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Sum and Product
Germany
Приєднався 29 лип 2021
Cutting Food With Maths
Many of the ways mathematicians discuss subdividing shapes revolve around food. Let's create a carb-heavy menu of five examples for that!
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Join the Sum and Product Discord server:
discord.gg/G8uXtfaMgu
Follow Sum And Product on Twitter:
SumAndProduct
...or Instagram:
sum_and_product
Or buy me a coffee:
ko-fi.com/sumandproduct
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Resources, references & further reading material:
Animations were made with CindyJS
cindyjs.org/
Emojis on Thumbnail by Openmoji
openmoji.org/
xkcd 804, Pumpkin Carving
xkcd.com/804/
Original Alice, Bob and Charlie busts created by Pablo Stanley
The Ham Sandwich Theorem on Wikipedia:
en.wikipedia.org/wiki/Ham_sandwich_theorem
The Intermediate Value Theorem on Wikipedia:
en.wikipedia.org/wiki/Intermediate_value_theorem
Stan Wagon (1985), "The Banach-Tarski Paradox", Cambridge University Press
Katie Buchhorn (2022), "The Banach-Tarski Paradox"
arxiv.org/pdf/2108.05714.pdf
Edmund Weitz (2023), "Das Banach-Tarski-Paradoxon" (German), Christmas Lecture at HAW Hamburg
ua-cam.com/video/fHE94q2g3hs/v-deo.html
Larry Carter & Stan Wagon (2018), "Proof Without Words: Fair Allocation of a Pizza", Mathematics Magazine, 67 (4): 267
doi.org/10.1080/0025570X.1994.11996228
Jeremy, Michael, Jeremy K., Andrew and Philip Hirschhorn (1999), "The Pizza Theorem", Austral. Math. Soc. Gaz., 26: 120-121
www.maths.unsw.edu.au/~mikeh/webpapers/paper57.pdf
Martin von Gagern (2014), Answer to the question "can anyone explain Pizza theorem?", Math Stack Exchange, accessed on 2024-01-24
math.stackexchange.com/a/866003
Rick Mabry & Paul Deiermann (2009), "Of Cheese and Crust: A Proof of the Pizza Conjecture and Other Tasty Results", American Mathematical Monthly, 116 (5): 423-438
doi.org/10.4169%2F193009709x470317
The Lazy Caterer's Sequence on Wikipedia:
en.wikipedia.org/wiki/Lazy_caterer%27s_sequence
Jack Robertson & William Webb (1998), "Cake-Cutting Algorithms: Be Fair If You Can.", CRC Press
Haris Aziz & Simon Mackenzie (2016), "A Discrete and Bounded Envy-free Cake Cutting Protocol for Any Number of Agents", 57th Annual Symposium on Foundations of Computer Science (FOCS) (pp. 416-427). IEEE.
arxiv.org/abs/1604.03655
Steven J. Brams & Alan D. Taylor (1996), "Fair Division: From cake-cutting to dispute resolution", Cambridge University Press
--------------------
Chapters:
[00:00] Introduction
[00:46] Ham Sandwich
[05:07] Pumpkin
[10:30] Pizza
[17:15] Pancakes
[22:33] Cakes
--------------------
Join the Sum and Product Discord server:
discord.gg/G8uXtfaMgu
Follow Sum And Product on Twitter:
SumAndProduct
...or Instagram:
sum_and_product
Or buy me a coffee:
ko-fi.com/sumandproduct
--------------------
Resources, references & further reading material:
Animations were made with CindyJS
cindyjs.org/
Emojis on Thumbnail by Openmoji
openmoji.org/
xkcd 804, Pumpkin Carving
xkcd.com/804/
Original Alice, Bob and Charlie busts created by Pablo Stanley
The Ham Sandwich Theorem on Wikipedia:
en.wikipedia.org/wiki/Ham_sandwich_theorem
The Intermediate Value Theorem on Wikipedia:
en.wikipedia.org/wiki/Intermediate_value_theorem
Stan Wagon (1985), "The Banach-Tarski Paradox", Cambridge University Press
Katie Buchhorn (2022), "The Banach-Tarski Paradox"
arxiv.org/pdf/2108.05714.pdf
Edmund Weitz (2023), "Das Banach-Tarski-Paradoxon" (German), Christmas Lecture at HAW Hamburg
ua-cam.com/video/fHE94q2g3hs/v-deo.html
Larry Carter & Stan Wagon (2018), "Proof Without Words: Fair Allocation of a Pizza", Mathematics Magazine, 67 (4): 267
doi.org/10.1080/0025570X.1994.11996228
Jeremy, Michael, Jeremy K., Andrew and Philip Hirschhorn (1999), "The Pizza Theorem", Austral. Math. Soc. Gaz., 26: 120-121
www.maths.unsw.edu.au/~mikeh/webpapers/paper57.pdf
Martin von Gagern (2014), Answer to the question "can anyone explain Pizza theorem?", Math Stack Exchange, accessed on 2024-01-24
math.stackexchange.com/a/866003
Rick Mabry & Paul Deiermann (2009), "Of Cheese and Crust: A Proof of the Pizza Conjecture and Other Tasty Results", American Mathematical Monthly, 116 (5): 423-438
doi.org/10.4169%2F193009709x470317
The Lazy Caterer's Sequence on Wikipedia:
en.wikipedia.org/wiki/Lazy_caterer%27s_sequence
Jack Robertson & William Webb (1998), "Cake-Cutting Algorithms: Be Fair If You Can.", CRC Press
Haris Aziz & Simon Mackenzie (2016), "A Discrete and Bounded Envy-free Cake Cutting Protocol for Any Number of Agents", 57th Annual Symposium on Foundations of Computer Science (FOCS) (pp. 416-427). IEEE.
arxiv.org/abs/1604.03655
Steven J. Brams & Alan D. Taylor (1996), "Fair Division: From cake-cutting to dispute resolution", Cambridge University Press
--------------------
Chapters:
[00:00] Introduction
[00:46] Ham Sandwich
[05:07] Pumpkin
[10:30] Pizza
[17:15] Pancakes
[22:33] Cakes
Переглядів: 504
Відео
Solving Matthew Scroggs' Puzzle Advent Calendar 2023
Переглядів 1544 місяці тому
I immensely enjoyed the 2023 puzzle advent calendar by Matthew Scroggs. And while I don't think I have a profound comment on each topic covered, you maybe still enjoy my thought process solving them. Please let me know in the comments if you would like to see an actual live solve for the 2024 edition! Join the Sum and Product Discord server: discord.gg/G8uXtfaMgu Follow Sum And Product on Twitt...
Cindy Build: Riemann Integrals
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This is the second of my Cinderella builds videos. It complements the first one, but is basically independent of it. With this, I hope you got a glimpse into the coding side of Cinderella. Please let me know in the comments if you are interested in more of this kind of videos and what kind of applications and visualisations you would want to see. Join the Sum and Product Discord: discord.gg/G8u...
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Переглядів 4278 місяців тому
Since some showed interested in learning Cinderella and CindyJS, I decided to do a little series of unscripted live builds of small applications in these programs. To start, I show you how to do calculations inside of perspective drawings or photos. Let me know in the comments if you are interested in more of this kind of videos and what kind of applications and visualisations you would want to...
Signed Distance Functions & Ray-Marching
Переглядів 27 тис.10 місяців тому
Tell me how far away something is, and I tell you what it looks like! This one took a while. Mostly due to other things in my life. But also because I stumbled over quite a few problems when trying to implement SDFs for the animation. As a consequence, I learned more, but I also had to rewrite the script a lot to convey the things I learned properly. Still feels much rougher around the edges th...
1000 Subscribers Special
Переглядів 727Рік тому
A big thank you to everyone who is watching my videos and subscribing to my channel! I can't produce videos as quickly as I would like, and so it's nice to see the channel grow nevertheless. If you are interested in more in-depth Cinderella/CindyJS tutorials, please let me know in the comments. Join the Sum and Product Discord: discord.gg/G8uXtfaMgu Follow Sum And Product on Twitter: twitter.co...
Complex Coordinates in Real Geometry
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Complex numbers usually appear solely as a way to solve polynomial equation. Here, I want to show you what that means for doing geometry. This video complements my other one about perspective drawings: ua-cam.com/video/mTw3o8-xMIo/v-deo.html While they can be understood completely independent from one another, they cover similar ideas from different perspective. It's definitely beneficial to wa...
Formal Concept Analysis
Переглядів 4 тис.Рік тому
There are many ways to do data analysis. One very descriptive version looks at nothing more than which objects have which attributes and goes from there. It is called Formal Concept Analysis and it is used in and for Chemistry, Theoretical Computer Science, Data Mining, Biology, Linguistics, Algebraic Geometry, Machine Learning and many more areas for all sorts of classifications. Join the Sum ...
Conway's Doomsday Algorithm
Переглядів 3,8 тис.Рік тому
Better sound, quieter music! I hope… Also, plenty of experiments with typefaces/fonts and other styles. If you want to impress people at a party with your insane maths skillz, ask them their birthday and immediately (plus/minus a couple of seconds) answer with the weekday it was. In this video, I show you how. Moreover, a while ago, I wrote myself a little app to practise this algorithm. Withou...
Introduction to Projective Geometry via Tic-Tac-Toe Grids
Переглядів 49 тис.Рік тому
My entry for @3blue1brown's Summer of Math Exposition 2022. It's my first video ever and there are a million things I would like to improve. Mostly the audio quality and my terrible, terrible accent. 😅 Alas, perfect is the enemy of the good, isn't it. So, here we go. #SoME2 Join the Sum and Product Discord: discord.gg/G8uXtfaMgu Follow Sum And Product on Twitter: SumAndProduct ...or...
how to do collisions tho ? you have source for that ?
I have nothing on hand, as I don't use it much for physics. (Except sphere-vs-SDF, which is automatic.) But there's a new paper, which looks interesting: doi.org/10.1016/j.cagd.2024.102305 Haven't had time yet to read it, though.
Finally found the time to watch the entire Video. Just BRILLIANT!!!! Thanks so much Bernhard!!!
Thank you so much! I'm really happy you like it!
S tier video 👏 Thank you!
Loved it 9.5, not a perfect 10 due to loud music
Do you know if this tool has been used for autoomatic text summarization techniques. It could probably be useful for it?
I know that FCA is used in computational linguistics. But I'm unfortunately not experienced enough to know for what exactly.
Very good video! 15:38 should the right side yellow equation be (1, inf | 2, 0)?
Thank you! Yeah, it seems I mixed up the order here...
Very good exposition. Really really clear, I thoroughly enjoyed it!
Nice video
this recipe video is looking a bit weird, but it should work!
edit: I removed the concept of mass from my food and I can now make a free copy of my pizza!
Amazing video!! Love it!!
Sehr interessant und hilfreich. Mehr zum Thema projektive Geometrie wäre wünschenswert.
Hab noch einige Themen zur projektiven Geometrie auf meiner Liste. Weil die aber nicht alle gleich verfeuern.
@@sumandproductlook forward to see your works.
projective geometry too hard for me .but you make it easy
This content is gold! I'm from Brazil and I'm developing a web platform with interactions, I saw that cindy.js would be ideal. Please keep bringing more videos like this, they are really good! A video showing how you do the animations, in addition to the github repository would also be great
Oh, I will definitely make more CindyJS tutorials! My animation code is very bad, so, I'm still a bit hesitant to make the repos public...
Both explanation and visualization are just perfect. Thank you!
3:13 This moment is brilliant and enlightening
intro is wrong. we do xyz/w (FOV * Z). Z used in an image to check if something is visible
German teachers are the worst….want to learn anything engineering maths,stem related…see Indian teachers…that’s why indianz have scaled maths and engineering education like crazy while these Europeans,Russians Germany teachers struggle big time
Amazing thank you very much ! just a little question your animations are really nice and smooth, how do you manage to do that ? is it a particular software or clever use of powerpoint + editing software or others ?
Thank you! All animations are made with cindyjs.org/ . Since it isn't designed as an animation tool, I do lots of custom code as well, though.
Could you provide a detailed proof (or reference) for properties of derivation operation provided in video?
The book "Formal Concept Analysis" by Bernhard Ganter and Rudolf Wille sounds contain everything you need.
amazing video and extremely informative!
Thank you for the amazing video ! I'm planning to use FCA to create concept heirarchies and this was a great primer on the subject coming from a non-mathematical background ! Cheers
Hello! Thank you so much for the video. Please clarify one moment: Is it mistake in 1st Concept Lattice? There is [ (Bo,Fa,Ti) (HDMI) ]. But should be [ (Ch,Fa,Ti) (HDMI) ] instead. Respectively, down in lattice: should be [ (Ch) (CD, HDMI, bio) ]. Pls, correct me if I'm wrong.
Yeah, you're right. Very weird... Guess I shuffled a few rows around late in the production of the video and didn't update the labels...
@@sumandproduct Got it, thank you!
I am struggling with februaries. Everytime i get it, i do 28 or 29 minus date to find what the closest doomsday is, but it gives me a negative number since 28 is larger than the number of the date. Am I calculating this wrong? Im not sure what to do with februaries at this step.
Oh, negative numbers aren't a problem! Say you have -4 at some point. That means you're 4 days before Sunday, which is the same as -4 + 7 = 3 days after Sunday, which would be a Wednesday.
@@sumandproduct Thank you so much, this is the first explanation i've seen. Subbed!
Oh wow, CindyJS looks awesome! I'd be really happy to see such tutorials :)
This feels interesting in theoretical circumstances, but I feel like it might need to be adapted in practical examples like the laptop one? With subjective reviews like this, I feel like having one "incorrect" review that overvalued or undervalued a laptop might change the results - real data is noisy and the mathematical rigor of this theory might be less suited to the problem than a more flexible neural network or genetic model. Worth looking into, though.
Oh, yes, this is pretty much a toy example. I assume that you can somehow "trust" the experts.
Great video! Unsolicited English tip: pronounce "v" as a hard, sharp sound to differentiate it from the softer rounder "w". Don't be misled by the "v" character! In English it is pronounced like start of "wasser" in German.
Very nice video! I learned a lot of new perspectives. As someone who works with spatial analysis tools on large datasets, I often wonder about how graphics approaches work versus the implementations of basic spatial relationship calculations. There is obviously much conceptual overlap, and I’m curious how many techniques are shared directly.
Everything is clearly explained @sumandproduct, great video, congratulations! I just wonder which software did you use for the figures and animations? Thank you
I'm using cindyjs.org
@@sumandproduct Thank you!
Lovely introduction! It’s nice you not only mentioned what SDFs are but also derived a couple of basics ones 👏🏻 P.S. Though ray marching usage is pretty rare, typically we render “pre-baked” 3D geometry (stored as a list of vertices and normals not its mathematical description) with some sort of ray tracing.
Is Cinderella free?
Yes it is!
I’m confused, is <A, B> just a different way to write the dot product, or is it something different?
It is the dot product!
How does ray marching work in spherical or hyperbolic geometries?
I imagine it works similarly whenever you have a metric.
Ray-marching for collision-detection? Sounds useful. How to handle the different shapes of the objects involved, though? Could be a topic for a new video.
I've only used it for simple physics simulations where the objects moving around are small circles. But it's probably worth looking into more...
@@sumandproduct Yeah, I can see how you'd do sphere-anything collision. Subtract the sphere's radius from the SDF of whatever it might hit and then ray-march the same as if you're rendering the scene. Basically, making a rounded box and doing sphere-box collision would be the same process. The real headscratcher is how to handle other shapes and the potential rotation that would be introduced when they collide.
@@Roxor128 From what I learned during the research for this video, I wouldn't be surprised if checking for collision of two arbitrary SDFs is just not possible.
@@sumandproduct it is actually possible, but it requires gradient descent on the sdfs, and likely only works on true sdfs, not ones created using the unions of the shapes. it boils down to reducing the problem to the same one as colliding sphere sdfs, but by solving the system of equations formed by the two sdfs you are colliding. very possible, but time consuming and not very performant, as opposed to using bounding volumes to perform collision detection. There are several papers on it from Nvidia/other graphics researchers, and I think it probably does have it's place in physics simulations
@@bigfloppa9220 Oh, that makes sense, yes. Thank you!
If you look at the two points where all circles intersect, if you scale them by a complex unit then they rotate. one of them turns clockwise, one goes anticlockwise. you can see this if you look at the real part.
BEAUTIFUL! I can finally get a proper introduction to projection geometry without the need to look up all different books on the internet! Nice!
finally a french competitor to 3blue1browm
@@markusa.stokkenes2271 He IS German ;-)
@@whannabi no they mean the COMPETITOR is french not the competitor's VIDEO
although, i think they're german, since the channel is in germany
Doesn’t sound French at all. German. France produced Fabrice Bellard and that’s enough for a few centuries 😅
Great video!
Thank you!