That weird light at the bottom of a mug - ENVELOPES 

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.

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!

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.

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

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

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!

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!

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

This guy is about to be blessed by the RU-vid algorithm Excelent video

massive W

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.

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"

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 💜

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

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

Agrh! You tricked me to learn math!!!

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.

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.

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!

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

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 😭

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!

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

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.

-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

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.

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.

My prediction: this is going to get really big really fast. This will be one of those videos that will randomly appear in your RU-vid 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.

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!

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.

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

So cool! Love the explanation and humor.

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.

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.

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 🥺

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

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

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 :)

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

Fantastic illustrations! Hope to see more from you.

Thorough and efficient explanation. I enjoyed this a lot.

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?

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.

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

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

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!

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!

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.

Ah yes of course. Heart shaped, yes.

This is such a great educational video, I was shocked to find it's your first one. Thank you and please keep on making videos like this one!

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!

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!

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!

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!

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

I love it, the math, the jokes, the animation. Please, make more videos.

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

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

I had to do a double take when I saw you only have 550 subscribers. This was a beautifully constructed and explained video, well done. Hope you continue to make more of these.

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

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

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

This is very informative and interesting, I had to draw "those cool math lines" in an art class some years ago but they didn't explained what they were so your video opened my eyes, thank you

Very well done. Lots of enthusiasm.

Great explanation and style! I hope you create more videos like this :D

Very nice job!!! Loved the explaination and the math behind this natural phenomena

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

One of the most pleasing videos I have ever seen!

Light absolutely fascinates me, thank you so much!

Immediatley subscribed. High quality content at it's finest. Keep it up! 💪🏻💪🏻👏🏻👏🏻

I have always wondered about it, thank you so much for explaining it. I didn't understand it completely but now I know where to look to learn more.

Thank you very much and you have just solved my mystery rested in my mind for two decades!

ur quality, amount of knowledge and insight is out of the world man, keep going ..............

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

Actually amazing video bro, im glad to say i was here before it blows up. Good luck with SoME1, great video!

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.

What a professional video! I loved the end, this is a million-sub worthy channel.

Delightful. Congrats on being a big winner.

I've been wondering about this for a long time. Great video! :D

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

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

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.

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.

Simply delightful! What a succinct and intuitive explanation!

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

Nice! I am not the only person who looks at the bottom of an empty cup! Great and simple explanation, easy to follow and absolutely loved the humour!

Dude I love this! Make more videos!

Thank you for sharing this! I have worked with convex envelop of functions before, but never did I understand what only does the points on the contact set matter. This video gives me some insight of it.

Nicely done! MOAR WE NEED MOAR

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

I can really see the efforts you put in this video, no one can stop this channel from going viral now, good luck dear

Awesome vidéo ! Each of the aspects fit perfectly to create some enjoyable, understandable and detailed content !

I loved the line rider analogy. Very clever!

Reached here from 3B1B I have always wondered about this exact topic. I even sent images to my friends as a fun question. So delighted to see this video Thanks a lot!!

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

Im surprised this channel doesn't have 500k subs. Amazing content!:D

I love this video so much, I learnt so much and paralogical's explanation is so engaging

we need more of this type of stuff!!

Okay I looked at this channel after watching the video to watch some more videos like this. But there aren't any other videos posted. This was such a good video! Please, if you see this, continue your RU-vid channel!

I was thinking about that cardioid the other day, good thing I found this!