That weird light at the bottom of a mug

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That weird light at the bottom of a mug - ENVELOPES

Рет қаралды 408,997

Күн бұрын

Envelopes are cool.
Interactive Demos on Desmos:
www.desmos.com...
www.desmos.com...
Did I get anything wrong? Let me know! I swept over a lot of interesting things here to keep it short!
This was made as part of 3Blue1Brown's Summer of Math Exposition: www.3blue1brow...
#SoME1

Пікірлер: 697

@kiranaun9593 2 жыл бұрын

I think the beauty of this video is that when you got into the numbers, I had no idea what you were talking about, but I could still fully understand the actual meat of the puzzle. You told two stories at once, one for the average viewer and one for those interested in the technical part. Even with intuition alone, you managed to beautifully demonstrate the math to someone who doesn't really know much about the numbers.

@suyunbek1399 8 ай бұрын

its not about the numbers, but rather about notation and operations

@tongpoo8985 8 ай бұрын

Math is beauty. I'm sorry you weren't blessed with teachers that were able to demonstrate it to you.

@kiranaun9593 8 ай бұрын

@@tongpoo8985 I'm fully aware of the beauty of math, I just appreciated the way this video presented it. I don't remember writing this comment but I'm sure I understood the technical part as I have always been a nerd.

@tongpoo8985 8 ай бұрын

@@kiranaun9593 oh ok no offense meant, I figured since you said you didn't get the mathy parts you may have been one of the unfortunate victims of the education system who didn't learn math because they had bad teachers, whether they weren't good at teaching it so you thought you were bad at it or they didn't demonstrate the beauty and perfection of it so you didn't have an interest. Many such cases

@spacecowboy5274 7 ай бұрын

@@kiranaun9593 At 6:22 why is the numerator 0 but not the denominator?

@LookingGlassUniverse 2 жыл бұрын

This was such a joy to watch! I loved the humour, that you tackled an everyday mystery and the intuitive way you explained it. I hope you’ll make many more!

@no-one-1 2 жыл бұрын

@freze994 2 жыл бұрын

Thanks I was thinking of commenting this

@williamhowland9977 2 жыл бұрын

I'm not saying I'll be disappointed if your future videos DIDN'T illustrate concepts with line rider, but I would be thrilled if they DID.

@cross.workshop11 2 жыл бұрын

Agreed

@aziztcf 2 жыл бұрын

I want cosmic inflation theory.

@CALIGS 8 ай бұрын

You’ll be disappointed to know Line Rider wasn’t used, but pleased to know it was just as incredible.

@jasonrubik 8 ай бұрын

@@CALIGS Exactly. i may not have my little beanie hat, but I am waiting in LINE to go along on this RIDE.

@saadahtsham7 7 ай бұрын

im not saying im a DUMBASS, but i literally understood NOTHING in this video (this is a joke)

@Jhawk_2k 7 ай бұрын

I was so proud of myself back when I discovered that same technique for smooth lines in lone rider. Knowing there's a connection to an underlooked everyday phenomenon is so cool

@niuniujunwashere 7 ай бұрын

absolutely astonished by how you were able to PROVE that the shape at the bottom of the cup is a cardioid. limaçons just flew over my head when i was learning precalculus, but this finally convinced me that they are not made-up shapes! i am deeply inspired by your lucid presentation and visualization, it really helped visualize the math. thank you for this video!

@jackwisniewski3859 2 жыл бұрын

BRO, you have no idea how exited i am by this video, ive been thinking about those shapes for YEARS now ever since i figured out how to make them in notebooks in school, ive been so curious, wondering if anyone ever described them but i never had any clue what to call them and thus i had no way of searching for information on them, i could only keep making them in my notebooks. Thank you so much for this enlightenment you have brought me!!!!!1!

@slinkyslider 2 жыл бұрын

Heart shaped limicon

@mals62 2 жыл бұрын

try putting in r= 1- cos theta or r= 1- sin theta and other variations of these equations in demos. you will get similar shapes!

@frettchen1498 2 ай бұрын

In some way I'm glad you only have 3 Videos up, else I would spend my whole afternoon watching and I really need to do my Machine learning assignment 😭

@megalofyia9280 2 жыл бұрын

Expected to look down and see a few hundred thousand subs - you’re going somewhere, keep it up :)

@pvic6959 2 жыл бұрын

meanwhile only at 300ish! wow! this is the first time ive stumbled across such a small channel and it feels magical! The quality of content was soo high. This guy is going places! cant wait him to go from 300 to 300k to 3M!

@JoBot__ 2 жыл бұрын

@@sdrawkcabmiay Wow, in 16 hours it went from 900 to 1150? That's crazy, right? I only gain like 5 in a whole week.

@I_LOV3_LASAGNE 2 ай бұрын

@@JoBot__2 years later, on 16/6/24, this guy has 53k subs. Well deserved.

@JoBot__ 2 ай бұрын

@@I_LOV3_LASAGNE Ooh, awesome. In this same time, I've only gained about 200 I think.

@trbone76 7 ай бұрын

I used to doodle what you drew in line rider all the time when I was a kid! I never knew they were called envelope curves or thought about how they relate to the mug shadow. Definitely a highly underrated facet of reality

@russiandollie 2 жыл бұрын

As a person who suck at anything related to math, you've made this video wonderfully easy to understand and made me understand the basic concept of the question I have always asked myself while enjoying a cup of tea. Thank you 💜

@pirogporig 2 жыл бұрын

This guy is about to be blessed by the KZfaq algorithm Excelent video

@pvic6959 2 жыл бұрын

thats what i was thinking! Such high quality content and only at 300 subs? nah, we gonna get him to 3 mil!

@splendor10 2 жыл бұрын

@@pvic6959 yah this guy have high potential to serve our curious mind.

@k-leb4671 6 ай бұрын

And then he doesn't upload anything else for 2 years.

@prayinq 6 ай бұрын

indeed

@wege8409 2 жыл бұрын

Thank you dude, seriously mind blowing video. I found this by googling "light at the bottom of mug". I remember trying to google this like a year or two ago and not being able to find it, and I googled it again on a whim today. The Line Rider comparison was a stroke of genius. Haven't heard about that game in years, used to LOVE Line Rider. It felt like one of the first real creative games before that Minecraft stuff. It's great that you have the laymen explanation as well as the nerd stuff. Maybe someday I'll understand stuff like partial derivatives but the best teachers put accessibility first.

@WaluigiisthekingASmith 2 жыл бұрын

Congratulations on "winning" the contest. I can absolutely see why you did. When you mentioned bezier curves at 7:30 I thought "now I have to watch the bezier video"

@sezelle 7 ай бұрын

This was a great video. I certainly didn't understand the bits of maths briefly explained over here but I went right to the wiki pages and learned the whole origin of bezier curves and how to calculate them. I never had a good relationship with mathematics ever since I was a kid but these types of videos help a LOT. I'm still a teenager so I have a lot of time to establish a healthy curiosity in my studies and through these, I'm glad to say that I'm getting way better not only in factual subjects such as maths and science but also in artistic philosophies. Im glad Im in this side of youtube now and saved my future self from possible bad outcomes.All thanks to channels like yours. Thank you.

@kariduanimations 2 жыл бұрын

There’s just something so satisfying about how light can be calculated

@tadaiyoradima 2 жыл бұрын

I had watched Freya's videos on bezier curves earlier and used these envelopes to write some rudimentary code to generate bezier curves, I wasn't sure if I was right, but yeee. Also, I remember drawing curved lines with the envelope things in my school notebooks, never knew they were this cool.

@ididsomestuff7786 2 жыл бұрын

I've always seen this and kinda suspected it had something to do with lines intersecting like that. I've had that image of the intersecting lines in my head for years (literally) trying to figure out precisely how it works. Now, I finally understand, and my brain has been set free (for now at least). Awesome video, I'm waiting for the next one.

@YHK_YT 2 ай бұрын

Agreed and also agreed on the choice of your profile picture

@terdragontra8900 2 жыл бұрын

I have played around with envelopes many times, and noted the envelope is the limit of the intersection of nearby curves, but I never used something like the F(x,y,t) notation. Very cool and enlightening to see it from that perspective.

@whatelseison8970 2 жыл бұрын

This was fun, funny, engaging and informative. The video is also really well paced and edited. You have a real talent for this! I hope you make more videos in the future!

@TheGoldenFluzzleBuff 2 жыл бұрын

My prediction: this is going to get really big really fast. This will be one of those videos that will randomly appear in your KZfaq Recommended’s along side History of Japan and Vsauce. Commenting now when the views are in the thousands and your subs are in the hundreds, not hundreds of thousands.

@RuleOfThrees 6 ай бұрын

I realized right away that this was going to relate to Bézier curves, but I did not understand how those work. Now I understand exactly what is happening, which makes using them in graphics tools so much easier. Thanks for the cool video!

@plutototoh 6 ай бұрын

This explains such a simple mystery from my childhood ❤ very cool. Thank you!

@KaiCyreus 8 ай бұрын

it just hit me that "cardioid" is "heart-like"

@ViktoriaMagrey 2 жыл бұрын

-I call it "Butt light", because to me heart-shapes are just butt-shapes.- ...Anyway, thank you! I hadn't come across this channel before and I love getting answers to questions like these! :D

@Eclecktro 2 жыл бұрын

💡+ ☕️=🍑

@rosuav 2 жыл бұрын

Wait, are you saying that you drink Butt light? I'm not sure if that's better or worse than the beer....

@yash1152 11 ай бұрын

_"heart shapes are just butt shapes"_ glad i aint alone. and thats precise reason why i dont use heart symbol as reaction wherever i have choice.

@Sreeto 8 ай бұрын

Okay you say this but the heart symbol is actually based on a woman's butt

@Circe-your-mom 8 ай бұрын

@@yash1152 Funnily enough, I use butts as reactions instead of hearts

@CALIGS 8 ай бұрын

Paralogical, your style is so nice to watch and your humor is refreshing.

@Ganerrr 2 жыл бұрын

massive W

@smoothbrained4channer976 2 жыл бұрын

indeed, this W appears to be one of of such great massiveness that it creates an easily detectable curvature in space-time

@glenncurry3041 2 жыл бұрын

Interesting to see the results of the light in the mug is the same pattern as an FFT graph in audio! There must be some form of wavelength component.

@ethanlane9873 2 жыл бұрын

Kinda! Because the shape of the mug is circular, it can be expressed with sinusoidal functions just like the wave motion of sound :)

@sankalp2520 2 жыл бұрын

They are related actually. The cardioid curve can be plotted in polar coordinates using a simple sinusoidal function r=1-cos(θ)

@hindigente 2 жыл бұрын

Absolutely brilliant. The way you speak is really soothing and reminiscent of Grant Sanderson. I particularly liked the envelope tangent (and the humour). Subscribed just in case you intend to make more of these.

@al6243 2 жыл бұрын

Based on the quality of this video, I was actually surprised to see your subscribers were only in the hundreds. I guess this is just how it is since your channel is still young and this is the only video so far. I am really looking forward for the future of your channel. This comment shall serve as my time capsule. I hope when I run across this again, your channel will have the views and subscribers it deserves.

@jasboi8642 2 жыл бұрын

1.6k subs now😄

@NipeHero 2 жыл бұрын

@@jasboi8642 3.25k subs now

@colouratura26 2 жыл бұрын

4.22k subs

@NipeHero Жыл бұрын

5.84K subscribers now

@yuez3555 8 ай бұрын

16k

@wege8409 2 жыл бұрын

It's kind of peculiar to me that even when the opposite side of the thorn is shrouded in shadow, it's still a strong enough effect to produce those rays of light.

@0r94n1z3r 7 ай бұрын

Agrh! You tricked me to learn math!!!

@josephjackson1956 2 жыл бұрын

Imagine being bad at eyeballing stuff, but using math to be super accurate

@linguamathematica2582 3 жыл бұрын

Thorough and efficient explanation. I enjoyed this a lot.

@tirone7520 6 ай бұрын

i like how you brought us from simpler concepts to harder ones the journey was good

@fili275 3 жыл бұрын

So cool! Love the explanation and humor.

@randomz5890 2 жыл бұрын

This was an amazing video. I'm astounded at how much style and humour it had. As well as a detailed and decently in depth explanation of this concept which yet wasn't too detailed that it became confusing. Keep up the good work!

@silverharloe 8 ай бұрын

Thanks for this, it took me back to a time in high school when I learned about radial coordinates (r, theta instead of x, y) and wrote a program (in Turbo Pascal - there, that dates me) to tick along 360 degrees of theta and calculate r based on whatever formula the user put in and then plot the resulting points. Most of the results were boring, but with some sines and cosines in there, we got nice cardioid and limaçon shapes. Your image of the lights in the bottom of a mug reminded of that time and gave me a sweet nostalgia hit. Thanks!

@maiamaiapapaya 4 ай бұрын

"I have bad eyeballs" really got me for some reason. 🤣

@bartdenturck5134 7 ай бұрын

First year engineering at the university in Ghent, Belgium. The prof called it a kidney line. Don’t remember how he did it but this explanation of yours is real elegant.

@MrGnome-ng6jv 2 жыл бұрын

I would love to see a more detailed breakdown of the derivation of the formulas you have presented. Not all of us can read those mathematical scribbles so easily as others! Excellent video.

@_Just_Another_Guy 7 ай бұрын

I have question: does the "envelope" shape change depending on the shape of the cup? What if it was a square cup with perpendicular and flat surfaces instead of a curved surface?

@anachronismic 2 ай бұрын

A square cup wouldn't reflect light in the same way, my intuition is that it wouldn't show much shape at all. It might be better to think about what happens for like an oval cup, or something with a smoother shape.

@rtoole13 3 жыл бұрын

Fantastic illustrations! Hope to see more from you.

@Sraktai 8 ай бұрын

Love your videos. Hope to see more. Never settle for speed over quality, these videos are masterpieces.

@shewittau 2 жыл бұрын

I've always wondered about the "cup bum" since I was a child. Thanks for the video.

@sentienttapioca5409 7 ай бұрын

I used to marvel at these 'mug butts' as a kid! Cool to know I'm not the only one to wonder how they happen.

@pramitadas7814 6 ай бұрын

This video gave me the 'excited grin' that's been missing for a while now. Please upload more such videos. Thank you for reminding me how much I love Maths 🥺

@seneca983 2 жыл бұрын

0:15 "heart shaped curve" I've always thought it's shaped like a butt.

@navidryanrouf441 2 ай бұрын

Optimist: the light makes a heart shape Pessimist: butt

@rigierish3807 2 жыл бұрын

It's fascinating : you start from simply questioning why there is a weird light shape in the bottom of a cup, a question a 7 year old could ask, and you end up with envelopes, derivatives and trigonometric tools. Plus, the video is pretty well explained and easy to understand even if you don't have any base in mathematics thanks to the illustrations. Anybody who claim hating math would probably reconsider it after seeing this video.

@emj-music 8 ай бұрын

i liked this video a lot! i’ve always wondered what that reflection was, and tying it together with calculus and limacons really was the cherry on top for me. bravo!

@adheensheikh6896 2 жыл бұрын

What I never understood and what even this beautiful video fails to explain (or even talk about) is how is it intuitively clear that the envelope is tangent to each member of the family

@starlight8554 2 жыл бұрын

I love coming across videos like this. These little gems make wading through the cesspool of the internet less insufferable. Bless you.

@marcel1372 7 ай бұрын

the lines making a curve thing blew my mind in middle school math i would do exactly what ur doing in line rider all the time

@123xlash 7 ай бұрын

When I was a kid, I would mess around with functions on my graphing calculator, and when messing with polar coordinates, I drew the graph for r = 1 + cos θ. The shape stuck out to me. I've since then noticed this interesting pattern at the bottom of a mug. I always figured they were the same shape, but I never knew why until now. Thanks for the great explanation of envelopes!

@movax20h 2 жыл бұрын

It is really cool, that it works without knowing explicit form of the family of curves. Implicit form is sufficient.

@echognomecal6742 7 ай бұрын

One of the mysteries connecting all kids with mugs: The mug heart-butt, examined, mulled over, wondered about, appreciated (especially on particularly sunny days or right under the "good light"). Clicked on this ImMeDiAtEly then got distracted with Line Rider.

@johnrickert5572 2 жыл бұрын

Back in the 70's, string art used the idea of envelopes, e.g., to form a ship's sail. I've wondered about the mathematics for quite a long time. Thanks for this very fine answer.

@Hyraethian 2 жыл бұрын

0:08 me, "oh no that looks a lot like the Riemann zeta function ." also me, fastens seatbelt and prepares for math. edit: I realized I untensed when It became clear this was not where the video was going.

@jonathanlister5644 Ай бұрын

The light pattern we see is called a Caustic, they are concentrations of light rays that form bright filaments, often they display cusp singularities.

@IshaaqNewton 2 жыл бұрын

A man of culture: that's an ass right there

@DanteYewToob 4 ай бұрын

“We get this… perfect little…” Me: “Butt. A little light booty.” “Heart shaped curve.” Me: “Yeah. A heart. That’s what I see too. A nice, plump, and thick… heart.”

@JohnSmith-yn4uu 2 жыл бұрын

Such an underrated channel. Don't know how I got recommended this channel but I'm hooked.

@tamimyousefi 2 жыл бұрын

I've always wondered about that since I was little, and never bothered figuring it out since I thought I would be to complicated. What a magnificent way to break this down. - Regarding the calculation of the envelope, something is fishy about F or its time derivative. I haven't done a derivation of F myself, but if F is correct the last term in the derivative is wrong.

@Psycandy 7 ай бұрын

in 3D rendering we call these light interactions *caustics* as the effect has its own render engine. Works well with glass.

@dearlantsov 2 жыл бұрын

i remember as a kid while drinking milk, i would look at this light, something about it just fascinated me so much and i would stare at it for hours

@jakobjonsson1474 8 ай бұрын

Interesting! I just learned about Nuquist curves in control theory and they look just like this!

@mohdanzar1895 7 ай бұрын

"When Daddy curve and mummy curve..."🤣🤣

@Foxxey 7 ай бұрын

I already knew about the quadratic-bezier-curve-like shape and cardioid but I never quite understood how to formalize these concepts. Your video is amazing at explaining the connections to envelopes and has some cool visuals! Keep it up!

@NobodyYouKnow01 7 ай бұрын

I haven’t even watched the video and I’m fascinated that you found a cardioid of light.

@0MVR_0 2 жыл бұрын

Looks similar to the right side of the Mandelbrot set where the geometry acutely converges due to the 'curling' imaginary influence.

@SkyfishArt 2 жыл бұрын

does it have anything to do with mandlebrot? i know nothing about math but they look similar.

@0MVR_0 2 жыл бұрын

@@SkyfishArt Yes, in the way that real and imaginary numbers interact, by convention, is through perpendicular projection. The envelope discussed in the video is essentially a concave 'focus of rays'. Something similar happens with the Mandelbrot Set since y=x^2. If one draws a circle centered on zero with radius 2 the circumference is the limit of interaction, the 'cup' of mandelbrot. Yet the shape happens because the bound of convergence occurs within that circle, one might call the effect an 'envelope' where the growth of the series x^2 is static. The most intense 'reflection' occurs near the horizontal real line x=1. However, the influence of an imaginary number, the root of -1, is to rotate ninety degrees the same way the cup is round. Hence the curl with a sudden intrusion.

@DavidMartinez-ip1dr 2 ай бұрын

I loved the line rider analogy. Very clever!

@ittixen 2 жыл бұрын

I had an intuitive understanding of what was going on, but I had no idea there were such elegant formulas to describe it. Beautiful and clear illustrations!

@antonk.653 2 жыл бұрын

Incredible. The video quality, the editing, and the near flawless marriage between nerdy math and intuition. You deserved my subscription, I wouldn't want to miss more of those that are (perhaps) to come.

@sage5296 8 ай бұрын

I figured it would be a cardioid, but I didn't know a cardioid could be represented as an envelope like that, that's cool!

@dyllanusher1379 2 жыл бұрын

Great video! I used to draw envelopes all the time and now I have a piece of art of some lines forming a complicated envelope that I found at a thrift store!

@susmita5548 7 ай бұрын

Same , i was too fascinated by this reflecting designs

@ThinkBeyond01 2 жыл бұрын

I too have participated in SoME1 and found this absolutely fantastic video on KZfaq while scrolling my recommendation. The way you have presented the complete video with every single point is phenomenal! Keep it up!

@LeoStaley 2 жыл бұрын

I've watched 3 dozen of these entries and this is easily in the top 2.

@chessematics 2 жыл бұрын

I tried to crack this exact nut a few months ago, but I never thought about relating it to partials and parameters. A million thanks for clearing my doubt. I can write a complete proof (rigorous) of it, if you wish.

@Krail1 2 жыл бұрын

I love when we can get cool math examples out of something I used to doodle in the corner of my school notebooks!

@_rodr11go 2 жыл бұрын

Dude this video is so good, wish get a lot more visibility because you definitively deserve, keep going with the content

@J-W_Grimbeek 2 жыл бұрын

Even though i understood almost none of the maths, your voice was soothing enough to listen to to enjoy the video. Not to mention I've seen lots of art made with these curves, so knowing they're called envelopes is cool. Not to mention the cardioid is a really cool shape in general

@MrSaemichlaus 2 жыл бұрын

This is a phenomenon that I've just barely noticed on the side of things, and I never bothered to find out why it happens. Thanks for showing me!

@rfldss89 8 ай бұрын

Ah yes of course. Heart shaped, yes.

@beaub152 8 ай бұрын

Very cool concept. Never learned this specifically in calc 3, but recognized that shape. Keep making new videos, i came from your new one

@masaltzero 2 жыл бұрын

The Mandelbrot Set: “hey don’t make me me my ima borrow this”

@AlmostOffline 11 күн бұрын

This is a fantastic video, love the visual explanation of everything. Consider adding a link to line rider in the description!

@wernerkarlheisenberg2182 7 ай бұрын

Now this was a video I didn't know I needed

@dadabranding3537 7 ай бұрын

You are a freaking rockstar educator. A league of your own. Keep making these.

@icyspectre 2 жыл бұрын

Wow, it's crazy to find out that envelopes are a "thing". I used to doodle these all the time at school, usually as a circle within a square.

@jun1408 2 жыл бұрын

omg love your video, not only show formula, it let me know math is in our real life in the cup but no much people notice, hope you keep doing it

@yasminh 5 ай бұрын

this has bothered me for so long I feel so relieved getting to finally know this thank u sm

@sayantanray6608 Жыл бұрын

I was looking for someone to explain it logically and Im fascinated by the way you represented it. Best explanation on envelope Ive ever seen. Im looking forward for your next videos :)

@yasminh 5 ай бұрын

2:10 we used to draw these curves using straight lines in primary school but I never knew the maths behind it

@lawrencewhitfield8155 3 жыл бұрын

Very well done. Lots of enthusiasm.

@Green0Photon 2 ай бұрын

7:28, when he was explaining envelopes at the beginning, i shouted that's a quadraric bezier curve! Mentally. In my head. And i was surprised when he said it was an envelope. But no, i was right, it was the definition of a quadratic bezier curve! It was just an example. This was a very good video.

@sahilnavale6558 2 жыл бұрын

Glad you didn't only show the concept but also the math therotical side.

@priyanshugoel3030 2 жыл бұрын

Real nice way to understand something which comes up in differential equations but is often not explained at length.

@Stilllifebutwhy 8 ай бұрын

Interestingly, yesterday KZfaq recommended me a video about this very effect. The video was a 2-year-old from the small channel with 2 videos and explained the mathematical term envelopes and how such a pattern is formed by light in a cup. And now KZfaq has decided to show me this shorts.