≥3, but base 3 is unique in that each succession lines up nicely with the last one. For larger bases, the pattern is offset. In base 5 for example: [1], [3, 4, 5, 6, 7], [13, ... 25, ... 37], [63, ... 125, ... 187], et cetera. Honestly, I'm surprised I never noticed this property of powers of three until now!
This relates to something I have a personal fascination with: balanced numeral systems. These are number systems with both positive and negative digits, centered around zero. So balanced base 3 has the digits [-1, 0, 1], balanced base 5 has digits [-2, -1, 0, 1, 2], and so on. I first noticed that the base 3 pattern was simply counting in balanced ternary, with each succession of numbers being all positive k-digit numbers. This made it quite obvious to me why the pattern behaves as it does. In larger bases, the pattern doesn't cover every k-digit number, only numbers with a leading digit of 1, which is why some numbers are skipped.
@@areadenial2343 I'll need to revisit this comment when I'm not as tired. And once I've rewatched the video because apparently it was 2 years ago and I've forgotten the context since.
Hello I have something to add on that I have thought of, and pardon me if you had already noticed, but in all of the shapes of the mods, all polygons outlined by the lines in the different mods all seem to make triangles. Maybe figure out a pattern in the variations of degrees that may relate to the fibonacci pattern itself? Like figuring out the laws to the fibonacci sequence, which I think of like a factor. The fibonacci sequence, something about it makes me think about factors. Not coming to mind right now.
First thing in my recommended after waking up in the morning. I absolutely loved the style and message of it. Looking forward to seeing more beautiful productions like this.
I slightly chuckled when I saw the words "Pisano period" was written in red text over a yellow background.
I will never grow up.
I don't get it can someone explain
@@4ltrz555 Think "urinano menstruation".
@@egilsandnes9637 oh lmao
indeed
I heard beavis in my mind
This is excellent both in concept and execution, thank you. I’ve been drawing patterns like these for years but without any sophisticated math(s) underpinning. I will be experimenting with the generative sequences you have described so clearly.
Yes! This is exactly the type of math visuals I have been sketching for some time now, mostly experimenting with star polygons. I'm so happy this was recommended to me. Great work, you have opened me up to new knowledge!
Love that you had the Lateralus reference in there at 10:23. Great job. Thank you.
First of all, since you were wondering: this was in my youtube recommendations
Second of all, wow. This video was amazing. I can see just how much effort you put into animating everything and I’m honestly shocked it has this little views.
Keep it up!
0:50 the mandalas - I love how the even numbers have a centerpoint, while the odd numbers have a center area/polygon. I never realized that until now, thank you.
Hey man you are setting a great example. I appreciate that you are inspiring people to explore in new ways and not just giving answers - I don’t want to find the answers to life’s mysteries in a RUclips video. I say, let people discover things on their own - that path is sacred. 🙌
On a side note, towards the end of this video you mention the Fibonacci series X2, etc.... I use this idea extensively in setting up modular compositions. There is a particularly elegant group of these multiples which can be used simultaneously - a fruitful rabbit hole to explore and interesting lessons to learn there . I call these the “Fibonacci Canons”. And again, thank you for doing it right. I shall subscribe!
- tommy
Do what's right for YOU - don't make decisions for others
If you don't want to find answers thru You tube or any other medium that is YOUR choice and let it be yours alone
Was that aimed at me? Cause it sounded like you have an issue with my comment. Perhaps something was lost in translation. I was simply admiring this persons method of teaching. If you have an issue with me please let me know.
@@TommyHoppeArt "I don’t want to find the answers to life’s mysteries in a RUclips video. I say, let people discover things on their own - that path is sacred" and then in the rest of your comment it seems you do like learning things via you tube
Whether a person with a full beard stands before you and talks or talks thru a video makes no difference
There is a lot to learn just by watching and listening - maybe not right for you but please don't speak for others
@@ramaraksha01 Fair enough. Good luck:)
The pedagogical outlook expressed in the introduction actually hooked me.
Multiple/alternative modalities, recognition of many possible representations, lovely! Math content that treats students as curious humans rather than the "show your work" automata i recall from my school days.
Thanks for sharing! One of the most interesting patterns I have found related to phi is Penrose tiling.
THAT is a great explanation of modulo
This video was amazing. I loved it so much. Best video I have watched on RUclips in a while. Please let me know if you have more video like this!
This is the video I need! Thank you foe the in-dept explanation of fibonacci sequence, an oddly favorite sequence!
While exploring my interest in number theory, I was trying to think about what Fibonacci numbers would like like under mods. I saw the odd repeating patterns and decided to do some research, finding pisano sequences and then later stumbling upon this video. This was very insightful and I have learned a lot from this, one of the best math videos I've ever seen. Nerding out so hard to this one
My new favorite channel 🤩 thank you so much 🙏. Amazing work here!
Almost sixteen minutes of bliss. Superb fun, highly creative. Thanks uploader!
LOVING that you added TOOL!
Having all the circels on a big poster would look sick!
I would spend many hours just studing the designs... hypnotizing!
Thank you for explaining the Pisano Period. This is yet another concept that I discovered independently while thinking about math, along with continued fractions, integer partitions, Hasse diagrams, and rep-n-tiles.
This needs way more views. Blew my mind
This is fantastic, just beautiful. Thank you!
Absolutely fascinating! I had just been wondering about creating digital art and this hits quite the spot
Wow, great video! You've provided me with lots of food for thought. I'll have to explore some of these designs myself and see if I can come up with some new results. Thanks for sharing this information!
This is the same hobby I do! (Exploring math especially visually) I plan on making some math videos but Ill probably make a dedicated channel for them.
Applying a modulus to an infinite sequence is such a brilliant idea, glad I saw it! Love this video a lot!!
I just love stuff like this! Great work!
Glad I found this channel by accident...keep up the good work!
This is so good, been trynna find a way to get back to math. Amazing stuff! Hope you grow fast!
I came up with an interesting symmetrical Fibonacci-generated pattern along a similar route once.
I started with a 10x10 grid of squares numbered 0 to 9. I then colored in every square whose coordinates corresponded to a Fibonacci pair of numbers, mod 10.
So the first few squares I colored in were (1,1), (1, 2), (2, 3), (3, 5), (5, 8), (8, 3), (3, 1), etc.
What I ended up with was a pinwheel-ish pattern that was beautifully unexpected.
I tried to pursue this further to a 100x100 grid, but I was doing it by hand and didn't get very far, to be honest.
I've always wondered if other mods made the same cool pattern that the mod-10 one did.
@@jacobyatsko Thanks for the share.
You can totally take mod fractions! You just can't factor them, and they don't behave as nicely under stuff like exponentiation. It all depends on what field of math you're working in - number theory, where moduli live, is usually only concerned with integers (and integer-like objects) anyway.
Damn I at the end I looked at the subscribers expecting 300000+ , keep up the good work!
Great video, made me understand pisano period
The fact that the Fibonacci one makes a plus sign is so incredible to me.
So cool! I’ve been working with Fibonacci in rings for years, not having any idea about Pisano!
I came up with another visualization technique - rather than treat each pair of numbers as a line, treat them as Cartesian coordinates. So mod 13 gives you a 13x13 grid, color in the coordinates as they come up in the series. Then, if you start with a different pair (say Fibonacci x2, or Lucas), it will fill in either exactly the same squares, or a completely different, non overlapping, set of squares. Keep going, and you can tile the square with a small set of nonoverlapping patterns. Striking symmetries appear with prime modulo bases!
And with the Tribbonaci series (and variants) you can tile a modulo cube with symmetrical, no overlapping patterns as well...
@@snotgarden4423 Interesting ... I'd be curious to see some examples ... .
@@TheSwircle987 Not sure how to share contact info on YT, but you can find me on twitter @billandtuna , I'd love to share!
You might be interested in the paper "Symmetries of Fibonacci Points, Mod M" by Flanagan, Renault, and Updike, if you're not already familiar.
Looking at the graph of modulus to Pisano period length reminded me of the output of my master's thesis/research on strongly non-repetitive sequences. They look surprisingly similar!
at 9:42, the reason that (ignoring the first 3 numbers) it’s always 1,3,7,9 is because mod 10 is the same as only looking at the last number, and all prime numbers after 5 only end in 1,3,7,9 due to the fact that ending in an even number makes it automatically divisible by 2, and ending in a 5 makes it divisible by 5
I'd say that this is very much art. The procedural nature of these designs reminds me of the the Library of Babel. The creativity doesn't lie in the procedure itself, but rather finding it among the infinite sea of other ones.
Excellent! I deeply appreciate your hard work and so very interesting and rare information! Keep up the good work!
"Tell him that this video is lit, Johnny."
Ah yes, finding this comment would complete my any% jojo refrence comment speedrun in 5.27 seconds
love that Tool reference, lol, great video
thanks for doing this! i appreciate your efforts very much. in community college, i submitted a spirograph drawing for display. they’re beautiful and remarkable. it was accepted. so, it’s art.
love,
david
What a time to be alive and curious. Thank you for sharing your work
RUclips recommended this to me randomly. Fascinating stuff!
I didn't know I needed to know this.
Really cool.
your animations and explanation are awesome!
This video is awesome. Thank you very much
this is the best video i've watched in a long time. thank you
Incredibly well crafted video!
V interesting, lot of cool open problems too! I may come back to this...
This is actually related to and helps understand polyrhythms and cycles of rhythms in general in music a lot more efficiently.
Amazing video and depth of content. Thank you.
9:44 The only possible remainders are actually 1, 3, 7 and 9, because since we’re dealing with prime numbers, suppose p = any prime number, p/10 will always give an uneven remainder inferior to ten, and the reason we don’t get 5 (the only missing uneven number) is because all numbers ending with 5 are multiple of 5. Therefore, we can only get the remainders 2 and 5 at the beginning (2/10 = remainder 2, 5/10 = remainder 5)
Great stuff, lovely graphics
I quite enjoyed this. Good job!
That egg art piece you made is sci fi af and incredible!!
Thank you for creating and sharing this excellent video.
Thank you for making this!
The way I use to obsess over the triangular numbers in my own head without ever knowing there was a name for it.
Thank you so much for uploading this video. It is helping me to get through the pandemic!
Great video, and interesting new perspective to better understand the mystery. 😊
I deliberately searched for fibonacci sequence looking for which items I could apply this sequence to, mostly which plants. Most of the videos appeared lecture-oriented or copy and pastas of other content in a v ambiguous higher-power way. The title and visual both are why I clicked on this one. I'm in a math class that touches on this and I want to expand my breadth of understanding how this connects and to what. Thanks for transparency on how this was made, too.
Totally awesome. Right up my alley.
Man this must've taken FOREVER to make with adobe software. Bloody amazing work, it looks beautiful! And very clearly explained
Fascinating analysis.
Excellent! Good results to you in your next projects...
Thanks, this helped me understand some of the patterns I saw in the trigonometry of an 11-pointed star
This video really made my night.
Thankyou for making sence!
4:32
To be honest, I'm more interested in those outliers around 250, 620, and 740.
Take a peek at around 990 too! These periods are really fascinating
This video is wonderful! Very inspiring.
Nice work ! Keep going !
Thank you for a fascinating video -- indeed, it was exactly as promised: "A New Way to Look at Fibonacci Numbers" and i found it quite thrilling.
13:00
I would say 0 is an even number, you can decide it by 2 and get a whole number (0/2=0) and it is surrounded by odds (1&-1) that makes it even. So how would the shapes differ if you counted 0 as an even number instead of ignoring it? Maybe it gets even more beautiful results
Awesome and clear.
Very nice! Was recommended this video by RUclips after I watched a Numberphile video.
I have only one word: beautiful.
Cool art!!! Awesome video!! I subscribed!
Amazing! RUclips just randomly suggested this to me today and I'm really glad! I'm taking a class on Complexity Science so I have been watching a lot of videos involving the Fibonacci sequence and Mandelbrot sets. Seems the algorithm actually helped out me here.
✨ . . .. ... ..... ........ 💕🥰✨
Excellent video! I'd be curious to see what would result from looking at the designs' inverses, that is to say, the connections that /aren't/ made might give some insight on those that are.
BIG THANKS.. recently started the journey of Generative art.. this is very helpful
this is brilliant! you need more subs.
Ivan Jones often intelligent people are known not to sub or provide a thumbs up. They simply observe and consider. Then move on.
@@KateYagi I needed those wise words, thank you kind person.
@@donvandamnjohnsonlongfella1239 I Subbed just because of this.:)
Great Class!
Just wanted to point out, for those who have not seen it, the result at around 9:40 is a result of Dirichlet's theorem.
This is my exact thumbprint, the fibonacci sequence is truly everywhere.
You can extend the modulus operation as a mod b = {a/b} * b, so that {x} is the fractional part of x.
If you aren't familiar with the fractional number of another number, it is defined as x-floor(x), where floor(x) is the only integer n so that x-1
Excellent. Very helpful
Thank you so much for this video 😇
After 3 minutes I know it was gonna be a new sub for me.
Nice work. At least something new worth watching after the bar was set (too?) high by Mathologer and 3b1b !
Congrats for the good work, I’ll be following you from now on.