A New Way to Look at Fibonacci Numbers

finding ways to visually represent mathematical concepts can be a great way for people to further understand those concepts and to get people who aren't as open to looking at equations and numbers to give mathematics a second chance if their first approach in school didn't seem that exciting fibonacci spiral is one of if not the most referenced visualization of a math concept in contemporary culture you probably see it everywhere because it's relatively simply to understand take the Fibonacci sequence make squares that have side lengths as long as the numbers in the series cluster them around each other and draw arcs of a circle throughout and voila you have your golden spiral or perhaps you simply like to flip through the graphs of all the connecting lines also known as diagonals of different regular polygons which trace out mandala like patterns for your personal use or maybe you want to express the numbers in each row of Pascal's triangle as stacks of binary squares to produce vaguely alien-looking shapes that could inspire creatures and enemies for possibly an 8-bit adventure game there's a lot of potential creativity that can go into making math more visually accessible and interesting and it's always good to have more people appreciate math and understand more how it can help them in their daily lives for this video I want to show you another way we can manipulate the Fibonacci sequence to create some more interesting designs these designs are related to a concept associated with the Fibonacci sequence called a Poisson Oh period which I'll walk you through quickly before we get to the actual drawing of the designs so in order to understand Pisano periods we have to understand the modulo operation the modulo operation finds the remainder after you divide one integer by another so if you remember when you learned about division you probably learned about dividing a number let's say 11 by a divisor let's say 5 the result also known as the quotient was 2 because you could subtract 2 fives out of 11 and the remainder or whatever's left over is 1 so the remainder is 1 after dividing 11 by 5 we can write this another way by saying that 11 modulo 5 is equal to 1 the number that we're dividing by to find the remainder which we previously called the divisor is now called the modulus keeping this in mind we can something like 14 modulo 3 we subtract 3 as much as we can from 14 and when we're done we have 2 left over so 14 mod 3 is equal to 2 if we try to evaluate 14 mod 5 we can subtract two 5s from 14 and get a remainder of 4 but if we evaluate 15 mod 5 we can subtract 3 5s and get a remainder of 0 this shows us that the remainders value is always going to be between 0 and 1 less than the modulus we can see this if we took a regular number line and applied a consistent modulus to every number on it let's say 3 we would notice that the resulting sequence would cycle through the remainders 0 1 2 and then repeat itself forever and this should be expected since on the number line we're just adding 1 to each next step you can see how the remainders form a natural loop of sorts and this holds for any modulus you use on the number line [Music] now this is where the Fibonacci sequence comes into play the Fibonacci sequence is made by starting with 0 and 1 and then each next term is the sum of the two terms before it so the sequence continues with zero plus one equals one one plus one equals two one plus two equals three two plus three equals five three plus five equals eight five plus eight equals 13 and so on let's substitute our number line with the Fibonacci sequence and see what happens when we divide by a common modulus of three as before we get another repeating sequence and this time the sequence is 0 1 1 2 0 2 2 1 this also works if we take the Fibonacci sequence mod for the resulting looping part is 0 1 1 2 3 1 and it works for 5 & 6 & 7 & 8 and actually it works for every modulus just like with the regular number line every modulus results in a sequence of looping remainders these sequences are said to be periodic and the length of each looping part of the sequence is known as the pisano period the Fibonacci sequence mod 5 with the resulting loop of 20 turns would be said to have a Bassano period of length 20 if you were to graph the Pisano periods of different moduli you will notice that the periods tend to get larger and larger as the modulus increases though the actual values still fluctuate up and down quite a lot you may be interested in just looking at the lengths of the periods themselves and try to figure out why these lines appear for example if you really wanted to but we want to get back to these designs which are all generated using the fibonacci sequence with moduli of different integers let's start with this group over here remember how before I traced out the sequence for the number line mod 3 I basically put three points evenly spaced on a circle labeled them with all the possible remainders and drew lines between the points according to the progression of the sequence you can see how this would create some pretty regular looking polygons inscribed in circles for any modulus we used but with the Fibonacci numbers suddenly the sequences are more seemingly irregular with the FIB series mod 4 for example we start at 0 hit 1 twice go to to go to 3 then we jump back to 1 again before returning to 0 or starting point it's a pretty irregular looking shape yet if we used the FIB series mod 5s sequence with five points around the outside we jumped from zero to one two one two three back to zero over to three twice then to one four zero four four three two zero two two four one and then finally the sequence restarts this time we happened to cover every possible diagonal which again are the lines that connect any two points with mod six the resulting shape doesn't cover every single diagonal but it still has vertical symmetry and mod eight gives another shape with an asymmetrical design filling the other designs for moduli one through nine seemed to yield a different design for each one then there's the design for the fib series mod 10 with a period of 60 it's a pretty long period compared to the others shown thus far and therefore has a more complex arrangement of intersecting lines yet it still ends up being symmetrical and comes across as a relatively nice and tidy design compared to some of the other ones this one is my personal favorite of course since we have an infinite amount of integers to use as moduli we can generate an infinite amount of designs some of the designs have many connecting lines due to having a longer Pisano period and some are really simple some manage to pull it together and create a nice symmetrical balance while others descend into a chaotic mess and while the designs tend to get more complex as the periods tend to increase there doesn't seem to be much rhyme or reason as to why they're so different from each other or what determines if a design is symmetrical or not at least not at first however during my research on these designs which included watching a few great videos by the channel number file which you should totally check out for supplemental material links are in the description I picked up on the fact that each looping mod sequence contains either 1 2 or 4 zeros now there are proofs that exist as to why the number of zeros is always 1 2 or 4 but I was more interested in their correlation with the symmetry of the resulting designs I noticed that sequences with 4 zeros always seem to make symmetrical designs sequences with 1 0 always made asymmetrical designs and sequences with two zeros seem to go either way so perhaps it would be nice to figure out why that is and see if there's a way to prove that those patterns hold up for any integer modulus that we use the other thing that I noticed after a bit is that while these designs look mostly disconnected from each other on the surface there are some designs that are actually quite similar and only differ by a couple of added lines for example the designs for mods 13 34 and 89 all are almost identical and so are the designs for mods 821 and 55 you might notice that all of those numbers are themselves Fibonacci numbers and the designs alternate between the two sets as you count down the line so perhaps there's a pattern forming here with these designs one subset of which looks like it's the other but copied and flipped over onto itself it seems too good to be a coincidence you may also notice that the same thing holds true for mods 11 29 and 76 as well as 7 18 and 47 these numbers are actually all part of another sequence in math called the Lucas numbers which is a sequence that is generated by the same rule as Fibonacci but instead starts with 2 and 1 instead of 0 and 1 and again I think it's pretty easy to tell the similarity that these two groups of designs have with each other in the visual sense unfortunately there doesn't seem to be many other ways of looking for visual connections between the rest of the designs at least from my point of view at the moment however maybe you will find similarities in the rest of the shapes that reveal more hidden ways of grouping these designs together but I could not find let's step back a bit and look at the big picture for a second we're essentially picking and choosing which points we connect in potential polygons for all of these shapes and the thing that mainly determines how the lines will turn out besides the modulus is the sequence of numbers itself we might get this shape through the Fibonacci series mod 10 but what if we changed our base sequence from the Fibonacci numbers to something else if we decided to use Lucas numbers mod 10 for instance that would give a different set of remainders that happen to loop but it makes a different design the same would happen if we use triangular numbers which use a completely different rule of producing terms compared to the Lucas or Fibonacci numbers the rule for the triangular numbers is basically just to always increase the difference between two successive terms by 1 if we decided to use prime numbers the sequence of remainders does not loop yet it just happens to contain mainly only 1 3 7 & 9 for whatever reason and of course nothing's preventing you from just picking random remainders and seeing what shape they happen to make the point is that using the fibonacci sequence is just the ingredient we decided to use in the recipe for making cool combinations of diagonals and maybe the Fibonacci sequence is really special to you and for that reason the specific designs you get from using the Fibonacci sequence hold more meaning to you but this also lets you know that you can extend this concept of using these modular frameworks to interpret various other sequences not just Fibonacci numbers this can open up a whole new world of exploring why different moduli in combination with different number sequences make specific designs one last thing that I want to show you before we move on to other things I want to show what happens when you substitute the regular Fibonacci sequence which starts with 0 1 1 with a Fibonacci sequence multiplied by another number 2 for example so all the terms are now twice as large when you make designs for the different moduli of this sequence you'll get a bunch of designs that we had before but you'll also get several new ones in fact you can probably see that all the designs of the regular sequence are still there but they now appear in every other modulus used if we were to use the Fibonacci sequence multiplied by 3 to start the resulting designs of the moduli would show the original series cropping up in every third design likewise if our starting sequence was multiplied by 4 we would expect that the original fib sequence modulus designs would appear every 4th design and indeed they do the designs centered around the fib sequence times 2 also appear every other design in this case it is almost like we're dealing with fractions of the original moduli with the way that these designs appear in relation to each other if in the original fibonacci sequence we were to label each of the modular designs with the number correlating to the modulus used we would just get a list of integers for the labels so when we switch to using the Fibonacci sequence times two we might be tempted to label the first one here with a number of 1/2 and the next new one 3 halves but we should be careful about this as far as I can tell moduli can only be integers so in labeling these modular designs we should be careful not to say that they were produced by evaluating the Fibonacci sequence quote mod 1/2 or mod 3 halves instead I think it's better to label these designs more like design one designed to design 3 over 2 cetera to remove a bit of faulty connotation in this designation the top number is the modulus used and the bottom number is the multiplier of the original Fibonacci series and with an infinite amount of integers available we can basically make as many designs as there are rational numbers that is the bulk of work that I've done with these type of designs there is another thing that I'm currently working with attempting to match up circular designs with fractional approximations of some common irrational numbers but I might leave that for a follow-up video right now I want to make sure that we don't forget about the second group of designs that I showed you at the beginning of this video these designs are also built off of modulations of the fibonacci sequence like before but the rules for drawing the shapes are obviously pretty different for this we're translating these modulated sequences into paths where every step of the path is either a left turn and then forward one unit if the term in the sequence is odd or a right turn and then forward one unit if the term is even if the term is zero we don't turn or move at all I've arbitrarily decided that the starting direction we face is to the right and since every sequence starts with zero nothing happens let's use the number line as our reference sequence for a second since the number line alternates between even and odd numbers the resulting graph is going to first turn left and move for number 1 then turn right and move for number 2 then turn left again for 3 then right again for 4 and so on forever if we use the original Fibonacci sequence we end up cycling through the sequence of left turn left turn right turn because the Fibonacci sequence is numbers cycle through odd odd and even I'll let you figure out why that is interestingly enough the resulting design we get also loops on itself and as a bonus it just happens to look like the addition symbol which is fitting for Fibonacci numbers I suppose but now when we start substituting in modded Fibonacci sequences we get many different sequences of even and odd numbers which will lead to many different designs it's interesting to note that unlike in the first batch of designs we looked at several moduli will result in identical designs some designs are completely contained and loop on themselves while others repeat a pattern that has a general direction heading off to infinity so far I haven't looked into what may in flu one modulus to produce one type of design while another modulus produces something completely different and just like with the circular designs we looked at before you can substitute your beginning sequence in with anything and produce more designs that way for whatever reason it's important to take away that all the stuff we're doing may not result in incredibly complex or thought-provoking designs but the fact that these signs are motivated by a mathematical understanding is what makes them special it's also important to note the artistic and creative significance working with things like this has hopefully this video can be used as a springboard for you to think about how math might motivate designs in your everyday life and how the application of a certain instruction for building designs can be expanded upon to create more and more interesting artwork and cool visual things to look at it can be seen as a different way of making art in that it's much more procedural and you basically are creating a set of instructions that produces unknown final results again in many ways to the work of conceptual artists like Sol LeWitt or maybe you disagree that making designs based off of purely mathematical data is even art at all maybe math visualizations are in a class of visual performance all by themselves how does this discussion of visualization and artwork connect to other types of data and visualizations for things completely unrelated to core math concepts hopefully I'll have more to share regarding what we looked at in this video in the future please leave comments about what you discovered relating to anything talked about in this video whether they be mathematical proofs or patterns linked to the concepts discussed or other ways to expand the visualization process I'll be very interested in what we can come up with thanks for watching

Channel: Jacob Yatsko

Views: 146,389

Rating: 4.9562387 out of 5

Keywords: fibonacci, fibonacci sequence, math visualization, numberphile, mathologer, modular math, modular multiplication, pisano periods, pi

Id: o1eLKODSCqw

Length: 15min 50sec (950 seconds)

Published: Sun Feb 23 2020

Reddit Comments

[u/MrSnuffle_]

Wow that guy only has 8 subscribers

[u/TheDeviousLemon]

This video was so well done. I feel like this guy is going to be big one day.

[u/ghoof]

Excellent!

[u/A_ARon_M]

Building an app to draw these would be a fun project in Python or Matlab.