Huge thanks to Jane Street! www.janestreet.com/join-jane-...
Check out Ben Sparks's GeoGebra files.
Binet formula 2D complex output: www.geogebra.org/m/twvvzpga
3D imaginary output of Binet formula: www.geogebra.org/m/z6dy9cj5
3D plot of absolute output of Binet formula: www.geogebra.org/m/pb7hmxyd
My four-part series on Numberphile videos about Fibonacci Numbers (from 2014) starts here.
• Brady Numbers - Number...
Here is me going on about the square root of five (Numberphile 2018).
• Lucas Numbers and Root...
This was the Fibonacci puzzle video from Matt Parker's Maths Puzzles.
• MPMP: The 1 Million Ba...
Read a whole bunch about "Generalized Fibonacci Sequences and Binet-Fibonacci Curves".
arxiv.org/pdf/1707.09151.pdf
The zero I found was at -9.14202391817 + 2.80064954276i and you can see the exact form here: www.wolframalpha.com/input/?i...
Try it for yourself and put the Binet Formula (((1+sqrt(5))/2)^n - ((1-sqrt(5))/2)^n)/sqrt(5) in the Wolfram roots calculator: www.wolframalpha.com/widgets/...
This site has everything you'll ever need to know about Fibonacci Numbers.
www.maths.surrey.ac.uk/hosted-...
Buttercup - The original buttercupchallenge
• Buttercup - The origin...
CORRECTIONS
This was a long video and in hindsight there are a few things I wish I had phrased better. Here they all are:
- I misspoke around 01:13 when I said "negative one, zero" as it is clearly "negative one, one, zero".
- At 07:53 I mean the negative values -5 to 0. I said it a weird way.
- My language at about the 1D input to 2D plot from 09:17 is a bit sloppy. The real values going into the Binet function are not the horizontal axis shown; the plot onscreen is solely the output.
- I say "axis" when I mean "plane" or even "complex plane". The big flat thing.
Let me know if you spot anything else!
Thanks again, as always, for Jane Street being my principal sponsor.
www.janestreet.com/
Thanks to my Patreon supports who do support these videos and make them possible. Here is a random subset:
Loren Thomas
Richard Dickins
Barry Salter
Susan Moury
Sarah Gerweck
Ulrich Kempken
Piotr
Gary Martin
Euler
Daniel DeJarnatt
Support my channel and I can make more videos:
/ standupmaths
Filming and editing by Matt Parker
Music by Howard Carter (excluding Buttercup)
Design by Simon Wright and Adam Robinson
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
US book: www.penguinrandomhouse.com/bo...
UK book: mathsgear.co.uk/products/5b9f...
Nerdy maths toys: mathsgear.co.uk/
Whoa.
Wau.
Wow.
Wow.
Wow
You mathematicians with your ultra-technical terminology, haha!
I like the Fibonacci series where you start with 0, 0. Its easy to remember
I can even calculate any item in that sequence in my head ;)
The formula for a term in the sequence is the simplest I’ve ever seen
Math truly is amazing
Ah yes, the sequence that correctly predicts the exciting things that happens in my life
I love how you choose the simplest possible sequence and it's "golden ratio" is undefined
Well, I mean... the Fibonacci sequence was discovered thinking about the ideal procreation of rabbits, and it's pretty hard to have a negative rabbit mate with a positive rabbit
That's what mathematicians do... Extending simple ideas to random dimensions...
You can but then they mutually annihilate and you get a huge explosion
They say opposites attract, don't they?
Don't even get started with imaginary and 4D rabbits
All you have to do is swing the rabbits around your head at a moderate fraction of the speed of light, and you get a handy anti-rabbit
The moments when his amazed face perfectly merges with himself are really trippy. Nice touch =P
It's called the buttercup challenge, he links it at the bottom of the description
It made me look up the song Really good song
@@niccy266 1
Lol yes
Kinda freaks me out, tho
φ : Let's see what's at the end of this infinite sum... φ : π!? π : Hey. φ : What are you doing in complex space? π : I work here. It's my job to be here at all times.
π : I was here long before you got here, and will be here long after you leave.
*rational numbers in geometric sequences intensifies*
"Wait, it's all pi?" "Always has been"
C'mon. You've messed with complex numbers. How are you *not* expecting a π there? Also, mandatory e, this time wearing the disguise of 'ln'
sharpfang dude most people don’t intuitively know that pi has something to do with the complex plane lol. I’m sure you’re very smart. Here’s a gold star ⭐️.
Fun fact: Phi (1.618) is really close to the ratio between miles and kilometers (1.609) which means you can use adjacent Fibonacci numbers to quickly mentally convert back and forth between them. For instance: 89 miles is nearly 144 km (it's actually 143.2), or 21 kilometers is roughly 13 miles (13.05). You can even shift orders of magnitude to do longer distances! e.g., 210 miles is around 340 km (multiplying 21 and 34 by 10) which is close to the actual answer of 337.96 km.
🤯
OH MY GOD
I use this trick all the time, it’s so useful
I find it easier to just do x+(x/2)+(x/10)
Finally a good way to do it quickly, but I still think the imperial units are hideous, just a little less than what I thought before
Missed opportunity: you could have had your amazed face trace the path of the graph shown on the screen at the time.
I think he tried. It was awfully close, wasn't it?!
I thought he was going to
He sure Parker Squared that one!
I've always preferred the 0, 1 start. With these numbers often found in nature, adding a moment of creation feels profound.
it also feels even simpler than a 1, 1 start, like if you had to enumerate all the possible starts, you'd start something like "(0, 0); (0, 1)"
I think both are found in nature. In some spiraled plants, there is a gap in the center which is effectively 0. Others have something in the center which is effectively 1. Until your comment, I'd never considered that. I'm pretty sure that the vast majority of fib-nth() functions consider the 1st nth to be 0.
this aint no sit down maths. we standin up now
Rise up gamers
I think Matt isn't stand-up comedian, he's sitting all the video, he's more of sit-down comedian
Calm down Nolan
Ha. I’m filming in a small room at home during the lock-down.
He's doing the Parker Square equivalent for standing up (dead meme I know)
The synchronized "Matt Parker's Maths Puzzles" cards were... _chef kiss_
thankyouverymuch
I hadn't even noticed! Very nicely done!
6:49 this is so oddly satisfying
MammamiaDasAhSpicyMeatball
a channel after my own heart
I have to say, I'm a tiny bit disappointed that his amazed face didn't follow the graph. It even pointed at his face! 6:45
Love the name
The line looks like my Doctors Signiature
underappreciated comment!
Looks like my doc's prescription for ... well anything and everything.
I'm actually thinking of trying to align my signature to this plot just for my internal giggles :D Would also make a nice company logo.
The Fibonacci convention was huge this year -- it was as large as the previous two put together. ThankyouladiesandgermsI'llbehereallweektrythechicken
Tipyourwaiters!
@Idiot Online Wondering Aloud 👏, 👏, 👏👏,👏👏👏,👏👏👏👏👏,👏👏👏👏👏👏👏👏,👏👏👏👏👏👏👏👏👏👏👏👏👏.....
Wait there's a fibonacci CONVENTION??? When and where!?
Oh get the heck out, I just got that
@@stevemattero1471 Location: just add the coordinates of the locations of the last two conventions. Time: just add the dates of the previous two conventions to get the new date.
I was waiting for the line, "And so I contacted Ben yet again and for some reason he blocked me and stopped responding to my e-mails."
The Binet formula for the Lucas sequence is actually simpler than the Fibonacci sequence: (ϕ)^n + (-1/ϕ)^n = nth Lucas number
That's amazing.
This reminds me of an experiment I did with Conway's life. I started wondering what would happen using the standard life rules with a bounded game, but set a cutoff for how many steps the game would iterate. I then took the union of each iteration of the previous game to create a seed for a new game, and continued to repeat the process. I mainly was doing this to see if you could use GOL to generate interesting height maps when I found an interesting property. For some reason if my iteration value was 2 meaning 2 distinct steps after the initial state to create a new input, the mean value of my bounded inputs approached pi. When they surpassed pi they would eventually trend back down to pi. I have no idea why pi arose because I am not that skilled at math, but I still wonder why that generation of inputs for a board state would trend towards it. The most I discovered was that method of generation retained symmetry if it existed in the initial board state meaning a blob in the very center would create symmetry along the diagonal, horizontal, and vertical axes.
Okay I lack the mental capacity to imagine what you did but I'm really interested in why would Pi appear there...
Yo idk what you are saying but that looks exciting let us know if you find anything
Interesting. Would you provide more details?
you did an experiment with 'Conway's life' lmao what PS: ik what GoL is
@@pranavkondapalli9306 wow, I didn't even notice that first read through. That's an unfortunate typo for OP to make.
Now I want to see a 3b1b style animation of the 2d inputs moving around to their 2d outputs
One of the “results” of the 3b1b videos is exponentiation moves complex numbers around in circles- so presumably like that? But maybe not since there are 2 exponentiations being added
Also just all the colours ordered as inputs mapping to their outputs
He commented on this vid. You could comment on his commeng
To explore this further would clearly require a large investment of time and effort. I suggest you apply for a Grant. Sanderson, ideally.
I see what you did there and I approve!
For those who don't know, Grant Sanderson is the host of 3Blue1Brown
@@anirudhranjan7002 he already comment
The positive only values look like a growing spiral from the side, while the negatives create a spiral we serve head-on. If you used them as different POV, you could maybe plot out the tips of leaves or the sharp bits of a pinecone. It's really neat..
This is just such a cool maths revelation, with an amazing payoff and one of the absolute best editing jokes I've ever seen. That's pi outta pi from me, even if I apparently can't read 3d plots very well.
I've always been a fan of the 0, 1 start, glad to see it got some recognition
I too like that start, although the 1,1 makes most sense with the origin story (breeding rabbits).
@@andymcl92 In the 1st generation you have one pair of young rabbits and no mature rabbits. So in the 0th generation you must have one pair of mature rabbits and one pair of young antirabbits. Then the mature rabbits give birth to the young pair we see in the 1st generation, but there are no mature rabbits left in the 1st generation because the antirabbits grow up and annihilate them.
But to me, it doesn't seem like it should work. The reason there is the two "1"s is because there's nothing before it. So if you start at 0, there's nothing before it, so you put another 0. "0, 0". But then, if you try to make the sequence by adding the two previous numbers to get the next, it just becomes and infinite string of "0"s.
I like to start with two zeros - makes the maths much simpler...
Why not make it completely general and start with the integers A,B ? So the series progresses A,B,A+B,A+2*B,2*A+3*B,3*A+5*B.... And we see that adjacent Fibonacci numbers occur in the coefficients. We can legitimately make A,B anything we chose including +ve and -ve values chosen at random.
Can we just take a moment to appreciate the editing involved for the amaze face
Its actually pretty simple, you just cut a still from a frame of the video and then move it to the time and place in the video in reverse. It is a cool effect though.
I love that you've made a living of messing around with interesting numbers and sharing it with us. I used to do things like this on my TI-86 graphing calculator, but never got far enough to make these kinds of incredible graphs (it was far beyond my mathematical understanding). Thanks for sharing your passion!
That was really sweet. I saw the title and started trying to imagine an equation describing a curve like that, with zeroes where the Fibonacci numbers are. Didn't realize that such an elegant parameterized version already existed.
Heh, that random pi at the end. That's something I love about maths, if you're ever hungry you never have to go far to find a delicious pi.
I want to know why though. Is it because every periodic system has a (circumpherence/2r)*dt relation? What about an 'oval', it can always be projected back to a circle right? Giving you a pi in every periodic system somewhere?
V Blaas I’m not sure exactly why this particular pi shows up, but complex analysis is absolutely riddled with pi so it isn’t that surprising. In particular, this function is made of exponentials, and complex exponentials are inextricably linked with pi.
2/5ths make it sound he could've used τau and get rid of the pesky 2.
@@ottolehikoinen6193 2/5 * 1/π =4/5 * 2/τ though.
@@TheBasikShow I remember when I took Complex Analysis in college, the answer to the exercises we did was almost always pi. If not, it was zero, 2pi, or pi/2.
19:44 Here comes Matt’s π day calculation 2021.
I thought the same thing. Use that absurd formula for area under the curve to calculate pi
Matt just looks so happy, and it makes me happy. This is actually a really cool find! Well done!
I love your enthusiasm, my dude. Keep learning, growing and challenging yourself and others! :)
Find someone who looks at you with the same excitement that Matt gets around numbers.
with detached heads floating in space? no thanks!
The ‘face’ bits were great. Nice effect.
Very interesting, that plot of the Binet sequence appears to spell out 'Jeremy Bearimy'...
Jeremy Bearimy you say?
6:44 actually looks like an inwards spiral beeing (exponentially) accelerated to the right
That was my thought too. Could help explain why pi shows up a few times. The Fibonacci numbers may just be a 1d slice of a 2d projection of a 3d spiral.
Wow interesting observation. Meanwhile the negative numbers in the Binet formula formed an actual spiral 7:09. If the positive inputs can be described as an "inward spiral" then the negative numbers would be an outward spiral.
Yeah it is!!, the (-phi)^-n term acts as a spiral exponentially decresing in radius and the phi^n acts to push the centre of the spiral to the right exponentially
"What a stupid idea! Who wants a video about Fibonacci numbers at 3 in the morning!?" Matt Parker: "Oh boy, 3 AM!"
Now me at GMT+2, knowing sleep is a social construct
Literally anyone awake at 0300 just wants something to do.
3Blue1Brown has a nice way to represent 4D graphs. What he does is draw the transformed gridlines of the input space. It's like what you did with the graph with the real number line as input.
Did no one think of using colour for the 4th dimension?
He also used colour gradients in another video (about finding the zeros of a complex function I believe)
Those graphs don't always look good, and they can even be more confusing for non-injective functions. Watch 3b1b video on Riemann's zeta, the map looks cool but it doesn't tell you anything about the function. You can't really recognize slopes and shapes, it's a mess. Unfortunately, this function looks like the kind of function which would be too messy to represent as a grid mapping.
@@olmostgudinaf8100 Yes, and it's really useful from a topological perspective. For example, a Klein bottle is quite intuitive if you colour the overlapping part because you can see the neck part moving in the "colour dimension". It's less useful for complex functions because you can't really see slopes. It's hard to tell if a colour is shifting at a parabolic/exponential rate. It's still used a lot through a plotting technique called domain colouring, but it's still not a perfect way to plot complex functions. There isn't a perfect way unfortunately, you'll always have some drawbacks.
What about two overlapping 3d surfaces attached to the 2D complex plane? Like the thumbnail for this video but with one real surface and one imaginary surface.
Very informative, lots of effort put in. Some of the best math content I've seen.
5:34 this excites me uncontrollably it's impossible not to smile
“Ofc you’re dividing it by the sq root of 5, big fan!”😂😂made me happy made me smile nice 👍🏽
You could utilize time representing one variable. An animated 3D graphic may be used to visualize a 4D equation.
You can kinda already do that with his program by sliding the complex input value.
Or you can colorcode the complex plane and then color it according to the complex output.
That seems like something a physicist would do
Yes! I want to see this!
While this does work in theory, it's not going to be like what most may imagine. Since the full plot is a 2D manifold embedded in 4 spacial dimensions, a 3D cross section would just be a 1D manifold embedded in 3 spacial dimensions.
The "other" thing I loved about this was the "how we got there" story. A great example of the mindset to approach problems scientifically and what to look for :)
Yeah I really liked this video start to finish - but I *really* wish he’d done a domain colouring/colour wheel plot!! I find them so much more intuitive
This is so awesome. Love the energy and passion. ❤️
When I learned Fibonacci sequence in 99 (i was 15), I tried to extended backwards, but I lacked the math to understand this whole "bi infinite" sequence. Watching this video was a real time travel to the past. Nice work!
The way Matt's goodbye face's hand was animated was wigging me out for some reason. Does not tarnish the good maths though.
Really kept my attention while he did the sponsored portion. Very clever, that one...
That animation kept me on my toes! More intense than the bouncing dvd logo
Gordon Wiley Tom Scott did a video on green-screen perspective errors being creepy.
it seemed like it was both timed to the beat of the music and slightly off at the same time. Methinks he learned a new editing trick and enjoyed it a lot :)
@@kuromurasakizero9515 definitely looks like he's having a fun time with it
playing buttercup while he does the amazed face... LMAO
Started reading through the comments hoping someone had already ID'd the song for me. Thank you
@@zozzy4630 You mean Darude Sandstorm?
@@ALZlper I think he does indeed mean Sandstorm by Darude.
@@ALZlper The song is Buttercup by Jack Stauber
@@zozzy4630 kzhead.info/sun/mJaqd5mtrnZ5imw/bejne.html
Love the various faces. Nice video editing! The goodbye face kept me watching all the way through the Jane St. promotional-a first for me. Nice audience hook, Matt!
LOVED this video my dude! Thank you!
"A third" incorrectly stated as 0.333, yet time stamped at 03:33 is some fine trolling... 🧐
That's some fine detective work also, dang.
i just realised that the curve that goes through 1 twice is actually a spiral/cone looked at from the side :D
I think it could even be described as an 'epicycle'. (Oy, Ptolemy: no! I respect your attempt to maintain the geometric integrity of our planet's immediate locality, but if you was to come round here, and start arranging *my furniture* into a highly idiosyncratic theological exegesis, I would say - 'Ptolemy, nooo! Outside now! You are not in the bustling multicultural milieu of ancient Alexandria. This is Lambeth. Now get your pharaonic physog out of my impromptu courtyard knees-up, you stripy antediluvian muppet!' Etc). :)
Was able to graph the 2D slice with real inputs, working on the complex input/complex output graphs excellent project thank you Matt!!
I really enjoyed the graphics/effects this video, along with the content 👍
I “enjoy” math and this is WAY out of my understanding of math ,but I just love the content. Thank you!
I was going to say this is not complex at all but yeah, is a bit complex. Get it? Is easy, thought, except for the 4D visualization part.
@@Ragnarok540 4d visualization can be done a lot better when using colors. I've explained that in this comment section before so I won't do it here again. But if you search, you will find how it's done.
@@Ragnarok540 For someone who watches math youtube videos for fun, it's quite difficult. Glad you get it so easily, though
@@carrotfacts Could you explain which part(s) you find difficult? Just curious.
I mean, all he did was say “here’s a solution to a recursion. It’s continuous on C”.
90% of the budget for that amazed face effect at 6:47 Edit: I stand corrected 7:28 btw for plotting complex functions, I've been trying for a while to make a program the plots the path of f(x + ti) in 3D where t is just the time. This could be a 4D plot
What is the song called
Buttercup
I love your videos! I don't really understand the complex maths involved, and I don't think I ever will get to. But maths really spark an interest and curiosity in me, I love to learn more and take a peek into this otherworldly stuff!
1:34 I just love Matt's humor, where he randomly does stuff, never addresses it, etc. Plz never change
Matt! You are already in python. Take a look at the library "matplotlib" it can do zoomable/movable 3D plots directly from python.
What software could I code an interactable fractal zoom using python?
Seems crazy to me to rely on excel when you have matplotlib - or at least I wouldn't admit it 😬
Matplotlib, pandas, numpy.
Ruben Moor you underestimate the obsession of Matt with Excel
It hit me near the end how good of a job you've done of editing this. The virtual plot that you're actually pointing to points on like a weatherman. Also I suspect you just learned how to do the face thing and it's really cool.
This is so cool. Thank you for making this video
That loop in the graph is mind-blowing!
The amazed face absolutely cracked me up!!!
Matt: Uses Python for computing the values Also Matt: Uses Excel to plot the values computed using Python We need to talk about Matplotlib. Or should I call it Mattplotlib?
Mattdoesntplotlib
Matplotlib sucks, excel is far better if you want to be fast
@@MaxDiscere Agreed Excel is great for a fast and dirty first look. But it's no good at all if you want to be able to zoom, change point of view, etc
Mattplotlib will give you graphs that are interesting and look good, but if you happen to look at them diagonally one of the ways, they don't quite add up. Also, some of your numbers appear in two places for some reason.
Wow, lots of people here need to learn to use matplotlib which is arguably both faster and and more powerful than excel. Plus it is interactive and give nice looking graphs
Can't wait to see the quaternionic version of this video!
Quaternions don't really add much beyond more complex planes (which is very useful when doing 3D rotations). The dual and split-complex numbers on the other hand do have some interesting behavior, but neither can act as a square root of -1.
This was one of the coolest Fibonacci maths I've ever seen!!!
I love how excited Matt is about everything.
6:29 this is the shadow of a spiral (3D onto 2D plane). Then the next part of the video shows a spiral, which is still a shadow of the spiral, but seen from a fairly easily guessed angle in 3d space.
That's a good catch! It does look like a projection of a decaying helix.
i saw nonlinear damped mass spring (have a vid on using quadrature osc to appx sine and cosine) s0 = 1.f; s1 = 0.f; // init s0 -= w * s1; s1 += w * s0; // loop .. where w = angular frequency 2 * pi * hz / samplerate
This is one of the awesome channel in you tube and I love it and learn from it. Thank you so much sir.
Loved the 3D representation of a 4D concept, super cool Would love to see a follow-up video with bigger graphs!!!
If the surface have not a name yet It could be named "the Parker's Blanket"
I’m wondering what are the properties of the loop that the two 1’s form... I don't know why, but it was the part that I found the coolest
I wonder what the area of the loop is
@@theot1692 I was about to say this lol. I also wonder the area of the loop. And if you wanted to go deeper I guess you could also do analysis of curvature, length, etc... never know what you might find.
The real question is does the loop shrink in the complex plane, and if so where does it reach zero size?
@@gajbooks volume of the loop? o_o
I dont think the line crosses the x axis at all, I believe that from the point of perspective where you looking from the X/Y axis vantage point it looks like it crosses the x-axis, but it doesn't, it loops around it, just like a inverse spiral if looking from the vantage point of Z/Y axis. (I dont know, it just looks like it)
I loved this video!! THANK YOU!
Beautiful! I never thought about using anything other than positive real numbers in the Fibonnaci sequence until today.
What I would do is the way 3blue1brown did the display of the Zetta function: start with a grid in the complex plane, and animate distorting it
It's insane how often pi shows up in any level of math. Funnily enough it's the first example I given when helping students to better understand infinite series and what they're useful for (alongside Euler's identity). Very cool video that I wish I wouldn't have waited so long to watch.
always with a 2.
Amazing! Good work!!
Wow! That's amazing and cool! Those are neat looking graphs! 👍🏼😀
This guy went insane. Really maths “y”. Imaginative. I love how he opens he mouth to show his excitement.
6:50 - next level videoediting - I love it
Great video, as always!
This should have been in my complex analysis module. Also, the limit of the integral of the Binet function - mind blown 🤯
Gosh darn it, now I want to look at Fibonacci quarternions!
You give "domain coloring" a try next time you want to visualize functions of complex numbers.
This is sooo genius and beautiful! Wow!
That first graph made me the most excited I've been about math, *ever!* :D
In regards to where Fibonacci starts, I’d always been taught it starts 0 1.
Zero indexing... nice.
I can't believe you did all of this teasing and then didn't show the plot across the line containing the zeroes
This is one of my favorite channels specifically because I think this is the only person I've ever seen excited as I get for math
6:45 Liked and subscribed just for that meme. Good job, Matt :D
Nice new point of view, thank you :). Also, by the way, in the log abs plot, you can see the two binet terms as two planes, which I find constructive. Remark: Personally I like to plot the abs and use colors for the output phase, to keep it 3d. It distracts a little bit from the phase, but often you don't really need it, and e.g. with the log abs you can see the zeros and poles quite well.
That sounds awesome! 😸 It will both show the angle and make it 🌈rainbow, which automatically boosts the awesomeness of a mathematical plot by about omega! Sadly, no version of Geogebra I've ever tried can make multicoloured outputs 😿 so it's gonna have to be a new file, not an updated version of this particular interactive.
You’ve likely already heard of it, but you could also look at the 5-adic interpolation of the Fibonacci numbers; this yields a 5-adic continuous function in fact! Really cool stuff. Unfortunately, I think you‘d run into the same difficulty (or more) getting a visualization of the result.
The best I have seen in this series
ok i really like the buttercup challenge thing you had going on, I've been listening to a lot of jack stauber recently and I thought it was really cool to see one of his songs appear in one of your videos!
That amazed face to 'Buttercup' 🤣
Hey Matt, looks great. However, you should try taking the logarithm of the absolute value, when plotting, since the fibonacci series is an exponential series and thus diverges quite fast. That would also help showing the zeroes and the "waves" you can see in the function.
This works when the output is large, but for small values, log is a very large negative number. Furthermore log is undefined at 0.
That was a fun ride! Thanks you.
Love your energy 😄
First time I’ve seen e, pi, and phi all together like that
That equation at the very end reminds me of Euler's Identity. You could call it Parker's Identity!
This deserves a double like... Nice job Steve
Sooo fascinating!
10:40 I just realized that Ben Sparks was in the MegaMenger project!
Not only was he in it, but he was *in* it!
It also reminds me that I have a private version of that in the works which I should probably get back to some day…
I was surprised at how easy it is to graph in Desmos: \frac{\left(\phi^{t},0 ight)-\frac{1}{\phi^{t}}\left(\cos\left(t\pi ight),\sin\left(t\pi ight) ight)}{\sqrt{5}} Set your preferred boundaries for _t_ Or, if you want animation, restrict _t_ to [0,1] and replace every instance of _t_ with _at_ for some variable _a_
This is awesome!!
Hell yes! Been thinking about this for two years but couldn’t visualize it without the tools!