Coding the Collatz Conjecture
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- Опубліковано 15 тра 2024
- It's the second episode of Coding in the Cabana! Here I attempt to visualize the Collatz Conjecture in Processing. Code: thecodingtrain.com/challenges...
🕹️ p5.js Web Editor Sketch: editor.p5js.org/codingtrain/s...
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References:
🗄 Collatz Conjecture Wikipedia: en.wikipedia.org/wiki/Collatz...
💻 Collatz Graph: All Numbers Lead to One: www.jasondavies.com/collatz-g...
💻 Trying to visualize the Collatz conjecture: mathematica.stackexchange.com/...
💾 Primitive Data Types in Java (more information about the long type): docs.oracle.com/javase/tutori...
Videos:
🎥 Collatz Conjecture in Color: • Collatz Conjecture in ...
🎥 Golan Levin's Modulo Operator video: • Guest Tutorial #6: The...
🎥 UNCRACKABLE? The Collatz Conjecture: • UNCRACKABLE? The Colla...
Related Coding Challenges:
🚂 #14 Recursive Fractal Trees: • Coding Challenge #14: ...
Timestamps:
0:00 The Collatz Conjecture
4:10 Programming in Processing
6:46 Checking The Number of Steps
9:23 Visualizing The Collatz Conjecture
20:44 Rendering to a PDF File
22:24 Conclusions and Goodbyes
Editing by Mathieu Blanchette
Animations by Jason Heglund
Music from Epidemic Sound
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🔗 Processing: processing.org
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#collatzconjecture #modulo #processing
You truly are the Bob Ross of programming
Gijs van der Giessen definitly, and by the way: Happy Birthday Bob Ross...
***How many apples you have in an **#Apple_Tree**, so many **#Dead_Nodes_Will_eat*****
***I appreciate your work in generally . I think here you are making a mistake. **#Collatz_Conjecture** into negative axes isn't a Collatz Conjecture , and if you follow my work , can see for what. Anyway as promised , I solved Collatz Conjecture via **#Murgu_Lemas** **#Collatz_Conjecture_Logical_Dead_Nodes_Lema** **#Eternal_Triad** (which demonstrated every odd Integers , except Logical Dead Nodes and **#Collatz_Perfect_Numbers_Dead_Nodes** but with a dual sense), and **#Collatz_Conjecture_Murgu_Formulas** --- **#Collatz_Conjecture_Solved** as promised. I worked on it in any **#Rest_Time_Times** , then take it as it is , but is solved at: " **ua-cam.com/video/RPEGFZbNW2A/v-deo.html** "***
***Anyway , at this time **#Beauty_Math_Curiosity** (**#Mathematics_Curiosity** ) is **#Murgu_Conjecture_Vicious_Redundancy** " **ua-cam.com/video/vn3H7aUmftk/v-deo.html** " for which I don't have dare to start over, because don't have time, but because maybe never will can demonstrate that have to had only 3 roots (1,5,17)***
Coding train is the best. Nothing inspires me to go work on my own projects like you. It is the combination of your personality and seeing you stumble and occasionally fail but still figuring it out in the end. It makes me feel like I can do the same thing.
I’ve found the Collatz conjecture is most interesting when viewed in binary, where the even numbers dividing by 2 essentially becomes chopping off all the zeros on the right side, and the odd numbers multiplying by 3 and adding 1 essentially becomes adding a number to itself bit-shifted one bit to the right and adding 1. There’s patterns in binary that you don’t notice in decimal.
Thank you.
Someone has messed with prng algorithms. Either that or has an inherent talent for them and should go dethrone marsaglia, the current rnjesus
I've only recently happened-across this corner of iterative functions...
IMHO, you've hit the nail on the head with 'binary'... That 4-2-1 loop (surprisingly 'octal') is, I believe, KEY to understanding this whole business (but I don't have the math skills to express it as a 'proof'...
Notice that seeding the function with zero ALSO goes into this loop (3 * 0 + 1 = 1)... All 8 octal bit patterns (0-7), as seeds, are contained within the lowest 3 bits.
Adding another 3 bits (seeds ranging 0 - 63) and the trajectories will, like the infamous 27, occasionally take off toward the clouds, but will eventually decay to the loop...
Add another 3 bits (0 - 255, total 3 octal digits) and some trajectories go even higher... BUT, they all decay to the loop again.
There are interesting observations to be made about 'intervals' between some of the longer trajectories (ones that go for long excursions before landing on a previously 'used' value that is part of a path toward inevitable decay... 4n+1 points to some interesting seeds, like 32 - (4(1)+1) a.k.a. 32 - 5, our old friend "27"... Hmmmm
So, it's been shown that all values of the highest octal digit value combined with all values of the middle octal digit value combined with all values of the lowest octal digit value eventually collapse to the 4-2-1 loop...
Conceptually, trying 2 "middle" octals between the high octal and the low octal (ie 12 bits or 0 - 4095) should (will) express similar behaviour. Try 3 "middle" octals (total 15 bits) ... same thing...
Viewing the intermediate values in binary, it soon becomes apparent that 'bubbles' of multiple zeroes form, and the lowest octal is desperately trying to drag the value lower (4 is even and 2 is even, so divide and divide... The bits have been shifted to the right twice, and stratospheric numbers come back down to the troposphere... Often a 'bubble' of 6 zeroes forms tantalisingly close to the LSB... A few more steps and the entire value collapses several orders of magnitude. It's fate is sealed and destiny a certainty...
Veritasium recently claimed all numbers to 2^68 have been tried, and, for all variations of those 68 bits, all numbers eventually 'collapsed'...
Excuse me for thinking this... I recognise that infinity is a very, very distant ceiling, but even with my 32bit explorations (lots of bit pattern variations) the decay begins after not too many iterations of the function... There are 'rallies' along some paths taken, but the conclusion seems inevitable... With nothing but this evidence and intuition, I'm happy to conclude that "bubbles will form" quickly enough (in the rightmost bits) that will reverse any excursion toward the stars, and the strange attractor of "4-2-1" is the fate of all positive integers that enter into the Collatz Conjecture...
Collatz(x) {return x&1 ? x+x>>1+1 : x>>1;}
I love the feel of these. Coding in nature to master the nature of code. Awesome!
Thank you!
"I think that I've made a major error." Why does this make me happy ? Will he solve the issue ? How much time will he take ? I learned something today.
Dan please do more of these, they are awesome
I love this series so much ^_^ it's really peaceful.
And the Collatz Conjecture is one of my favorites!
Thank you! and Thank you for the support!
That is a great observation! It's like Daniel is the Bob Ross of coding!
edit: I just watched the first in this series and saw that UA-camr István Horváth already made the Bob Ross connection..
Where you people got this emoticon..I need one
haha!!! For me Collatz Conjecture is mysterious!!!!
I think this video merits some clarification:
1. We don't know if the Collatz Conjecture is true, it's just a conjecture.
2. Both methods of visualizing it are actually the same, one is just a different arrangement of the other.
3. 20:45 - "Just take _a minute_ to ponder the fact..." - Slight understatement: Stephen Wolfram has built a whole new kind of science on this idea!
For those who wanna have the pattern like the numberphile video add:
If (value%2==0) rotate(0.27-(0.0002*j))
Else rotate(-0.19+(0.00025*j))
Being j the index of value.
wrong
Yassssss! Always great to see videos inspired by Numberphile videos!
What amazing timing! This morning as I booted the computer, my goal was to program something creative and fun while learning something new in a casual, relaxed Sunday morning pace. My wish was granted when I saw the Coding in the Cabana notification. Right now the scene outside my window is a snow-globe world of gently swirling snow flakes. Thank you for transporting us all to the cabana and your garden. This series is a wonderful change of pace.
Yay, share what you make!
I really like the mood of this serie ! Nice mix of low and high tech, and covered subject are really good ! Thank's
Fantastic! Such a wonderful example of how algorithms can mirror the shape of nature
Your enthusiasm towards math, programming and the beauty of life overall 20:48 is something, that i think, most of the community shares and what makes you such a enjoyable fella to watch, btw love the new series of coding in the cabena😊
Yes episode 2
Love the chill atmosphere of the cabana
Love to see you so relaxed!
I am enjoying these videos so much :) - I saw the numberphile video, but having the perspective of coding it like this is a very nice addition! TY Dan
You inspire so many people and at the same time, you make learning fun! Great content
Coding in the 'Cabaña' technically speaking. Ñ (lower case ñ, Spanish: eñe, Phonetic Alphabet: /ˈeɲe/ "énye", About this soundpronunciation (help·info)) is a letter of the modern Latin alphabet, formed by placing a tilde (also referred to as virgulilla in Spanish) on top of an upper- or lowercase N. Gracias por tus super classes maestro :)
Indeed!
@@TheCodingTrain Thank you for keeping inspiring us every day...in the Cabaña ;) Gracias Maestro!
This series is the most beautiful thing on the internet
Algorithms feel less scary and cold now, thanks to you! Great video and idea
Woow this brought me memories! Nowadays I don't have time to code anything anymore. Great video as always
men that was a great ideia to code in an diferent inveronment that we mostly see in youtube, thanx a lot
This will be the last video in this series using the chalkboard, apologies, I already shot this one before I got all the feedback on the first! Get the code and submit your version here! thecodingtrain.com/CodingInTheCabana/002-collatz-conjecture.html
Thank you for this dan!
love this series!
LOVE your content. Please keep demonstrating projects in Java!
Love this, awesome style, makes me happy to learn! :)
so cool that it is possible to save your code output to a vector image in processing!
Simply amazing!
I like your cabana. Its so quite and makes you focus on doing.
Loving the plants!
Episode 2 is fantastic ☺️
I love your coding projects so much! Great work!
That was such a nice video, thanks!
I'm simply unable to ignore and not like these videos immediately
Keep ‘em coming please
Yey another episode in the cabana!!
you are my favorite UA-camr！
i love this series
That scratchy board gives me weird flashbacks to a brocken board in my old math class 😞
This is cool ! sometime nature seems so simply encodable it's quit astonishing. I'll try making art with this set of sequences. To me you should make a non linear progressive rotation to deal with the superposition you should even try color gradient. ;)
Such a peaceful place for programming.
Please do every video in Cabana and probably some in the garden too 😍😍
i love this format. :)
So nice and peaceful 😻
I really want to work on this now :) I'll see if I can find a way to export a p5.js project as PDF!
I'm loving the new series 👍👍
Next time do a "Coding in the igloo"
Coding in northern Ontario
Can't wait to make my own variation of this, will surely do this after i come back from vacation.
Looking forward to it!
@@TheCodingTrain yes 😄 I haven't made my own variations for a while because I haven't got the time to do so. 😄😄 but i will also do the #TeamTrees one and this one too ❤
#TeamTrees
I remember when the Numberphile video came out. I visualised it in p5, and was discussing it in my school with my friends. Another friend of mine was playing volleyball nearby and had given me his phone to broadcast the game on Instagram live. I set-up the phone near me and forgot about it and continued my discussion. The next day, my volleyball friend met me and said that many people messaged him and asked him what kind of nerd friends he has.
P.S: A third of my tea evaporated due to boiling while I wrote the comment
Oooo yeah, faster and more controllable than an Lsystem. To make it grow you can trail particles. So also size of the particle can be set to be part of the game 🎉
I am eagerly waiting for his 1 million subs.
Love your videos. I am trying to replicate bubble shooter game in p5. Saw the Minesweeper one and space invaders and tried to incorporate logic from their. Can you
please do bubble shooter game as a coding challenge.
You have always won our hearts as well as gave us a lot of knowledge... The way you define the topic before coding about it changes the whole game... If someone doesn't know much about the topic can also understand it... ❤
I love these simple visual excercises! They make me so inspired! :)
Love these ! Makes programming seem less stressful when youre in an environment like this
I am really glad that we have someone like you in youtube, i am wondering if i can challenge you with some coding with matter.js, can this plugin add physics to text and letters ? would it be able to add physics to it ? for example a circle hitting the text and it would explode ? i would really really appreciate it if you can look into it :)
Friend! Could you make a Coding Challange video where you implement the Bresenham algorithm for drawing a line in pixels between two points? It would be cool!
amazing series, was waiting for the 2nd one for soo long :D
Please do videos on postfx in processing they are really dope and important
I love processing and exporting my own pdfs
This is quality content
The dampening of the chalk sound makes it alot more watchable, loving this series so far!
Looking forward to some winter cabana episodes with a wood fire crackling in the background.
More likely just a warm coat and layers. . need an orioles beanie!
Next time try coding the chaotic movement of dust particles after step dancing in the cabana . ;) Good video with nice mood!
I hope one day I would be able to code like you.
Oh my God! I did it! I'm translating the code to godot and it's amazing!
Good old collatz conjecture. I think the second or third coding project I ever did was a program you gave in random numbers and it spat out if they still fit the conjecture (always building on the previous calculations to make the next one's take less time) but since there is no disproving example within the first 2^64 numbers the program was kinda useless xD
Perfect
Would make a really beautiful photoshop brush - very organic.
the cabin is perfect for processing , simulating nature in nature ,love it
TY, I hadn't seen these forms for CC before and for graphics the first thing I think is: Hair this is a way to cheat organic looking hair
now that I know a quick image search gives some really good results of graphing techniques for different visualizations besides the common graphs.
Orioles cap? Then I remembered you're from B'more...
That's a nice shed, but it's gonna get cold inna winter, and winter is coming! 8-) Keep warm, keep on making these things
delightful
you are the coolest prof i know.... 🤩🥰 ....
LOVE CHALK!!
Love it
This is great
thank you 4 showing us how 2 save output drawing as vector file (pdf)
i believe in order to add it to MS Word file (.docx) it has to be in .emf format.... do we have this format as an option in Proccessing or P5.js ??
@@ahmedhassanahmedhassan6495 You can save it as a .png file by using 'save("collatz.png")' where he shows you to put 'endRecord();' in the code. This means you don't have to put 'beginRecord(PDF, "collatz.pdf");' and 'endRecord();' in the code! This means it doesn't look as good when you zoom in, so if .docx documents accept .svg files, you can save it as 'collatz.svg' instead, I think.
Perhaps the issue is that you don't rotate back until you get to the next starting number? I think you should reset the rotation after every move. From what I understood, the tendrils should never be able to do full rotations. They'll all be within the spread -π/6 to π/6
oh! Yes, that makes sense!
I LOVE that you let your kids display their art in the background. No offence if it is yours. Ha ha. We all start somewhere. That's probably something I would produce also.
Yes, those are my kids' artworks! (Better than what I would produce!)
@@TheCodingTrain Encourage your kids do do more. Tell them your audience loves to see their art. I would especially like to see some type of artwork that they enjoy, not necessarily something they were asked to do at school, if that was the case. Looking forward to more from all of you. I watch your videos all the time. You are a great teacher!
Will you ever do a JS framework video? Like react maybe?
wow, hopefully you saved all that output!!
Dan, what places/books/sites would you recommend to find more cool algorithms?
Here's one of my favorites! amzn.to/2Wj9QQh
@@TheCodingTrain Thank you so much! Thank you for doing these videos btw, they make me like math. Also, you're a great teacher!
Please do a follow up video where you code a collatz tree.
really you can just check if the sequence ever reaches a power of 2 cause if it does then every successive number after it, will divide by 2 until it gets to 1. Im sure there are many other simplifications possible but i get it the point was to get the pretty diagram not efficiency
Wonderful. I'd already seen the Numberphile series about this, and it's fascinating.
One comment: Your optimization that changed 3n+1 to (3n+1)/2 might have screwed things up, as far as trying to match the Numberphile pattern, because now your step count (and branch length) will change by 1 for every odd number in the sequence. That would change the length and shape of the path for every branch.
By the way, I don't know what those little white fuzzy things are floating around in your cabana, but I hope you're not allergic. ;-)
Yes, that is a good point, thanks for the feedback! I think the floating things are just dust particles, the sun was shining directly in!
I wrote my "Facharbeit" in school on that topic :O
lol this is awesome!
cool, thank you
Thanks!
Thank you for the generous support!
Perhaps instead of descending to 1 from every number, you could try to construct the tree recursively starting from 1 instead. For any number, multiplying it by 2 is an option. But when the number is even, you can check if it can be expressed as 3n+1 for an odd n and if yes, you get a branching point.
yep!
That white board that's black. The sounds. Like needles in my brain. Pain. Pain.
Is the reason it doesn’t look like Edmund Harris’ visual because you haven’t factored in the branching?
Also, just to add I’m a recent convert to the Coding Train/Cabana and I absolutely love it! Thank you.
I'm going to delay watching this until Saturday so I can have a nice free morning with my tea and my coding train :D
recursive solution for p5.js (just the sequence):
function setup() {
const res = genCollatzSeq(27);
console.log(res.length-1);
console.log(res);
}
function genCollatzSeq(num) {
const seq = [];
function collatz(num) {
seq.push(num);
if (num > 1) {
if (num % 2 === 0) {
return collatz(num / 2);
} else {
return collatz(3*num + 1);
}
}
}
collatz(num);
return seq;
}
🎶Coding in the Cabana... Music and passion were always the fashion in the Coding Cabana... 🎶
What causes the sharp angle in the middle of the drawing?
you need to make a coding challenge from Numberphile's video "The trapped Knight"