The hidden geometry of SET (
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- เผยแพร่เมื่อ 22 เม.ย. 2024
- The family game SET is a quick-paced game of pattern recognition. However, there's more than meets the eye. It turns out that one can view the game in a geometric way. Through a clever correspondence the game is translated to finding lines in F_3^4! This connection allows for many beautiful constructions.
This video was made for the 3Blue1Brown second Summer of Math Exposition "competition" (#SoME2). See the following video for more information: • Summer of Math Exposit...
Sources I used for this video:
www.tandfonline.com/doi/abs/1...
homepages.warwick.ac.uk/staff/...
Music in this video:
"Deep Relaxation" Kevin MacLeod (incompetech.com) Licensed under Creative Commons: By Attribution 4.0 Licensecreativecommons.org/licenses/b...
Hachiko (The Faithtful Dog) by The Kyoto Connection | freemusicarchive.org/music/Th... promoted by www.chosic.com/free-music/all... Commons CC BY-SA 3.0creativecommons.org/licenses/...
Chapters:
00:00 - Introduction
00:23 - The rules
02:20 - Cap Sets
03:04 - Finite dimensional vector spaces
04:32 - F_3^2
06:25 - Explicit connection between SET and F_3^4
08:35 - Quick recap
09:02 - Magic Squares
10:48 - Remaining Cards
11:58 - Cap Sets in detail
12:32 - Maximal 2-cap
13:27 - Maximal 3-cap
14:49 - General d-caps
Made entirely in Blender:
www.blender.org/
I have some free add-ons, you can find them here:
github.com/WannesMalfait/Blen...
Any suggestions?
Leave a comment
Beautifully explained, I love SET!
Fun fact: if you take all the possible SETs (of which there are 1,080), 10% have only one feature different, 20% have all features different, 30% have two features different, and 40% have three features different.
Wow. I wonder what the breakdown of CLAIMED sets by real players looks like. I’d think everybody has a bias. Certainly the 1-difference has to be over represented
@@AlphaPhoenixChannel Interesting question. My guess is that 2-differences will be the most underrepresented. Those seem to be the hardest to spot.
So, the scan acceleration strategy should be start from "fixing" a single most prevalent feature(s) currently on the table and focusing on building a 3-diff set with this single shared feature, then fixing the next most prevalent feature and so on?
Haha I love it when the Pigeon hole principle gets used in high level mathematics - it's so simple it's hard to imagine it's useful when I first learned of it.
Oh, the pigeon hole principle is everywhere!
My stupid ass immediately assumed you were talking about the category SET not the game.... I was like damn he really gonna do a video on category theory talking about the geometry of a large category while just keeping it 15 mins lmaooooo
For that exact reason I wrote SET instead of Set, and also put the cards in the thumbnail. I understand the confusion though. Writing something about the game SET is very confusing. You have to juggle sets, SETs, Cap Sets and Set...
Sit down 3b1b a new show is in town
Watch out, Blender is the new Manim (don't do this to yourself, making programmatic animations in geometry nodes is a pain)
Had to do a double take at that username lmaooo I didn't expect you here
3B1B is way more elegant and better explained.
I looked at the first 3x3 nicely ordered grid and was like, "Hey, every SET in here lies in the shape of a term in a determinant equation of a 3x3 matrix. That's really suspicious."
That is how mathematics often works. You see a 'suspicious' pattern, then try to figure out why the pattern is there
Very nice explanation. I'd like to add one minor point (doesn't change a thing of what you said, just a curiosity)
When you look at the final six cards of a game, and pair them, and find that the cards that complete SETs with these pairs are themselves a SET, there is a possibility that you could find the same card three times. This is also technically a SET (cards would have same number, same colour, same shading, and same shape), albeit not one you can encounter in the game normally, since all cards are unique.
I hadn't thought about that, but you are right of course. Very interesting point to make. Another fun fact which I didn't include in the video is that you can always figure out what card is missing if one card was taken before the game. (Just calculate the additive inverse of the sum of the remaining cards at the end of the game)
@@WannesMalfait So actually a variant that I was taught and I really like, is to deal the last card upside down, and not reveal it until one of the players claims it in a set.
Multiple times during this video I audibly said, to the screen, "dude... theres no way!" Thoroughly entertaining and informative!
Set is one of those great things that feels like it *has* to have a deeper connection and I’m glad I now know it does. Great video!
Most excellent! they use this example when teaching abstract algebra!
Great video! I remember playing this game years ago, and watching a well made mathematical analysis was amazing!
holy shit. i haven't been this impressed by educational content in quite a while. this video is fantastic and you seamlessly combined more than just 2 mathematical concepts. absolutely fantastic!! thank you for putting this out, and I really hope you make more content in the future!
I discovered the "trick" with adding the card-"values" with modulo to write a set-solver some years ago. Fascinating to see the same method I discovered myself a little bit more mathematically correct explained :D
Oh hey, it only took me 6 months to finally watch this video 😅 and I loved it! From one of my favourite games and interesting geometry to your soothing voice and nice editing, it all comes together so well!
Y'know, I say that it's my favourite, but in reality, at some point it became too easy for me because I learned a certain trick. Now, watching your video, I realized that I have instinctively found the rule that every two cards have a unique third card to form a set, and now seeing the mathematical proof got me quite happy 😊 I also didn't know about the 20-cap rule because I never paid attention to the fact that there's always a set with 21 cards
Anyway, I'll make sure to check out your other videos, and keep up the good work!
Also I now need to learn about the pigeon hole rule, cuz that's the first time I heard about it 🙃
Geesh you are getting way to good at making videos. The quality of this one is amazing keep it up.
I have been waiting for someone to make a video on SET for ages
6:15 when the pieces finally fit! 😤👌
I used to play this game a lot as a child! This brought back some core memories
Thank you for this beautiful informative video
This is wonderful. I'm also highly impressed that you did this all in Blender. I've tried picking up blender but found it too confusing. That was many years ago though, so perhaps things have changed.
@loudgreenideas2 That's awesome! I'll have to check it out. I've been wanting to learn that type of thing for a while now and just haven't bit the bullet.
Amazing! As a TH-camr myself, I understand how much effort must have been put into this. Liked and subscribed :)
I'll be watching this several times thanks
Well done! Clear and entertaining video
Way cool! My first thought on the 2d group was 5, but hadn’t accounted for the PBC lines! That’s pretty wild
I got to the part past where you talked about the wrapping of the line, and then highlighted the points in the 3x3 and I literally paused the video and my jaw dropped.
That's freakin' cool, man. Thanks for posting!
Awesome video, thank you for sharing
My favorite set fact is the one he mentioned that for any two cards, there is(!) a unique(!) third card that makes a set with it. This fact is fairly clear on its face, but what it really is is a way for people to get better at the game. Take two cards and visualize what the third card is. I still do this when we’re unsure if there is a set on the board; run through the pairs, then check if the third card is on the board.
super lovely video. combinatorial geometry is so pretty...
Loved it!
Amazing game. Amazing video, especially for those of us who love the game.
Thanks for sharing! ☺️
Absolutely mind blowing vid!! The whole time I was waiting for the max 4-cap proof and ..."sadly the techniques discussed in this video dont work on 4" discrete math being discrete math XD
To be honest, the proof for the max 4-cap is not that hard, it just becomes more combinatorical and less visual. I felt that it was too difficult to include in the video, but I think a lot of viewers could understand the proof without problem. I linked to a paper in the description which includes a proof
Excellent
I would like a sequel with representation theory and Fourier transforms, since those are the parts of the argument I never understood enough 🙃
Very cool!
So i guess this is technically....
Set theory?
Get out
Very nice video with pleasant animations. :D
I'd been poking around in finite projective vector spaces like the Fano plane. My ears pricked when I heard "Given any two cards, there is exactly one card that [lies on the same line]."
Projective planes also turn up in "Dobble" / "Spot It!" as "Any two cards share exactly one symbol". The Fano plane is like a game of Dobble with 7 cards, each with 3 out of 7 symbols - then for any two cards, their shared symbol appears on precisely one other card.
The Fano plane is minimal with that last property, iirc. I haven't yet checked to see which larger geometries have the same property and could turn Dobble into a SET-like game of near equivalent size.
On second thought, any bipartite cubic graph will do - 81 cards each with 3 of 81 symbols. Grr. No geometry for me this time.
Unless biregular graphs happen to be intimately related to finite geometry. I guess with Levi graphs, you could add some geometric flair, but it's not necessary for Dobble.
Shouldn't it be called a "finite vector space" not just a "finite dimensional vector space"?
Hmmm, yeah I guess you are right. Although not wrong, finite-dimensional doesn't really describe the fact that we are working on finite fields. Good catch!
@@WannesMalfait The least ambiguous way of actually describing this is to just name the vector space itself.
It also turns out that for knots, one uses the formula 2a - b - c for crossings when developing the matrix representation. I doubt it is coincidence.
I haven't done much knot theory, but it does sound quite familiar indeed...
Learned this in MathPath 2022. I was so surprised to see this on my recommended list since I didn't even search up SET once.
EDIT: there was more stuff on affine geometry as well (including a higher number of possibilities per attribute.)
we ended up making AG(5,3)
Yeah, there is still a lot of interesting stuff I didn't touch on in this video. The afffine isomorphisms become especially important when talking about the combinatorics to correctly talk about "unique games" for example
I was fully expecting a video about the geometry of the category Set, not the board game SET (never heard of it).
Something like how the subset relation when used to generate a graph produces a lattice.
I tried my best to make the title, thumbnail and description as clear as possible that the video is about the game SET, and not the category Set. Hopefully thev video was still interesting nonetheless.
Great video, but I think you could expand on how the pigeon hole principle applies in this case. I understood everything in the video clearly apart from that. I'm stuck staring at the graphic thinking "why must the 4 points be on the 3 lines?". Something there feels unsaid. Also I've always imagined the blue line you've drawn crossing 7 boxes as coming in at a 45 degree angle through x2 and x3 and reappearing on the right to hit x5 and crossing just 3.
I thought about expanding the explanation a bit more, but I thought it would make the video too slow. The explanation in more detail is as follows. The 3 lines cover all the squares in the grid (if they didn't then there would be another line through that point, but we already have all the lines). So of the 4 remaining points, they all lie on at least one of the lines. Since there are 4 points which have to be divided over 3 lines, there is at least one line with at least 2 of the points, by the pigeon hole principle. The blue line is just visualisation. You could draw it in multiple ways
i love set but I got a bit lost in the end. oh well. good job :)
Yeah, the end is a little bit rushed. I didn't have a lot of time yet, so the last parts are less polished and maybe too fast
What happens if you expand the rules of SET in the other direction - requiring k values that are all equal to or all different from each other in each dimension instead of exactly 3. Does it work for all k of prime power order? It certainly fails to generalize in the most obvious way to non-field finite rings such as Z_4, where {0, 2} is a line with 2 elements instead of the requisite 4.
So you would have a game "k-SET" where exactly k cards form a SET if and only if their features are either all the same or all different. I don't really know what could be said about such a version of the game, but I assume the geometric interpretation will vanish or at least be completely different when working over different kinds of fields.
I think that example at 2:26 is incorrect since squiggles represent a complete Set (the other Set is a shapes that are ''stripped'). Are those a Set?
They are not a set. They have to be all different or all same in all comparisons. The shapes are fine, all squiggles. The fillings are fine, all stripes. But for color, there is 2 purple and 1 red. The color is neither all same nor all different.
@@guitarbrother1234 Ahh! Got it now. Thanks! :)
Am I the only person who thought it was about the category SET?
Are you sure Matt Parker from stand-up maths didn't already cover the mathematics of this game?
I'm getting a weird deja vu now
He made a video about Dobble if I recall correctly. That game has a projective geometry at its core, while SET has an affine geometry at its core. There do exist generalisations of SET which use a projective geometry, but I didn't look into those in any detail
@@WannesMalfait those adjectives to geometry aren't familiar
I mean, I know the words out of context, just not in tandem with geometry
But guess I'll be watching both videos again and try to figure it out(tho I'd appreciate definitions too if you could give me(iff it wouldn't bother you too much))
@@Ewr42 A sort of quick and dirty explanation is as follows:
An affine geometry is like the geometry you are used to, with lines planes etc. Importantly, there is no fixed origin (in comparison to vector spaces), and there are parallel lines.
A projective geometry is very similar to an affine geometry except that you add extra points at "infinity". For example, in the 2d case (projective plane), these points are precisely the 'intersections' of parallel lines. In general it is much nicer to work with projective geometries, because you don't have to continuously add special cases for parallel lines and so on
Lol the last week of school my calc teacher played this with us
The mathematics makes sense but why is the game still so dang hard? It took me a solid 10 minutes to find my first set in the exercise in the beginning.
Just need some practice 🙂. Knowing the maths won't really help you get better at the game, but does make it mor fun in my opinion
OMG IS THAT A VIDEO ABOUT SET???
Yes, yes it is
Why red and green 😭
Do you mean for color blindness? I'll admit that I hadn't thought about it. I just copied the colors lf the cards on the SET website. I think newer versions of the game use better contrasted shapes and colors. Hopefully it was still possible to follow the video. It's something I'll keep in mind for the future.
In the "Quick recap", right before the 9 minutes mark, if you replace the 2 squigglies which are at (2,1) to be at (1,2), then that set will no longer be on a line. How that gets along with the idea that is explained in this video?
The cards are layed out in a very specific manner. If you look at the coordinates of the cards you will notice that they match with those of the grid. We have
DIAMOND 0
SQUIGGLE 1
CAPSULE 2
3 shapes 0
1 shape 1
2 shapes 2
So the two squiggles card has coordinates (2,1) or (1,2) depending on whether you place the shape changes along the x axis or the y axis. In the video it has coordinates (2,1). Now if we want it to be at (1,2) then this coordinate change has to happen to the other cards too. The other cards in the SET (0,2) and (1, 0) would change to (2,0) and (0,1) for example (depending on the transformation you choose) and they would still lie on a line. Essentially you can change the positions of the cards as long as it's an "affine isomorphism" which always preserve lines (points on a line before the transformation are still on a line after the transformation). These affine transformations correspond to picking different numbers for the attributes or changing the order of the coordinates
@@WannesMalfait But during a random game of Set, I could get the cards placed as I described, without preserving the lines. I cannot force the game to adhere to affine transformations only. When viewing your video, which btw is very nice, I thought it will give me winning strategy: "follow the lines" - but sets could be found also outside the lines
this isn't a "how to play SET" or "how to win at SET", but rather simply a look into how and perhaps WHY the design of SET works the first place. they're not simulating games, they just demonstrating the mechanics behind it
@@CDCHexaku Yup, exactly that. If the cards were always layed out in one specific manner, then the game would become very boring as well. Maybe I should have put a disclaimer at the start like "This video will not make you better at SET. In fact, it might make you worse by distracting you with beautiful mathematical patterns in SET'. Sorry that that wasn't clear
@@CDCHexaku
From the video's transcript:
5:50 We have shown that three distinct points lie on a line, if and only if their coordinates sum to 0.
6:23 Doesn't that sound an awful lot like the rules for a SET?
6:28 Let us make the connection more explicit.
7:02 Here is the important part.
7:04 When I highlight any SET, the corresponding points lie on a line.
7:09 Conversely, any line is also a SET.
Is it one to one correspondence or not?
I'm not a fan of Set. Visual set finding in noise is not my jam. Set theory, graph theory, and geometry, now we can talk.
That's fine, not everyone needs to like the game 🙂
This didn't make me any better at playing SET.
That is true, it might have even made you worse at it. Now you'll be thinking about geometry instead kf actually finding SETs when playing the game