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!
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’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!
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
@@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!
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?
@@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
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.
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.
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!
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.
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!
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.
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.
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.
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.
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.
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 TH-cam 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 TH-cam 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
I'm interested in learning what type of a program was used in making and displaying the graphics in this video(the animation in the end made me realize this!)?
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)
Interesting video. I have been discovering the beauty of math and how it truly weaves its way thought all of creation. I can already see application in the arts and will be applying this to some musical ideas that I have been exploring. Contrary to what I believed my whole youth, I am finding math to be quite beautiful, useful and not as scary as I thought.
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!
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
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!!
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
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 reminds me of "Spirogragh" that came out in the 60's. I never got bored with it, yet I always felt a sadness come over me. You see, I have always loved math and geometry with a passion, but as it was, the two never loved me in the same way. My brain was never wired for it. As with some people who say that they are a woman trapped in a man's body, similarly, felt I was a mathematician genius trapped in a D- average mind. Nonetheless, this doesn't stop me from enjoying videos such as this one. And if I may say without qualification, this video was wonderful and fascinating to watch. Thank-you for the time you put into it. By the way, TH-cam recommended this video to me.
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.
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
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!
i would like to see the path designs but this time the angle is according to the angular fraction of the remainder and the modulus. so if we did mod 3 and the remainder was 1 then we go in a 120degree angle (anti clockwise) from the previous line. if it's remainder 2 then we go in a 240degree angle (anti clockwise) from the previous line. if the remainder is 0 then we go backwards (360degree angle anti clockwise) from the previous line.
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!
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
Yes, I wanted to "hear" the Fibonacci Sequence in high school about 30 years ago, so I ended up recreating the modular mathematics wheel. I had my brother code a program for me so I could generate more of these sequences, graphs, and sonifications back then. The graphs were an unexpected bonus.
@@luisramirezbass Thanks for sharing, although, yes I have been aware of it for many years. I actually use the OEIS rather frequently since my own mathematical research is highly focused on various types of integer sequences.
Yeah, when I first discovered these beautiful graphs of the Fibonacci Sequence 30 years ago in high school, I wondered if they might somehow be useful in detecting hidden patterns in data ... .
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.
I did this randomly last year and found there is a period, I didn't know it had a name and i did it for integers upto 10 and it was somehow very cool and felt the right thing to do without any goal. I'm surprised how this is really a thing
9:42 The sequence only contains 1,3,7,9 because 2 is the only even prime (i.e numbers ending in 0,2,4,6,8) and numbers ending in 5 are always divisible by 5.
Well this really got my old brain cells buzzing thank you Jacob! The first thing I want to say is that for some years I've been learning and using (privately) a schematic programming system called FLOWSTONE, by Dsprobotics. I'm sure it will be possible to make a program that will generate these visualisations exactly as shown without all the tedious video editing, and I'm going to make that a new project along the lines of specifying a sequence, or even entering the formula for a sequence, and how to modify it. I would like to send you the results once done. The main project I've been using Flowstone for over the last 12 years or so has been developing a music generation system. I basically use more or less randomly selected logic gates to generate three different repeating sequences of numbers between 0 and 7. The first pattern determines the order in which 8 oscillators tuned to a common chord sounds. The other two patterns determine how the pitch of each note is changed by a specified amount. Once I find a pattern that is musically interesting, I use the oscillators to generate further parts to make a complete performance. This video has inspired me to adapt what I've done, to use these sequences you've so beautifully described, instead of my semi-random ones - music generated by mathematics can be surprisingly interesting, dramatic and moving. Again, I would like to send you the results, but I suspect the music version will take somewhat longer than the visualization. Thanks again, Jacob, for sparking off so many interesting ideas.
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.
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.
just a small note, you can absolutely use modular arithmetic with all real numbers, but that tends to be more of a computer science approach. Look into the sawtooth wave; it is a perfect extension into real numbers, albeit not quite as easy to use now that I think of it, the Desmos graphing calculator allows the mod function with real numbers, so that’s a pretty good way to visualize it and hopefully understand it
So what about mapping out modular sequences along a three dimensional fib spiral using the spots where they intersect the cubes as the points for the diagonals?
Something neat about the pattern revealed by the Fibonacci sequence, is that you will get the pattern regardless of the starting numbers. 246 and 10500, 2,134,431 and 12, pick any pair.
This is pretty cool, I wish stuff like this was shown in schools. It wouldn't replace in-depth learning, but it gets students excited about math and makes grasping the concepts way easier
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.
Hi Jacob, first of all great video hats off !! A personal opinion at 08:05 the way the patterns alternate and repeat each second time looks like fractals, same pattern from small to big picture
There is ever so much material here that you all presented, and there is such an elegance in all of it, where do I begin? In several of your scenarios some mods possibly do crossovers, the 5's, the 10's and the 15's are ever so similar or exactly the same. What I noticed about these was in the direct center there was a near perfect pentagon. As we get into what can be noted as "PHI" Fibonacci is the more closely connected to the number 5 more so than any other and also the PHI more so than any others. Also in the three I noted I a triangle imbed in them. Using an onscreen protractor the angles at the base of this triangle were near or close the same degree of the base of the Pyramid at Giza. And I am hard pressed to discover anything Tesla in your presentation. The 3, 6, or the 9. And if any of your presentations could be tied in with "Sacred Geometry" that could fill another 4 or 5 VIDS. For I do see shapes imbedded in your drawings that could apply there as well. Now, I had to come back for I understand "Binary". Could Mod 3 be used to explain "Trinary"?
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.
Jacob - Hi ! I had been previously introduced to the concept of The Fibonacci Sequence and ways you could use it to illustrate the symmetry of math, but before this Vid, I didn't really appreciate it - just enjoyed the beauty ! This is the first presentation on the subject I've been able to follow ! Btw- It just came up in my "Suggestions" - I have more than a passing interest in such things, so will view a Vid on subjects like this fairly frequently. We MIGHT speculate that with the LOSS of SO MUCH of Y-T's Creator Base because of their STARTLING new (over the last couple of years) emphasis on de-platforming creator voices _they don't "like"_, a vacuum has been created that your Vid along with others (I've noticed a NUMBER of OLD Vids coming up lately ... ) have begun to fill . THAT may be why yours has resurfaced. I'm GLAD it DID, but am cautious at this possibly compelling explanation for it appearing in my "Suggestions" . Anyway, ALL the BEST !! -C.
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.
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*
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! :)
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 👍✌️🌍☮️
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!
6:18 This mod 10 design was brought to you by, brilliant
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) th-cam.com/video/BoQ6ZC8EUQ0/w-d-xo.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.😂
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
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 :)
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.
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.
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!
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.
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!
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.
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.
LOVING that you added TOOL!
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.
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?
This video was amazing. I loved it so much. Best video I have watched on TH-cam in a while. Please let me know if you have more video like this!
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 TH-cam 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 TH-cam 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:)
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
Thanks for sharing! One of the most interesting patterns I have found related to phi is Penrose tiling.
I'm interested in learning what type of a program was used in making and displaying the graphics in this video(the animation in the end made me realize this!)?
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)
This is the video I need! Thank you foe the in-dept explanation of fibonacci sequence, an oddly favorite sequence!
0:49 I can't see normal nonagon diagonals after kglw's nonagon infinity
My new favorite channel 🤩 thank you so much 🙏. Amazing work here!
Interesting video. I have been discovering the beauty of math and how it truly weaves its way thought all of creation. I can already see application in the arts and will be applying this to some musical ideas that I have been exploring. Contrary to what I believed my whole youth, I am finding math to be quite beautiful, useful and not as scary as I thought.
This needs way more views. Blew my mind
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.
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
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!!
Damn I at the end I looked at the subscribers expecting 300000+ , keep up the good work!
Almost sixteen minutes of bliss. Superb fun, highly creative. Thanks uploader!
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
Absolutely fascinating! I had just been wondering about creating digital art and this hits quite the spot
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 reminds me of "Spirogragh" that came out in the 60's. I never got bored with it, yet I always felt a sadness come over me. You see, I have always loved math and geometry with a passion, but as it was, the two never loved me in the same way. My brain was never wired for it. As with some people who say that they are a woman trapped in a man's body, similarly, felt I was a mathematician genius trapped in a D- average mind. Nonetheless, this doesn't stop me from enjoying videos such as this one. And if I may say without qualification, this video was wonderful and fascinating to watch. Thank-you for the time you put into it. By the way, TH-cam recommended this video to me.
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.
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
THAT is a great explanation of modulo
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!
This is fantastic, just beautiful. Thank you!
Glad I found this channel by accident...keep up the good work!
Excellent! I deeply appreciate your hard work and so very interesting and rare information! Keep up the good work!
i would like to see the path designs but this time the angle is according to the angular fraction of the remainder and the modulus. so if we did mod 3 and the remainder was 1 then we go in a 120degree angle (anti clockwise) from the previous line. if it's remainder 2 then we go in a 240degree angle (anti clockwise) from the previous line. if the remainder is 0 then we go backwards (360degree angle anti clockwise) from the previous line.
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!
Great video, made me understand pisano period
Having all the circels on a big poster would look sick!
I would spend many hours just studing the designs... hypnotizing!
Now i know where the brilliant app logo came from, ty
same
I just love stuff like this! Great work!
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
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 is actually related to and helps understand polyrhythms and cycles of rhythms in general in music a lot more efficiently.
all mod 12 can be associated with the chromatic scale in music and generate melodic or harmonic structures. Thanks.
Yes, I wanted to "hear" the Fibonacci Sequence in high school about 30 years ago, so I ended up recreating the modular mathematics wheel. I had my brother code a program for me so I could generate more of these sequences, graphs, and sonifications back then. The graphs were an unexpected bonus.
@@TheSwircle987 I don't know if you know this page, oeis.org/webcam, but you can listen to (midi file), the result of any mathematical series:
@@luisramirezbass Thanks for sharing, although, yes I have been aware of it for many years. I actually use the OEIS rather frequently since my own mathematical research is highly focused on various types of integer sequences.
@@luisramirezbass As for sonifying sequences, I tend to do it myself in my own music, but it is a very cool feature of OEIS.
I wonder if this method can used to make sense of data sets ?
Yeah, when I first discovered these beautiful graphs of the Fibonacci Sequence 30 years ago in high school, I wondered if they might somehow be useful in detecting hidden patterns in data ... .
love that Tool reference, lol, great video
Great Class!
What a time to be alive and curious. Thank you for sharing your 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
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.
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.
Thank you for creating and sharing this excellent video.
This is so good, been trynna find a way to get back to math. Amazing stuff! Hope you grow fast!
i wish you were still making videos :(
There will be more!
@@jacobyatsko YESSSSS
0:51 are the bottom right and the bottom left missing 1 line each?
Yep i saw that to
I did this randomly last year and found there is a period, I didn't know it had a name and i did it for integers upto 10 and it was somehow very cool and felt the right thing to do without any goal. I'm surprised how this is really a thing
V interesting, lot of cool open problems too! I may come back to this...
I didn't know I needed to know this.
Really cool.
9:42 The sequence only contains 1,3,7,9 because 2 is the only even prime (i.e numbers ending in 0,2,4,6,8) and numbers ending in 5 are always divisible by 5.
you have less than 7k subs??????? Criminal! This is a fantastic video!
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.:)
Well this really got my old brain cells buzzing thank you Jacob! The first thing I want to say is that for some years I've been learning and using (privately) a schematic programming system called FLOWSTONE, by Dsprobotics. I'm sure it will be possible to make a program that will generate these visualisations exactly as shown without all the tedious video editing, and I'm going to make that a new project along the lines of specifying a sequence, or even entering the formula for a sequence, and how to modify it. I would like to send you the results once done.
The main project I've been using Flowstone for over the last 12 years or so has been developing a music generation system. I basically use more or less randomly selected logic gates to generate three different repeating sequences of numbers between 0 and 7. The first pattern determines the order in which 8 oscillators tuned to a common chord sounds. The other two patterns determine how the pitch of each note is changed by a specified amount. Once I find a pattern that is musically interesting, I use the oscillators to generate further parts to make a complete performance. This video has inspired me to adapt what I've done, to use these sequences you've so beautifully described, instead of my semi-random ones - music generated by mathematics can be surprisingly interesting, dramatic and moving. Again, I would like to send you the results, but I suspect the music version will take somewhat longer than the visualization. Thanks again, Jacob, for sparking off so many interesting ideas.
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.
I have only one word: beautiful.
That egg art piece you made is sci fi af and incredible!!
TH-cam recommended this to me randomly. Fascinating stuff!
Amazing video and depth of content. Thank you.
this is the best video i've watched in a long time. thank you
Totally awesome. Right up my alley.
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.
Incredibly helpful for my current relationship with Fibonacci.
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.
The fact that the Fibonacci one makes a plus sign is so incredible to me.
just a small note, you can absolutely use modular arithmetic with all real numbers, but that tends to be more of a computer science approach. Look into the sawtooth wave; it is a perfect extension into real numbers, albeit not quite as easy to use
now that I think of it, the Desmos graphing calculator allows the mod function with real numbers, so that’s a pretty good way to visualize it and hopefully understand it
So what about mapping out modular sequences along a three dimensional fib spiral using the spots where they intersect the cubes as the points for the diagonals?
This video is awesome. Thank you very much
Why isn't THIS on Numberphile or 3 Blue 1 Brown
Something neat about the pattern revealed by the Fibonacci sequence, is that you will get the pattern regardless of the starting numbers. 246 and 10500, 2,134,431 and 12, pick any pair.
This video really made my night.
This is pretty cool, I wish stuff like this was shown in schools. It wouldn't replace in-depth learning, but it gets students excited about math and makes grasping the concepts way easier
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.
Hi Jacob, first of all great video hats off !!
A personal opinion at 08:05 the way the patterns alternate and repeat each second time looks like fractals, same pattern from small to big picture
There is ever so much material here that you all presented, and there is such an elegance in all of it, where do I begin? In several of your scenarios some mods possibly do crossovers, the 5's, the 10's and the 15's are ever so similar or exactly the same. What I noticed about these was in the direct center there was a near perfect pentagon. As we get into what can be noted as "PHI" Fibonacci is the more closely connected to the number 5 more so than any other and also the PHI more so than any others. Also in the three I noted I a triangle imbed in them. Using an onscreen protractor the angles at the base of this triangle were near or close the same degree of the base of the Pyramid at Giza. And I am hard pressed to discover anything Tesla in your presentation. The 3, 6, or the 9. And if any of your presentations could be tied in with "Sacred Geometry" that could fill another 4 or 5 VIDS. For I do see shapes imbedded in your drawings that could apply there as well.
Now, I had to come back for I understand "Binary". Could Mod 3 be used to explain "Trinary"?
your animations and explanation are awesome!
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.
Jacob - Hi ! I had been previously introduced to the concept of The Fibonacci Sequence and ways you could use it to illustrate the symmetry of math, but before this Vid, I didn't really appreciate it - just enjoyed the beauty ! This is the first presentation on the subject I've been able to follow !
Btw- It just came up in my "Suggestions" - I have more than a passing interest in such things, so will view a Vid on subjects like this fairly frequently.
We MIGHT speculate that with the LOSS of SO MUCH of Y-T's Creator Base because of their STARTLING new (over the last couple of years) emphasis on de-platforming creator voices _they don't "like"_, a vacuum has been created that your Vid along with others (I've noticed a NUMBER of OLD Vids coming up lately ... ) have begun to fill . THAT may be why yours has resurfaced.
I'm GLAD it DID, but am cautious at this possibly compelling explanation for it appearing in my "Suggestions" .
Anyway, ALL the BEST !!
-C.