I'm pretty sure you could continue using circles indefinitely if you also go up a dimension. So 4 spheres can intersect into 16 regions. And in general you can represent N sets with N number of (N-1)-Spheres since dimensionality grows exponentially too. It's a totally different direction than you took in your video, but interesting to think about.
@Joshua Johnson imagine four spheres arranged in a tetrahedron with equal radii such that all have enough overlap that atleast a point exists in the range of all four.
This is way higher quality than I was expecting a random home page recommended to be, I hope the interaction from a comment and subbing pushes it to more people because this was clean and well made
Keep Talking and Nobody Explodes has a Venn diagram of 4 sets. It’s in the bomb defusal manual for Complicated Wires. This is a nice insight on why that Venn diagram looks how it does.
oh my GOD i've been thinking about four-figure venn diagrams for months and this video just popped up, RU-vid knew I was desperate for answers, thank you!
One Problem, humans can't draw actual fractals with infinitely long boundaries. And even drawing a good enough approximation probably would be quite difficult.
5:02 I spent some time thinking about this statement, and realized that if any ellipse is removed, one of the opposite ellipses’ head will stick out from the center and create an extra unwanted region. For example, removing E does not merge the region EB (a red one on the left) into B.
@@CarmenLC yes, but it will be contradictory. Each region of a venn diagram is supposed to represent one single set, only, likewise, one set can only be represented by one region. This extra space will create a scenario where there’s two regions representing one set. Which shouldn’t be
@@bobh6728 think of it like making a cross with two ellipses. We’re only using two to simplify the visualization. The middle part is where they intersect which is fine, but the arms and legs of the cross that are sticking out is the problem. Both outter parts of each ellipse represent the same thing, yet they’re two different regions which shouldn’t be in a Venn diagram. Hence why that isn’t considered a Venn diagram.
My RU-vid recommendations be like: Here's some D&D videos, also here's a piano being thrown from s roof, the icing of the cake will be a diagrams video. Did I watched them all? Yes. Do I have use for them? Mostly no. Did I have a blast? Absolutely!
Week ago I saw Ikigai diagram consisting of intersecting circles "what I love to do" / "what i do well" / "what humanity/people needs" / "what i get paid for". Something was not right about that and now I know what exactly. Thanks!
I remember solving it with triangles when our math teacher gave that task to a few of us. We (like, 3 of us) learned from each other's mistakes, so we made similar solutions.
Years ago, I was curious how many regions would exist in a venn diagram with n values, and made a little spreadsheet with a formula to figure it out for me. Took awhile, but I figured it out. Neat that someone made a video about it. Really shows that I'm not alone in my random wondering.
@@manuvillada5697 Let us consider a diagram on n sets. Let us consider a set A, A shares a region with every possible combination of other sets and there are 2^(n-1) such combinations. For a different set B, we have to count combinations again, but exclude those containing A, so there are 2^(n-2). So, we want the sum of 2^(n-i) for i from 1 to n, or more simply, the sum of 2^i for i from 0 to n-1, which gives 2^n - 1 (or just 2^n if you consider the outer set) Another way to think of this is that any element can be in any combination of these n sets (potentially in none of them) so again we get 2^n (for any set it is either in it or it isn't so each set has 2 valid states and so there are 2^n valid states altogether)
This took me back to when I used to try to doodle symmetric 4-set Venn diagrams at highschool. I really enjoyed this video, the way you explain everything is so intuitive and enjoyable. Instant sub 👍👍
I don't really care for math yet this still managed to interest me, and it was a random recommendation. Gotta give credit where it is due, this is really well made and presented
I have absolutely no need of this information currently but somehow, watching the preview a bit, made me interested. When I finally got the answer, (Draw oblongs) I thought I would lose interest, yet for some reason I wanted to finish the video.
@@smalin but then the question is, what is the largest number of categories that spheres in 3D can represent. And what about a generalized answer? My conjecture of intuition and laziness is that for dimensions N, N-spheres can create an accurate Venn-diagram for N + 1 categories. (By N-spheres I mean the N dimensional equivalent of a sphere. A 2-sphere is a circle, a 3-sphere is a sphere, a 4-sphere is a hypersphere? Something like that)
@@evanmagill9114 While I have not put any thought to this question specifically, I have done some thinking about N-Dimensions, and I don't believe your intuition is correct. Look up the "sphere packing" problem, recent breakthroughs have been made that have reveal the crazy and very non-intuitive ways that you can pack spheres in higher dimensions. I would guess that the highest 3 dimensional sphere venn diagram would be 4, because of the tetrahedral formation, but I would bet that that number would grow more exponentially for Dimensions higher than 3.
@@phee4174 the embedding dimension is what you're referring to (the lowest number of dimensions of a simply connected ambient space in order for the subspace to also be simply connected), but the convention is for the intrinsic dimension (the dimension of the parameter space needed to exactly specify each point in the space partitioned by connectivity) (or the dimension of the tangent hyperplane of the manifold) (or the Hausdorff dimension of a space that happens to be smooth)
this is such a great video! you did an especially good job on writing the script and visualizing your points. i hope you get more recognition and continue to make both entertaining and educating video like this. much love!
Nice work! I thought this was going to go in the direction of higher dimensions, e.g. 4 spheres arranged in a pyramid. You can just keep adding dimensions, but of course beyond 4 spheres the usefulness of what you produce as diagrams is pretty questionable. :)
When I took a class that was part Boolean Algebra and part circuit design, they taught us about Veitch diagrams (which I see have now been replaced with Karnaugh diagrams). They work pretty well for up to about six sets, with each set represented by rectangles, some of them wrapping around the opposite edges of regions.
Spatially, you need 3 dimensions at least. 4 spherical volumes tetrahedrally arranged. In general, N-1 dimensions for N spherical volumes N-hedrically arranged.
Yknow those times when youtube will recommend something that piques your interest and then suddenly find a gem. Yeah that's exactly how I'd describe this. Amazing work! New subscriber (also helps cause I'm a maths student)
So many new math channel popping out! Keep up the good work! Good quality content will always find its way to the top ;) Math is fun that a lot of different domain often cross each other at a place least expected.
I once had a small crisis while high at 1am because I wanted to do a diagram with 4 sets and felt like a goddamn genious because I did one with triangles
Venn was a Don at Gonville and Caius College, Cambridge in England. Later, another Don (A.W.F. Edwards) from Caius wrote a very illuminating book about Venn diagrams entitled "Cogwheels of the Mind - The story of Venn Diagrams". In it he shows various forms of Venn diagram and, in particular shows a general method for drawing 4, 5, 6, ... etc set Venn diagrams. There is an Asymptote/Latex script for generating an example and I also (out of boredom as much as anything) wrote a script to draw them using the regular context line drawing commands and also using SVG in HTML/Javascript. I wish I could paste an example here.
I like your lisp! I used to do this in highschool and I saw the faults when I can't find places for some elements of my set. I then chose to change my graph into something different lol xd
Actually the construction described in the video allows any area to have the desired size, since you can either (if you want equi-sized regions) draw each curve in such a way that it splits each region from the prior step in two, or (if you want the regions to represent the "amount" of data-points lying within them) you can tally up before drawing an aditional curve how "much"/"many" data-points are in each of the newly created areas in total (including subareas) and devide the area accordingly. A related question, which I wasnt able to find an answer for is how to opimize for largeness of the smallest largest open balls contained in the subareas as well as the smallness of the largest smallest closed balls containing a subarea.
Or which constructions result in practically sized and shaped regions for actual use displaying data. :D Perhaps optimize for both minimal SD of the areas of each region (except the purely exterior region), and minimal SD of some function which assesses how similar to a circle each region is?
@@StrategicGamesEtc To be fair, it's the circle that got us in trouble in the first place. The radical solution is to optimise similarity in overall area, but completely throw out the devotion to circles: have some sort of chain or knot configuration with some interesting symmetry, but not a circle in sight. (And find different standard shapes for the number of shapes required.)
I wonder if a venn diagram for four sets can be drawn with circles if one draws on something that isn't a Euclidean plane, as I think two circles can intersect at least four times on a sphere
@@giveme30dollars A non-Euclidean plane is not adding a 3rd dimension though, you can still only move in 2 dimensions on a plane of a sphere, there's still no up and down as you're supposed to stay on the plane ;)
@@giveme30dollars Don't think of the mathematical plane as a physical object (or anything in math, really) while we usually depict it to understand it better, it really is only described by its characteristics, among them, that the plane only possesses two coordinates, two dimensions; therefore, any mathematical concept that can be expressed with only (and strictly) two coordinates is a plane. The surface of a sphere is a plane, for example.
I've been bothered by 4 circle "venn diagrams" for years. Have put a little thought on how to accurately represent the intersection of independent variables better, but not _much_. So this was both interesting to learn about, and satisfyingly vindicating for that minor annoyance lmao.
I cannot place his accent (hint: Burmese!), but this is definitely my first time hearing it used instructively, and certainly the first time hearing "circles" pronounced as "sheowkulls". Very pleasant to the ear!
To the 100k future subscribers, SheanMiki was here before 1000 subs! :D The quality of the video is really good. Well explained! Hope to see more videos from you :>.
When I attended statistics on college my teacher said "if you think Venn Diagrams are easy just try drawing 2 sets that belong to different universes and yet intersect" The solution was kinda easy and hard at the same time, you had to think of universes as planes that cut through spheres (the sets). So there is an infinite amount of universes where those sets existed intersected, an infinite amount of universes where only one or the other existed and just one universe where both where the same
I realized this when I tried to draw a 4 set venn diagram... using ASCII characters while commenting some code XD I wanted to use the diagram as a "quick way to visualize" some data, and ended up spending hours in a rabbit hole on how to draw them instead! Still, this video made everything much clearer
Solution: make a 3d venn diagram. The important thing here, is, if I'm not mistaken, to respect the symmetry of the final shape, in all directions. A 3d pyramid like structure (tetrahedron-like), made out of 4 spheres, will be symmetrical in all 4 directions. I don't know how best to describe it. But try it, it should work. Edit: okay I've come up with a way to explain why 3d works. Each sphere, from its perspective, must have the same relationship to all 3 balls, for the pure symmetry of a 2 circle diagram to be respected for every item in the diagram, to every other, with 4 items in your shape. That way, every sphere is connected to every other in the same way. This is not the case with the four 2d circles. Because A & C or B & D are in a diagonal relationship, but A & B, B & C, C & D, and D & A are connected in a "side" relationship. So not every item is connected in the same way to every other. 3d Venn diagrams solve that. A rule could be: if you can connect every item to every other in the same way, you can have a Venn diagram with identical items and as many items as you want. Ps: you do not break the pattern! 2 to 4 to 8 to 14 to 22... is 2+(2) then 4+(2+2) then 8+(2+2+2). You expect to see a doubling behaviour, and so did I at first, but your further patterns show the issue is breaking the shape-to-all-other-shapes-in-the-diagram similarity. That fourth circle is just added on top of a triple symmetry. And so is every shape you can think of. Actually, the real fundamental issue is the amount of unique ways to connect two things, here two shapes, together. In a 2d world there are only two ways to uniquely connect objects. From the side/side and from the up/down direction. In other words, you can connect your object to two other objects who in turn are connected to each other and create a Venn relationship. In a 3d world, there are 3 unique ways to connect objects, side/side, up/down, and back/forth. So you can connect your object to 3 other objects without any one object ruining the relationship between any two objects. In a 1d world? Only 2 line Venn diagrams are possible. In a 4d world? I suppose I haven't thought about it enough. Very interesting video! Thanks! :)
I got a real kick out of the way that you showed us the most elegant solutions that are currently known, and then went back and called Venn's original solution a cheat haha 😂
We draw 4 set venn diagram with a 3×3 square matrix and then connect alternate rows and alternate columns with semicircles, outside the matrix (I don't know if that's the right description). It makes it way more easy to understand than circles and Ellipses.
Make the opposite one for example A and C, intersect the other in a way that no other intersections are made, like a placement of the circle to be in the opposite corner. Though this may not be the point of this as this requires two of at least two circles/regions. This is the easiest way I believe
Another diagram idea: For any diagram in dimension k of n sets, to make a diagram of n+1 sets, increase the dimension to k+1, duplicate the diagram in the k+1 coordinate, make sure the two diagrams are separate, and name the resulting k+1 "plane" as n+1. For example, take a 3 set diagram in R^2. Change to R^3, setting the z-coordinates of the 3 set diagram to 0. Copy the 3 set diagram onto the plane z=3, and name the plane at z=3 what the 4th set is. This quickly becomes hard to visualize, so here's a coordinate expression: Since there are a countable number of sets, order the n sets from a_1 to a_n. For each set, choose 1 to include it and 0 to exclude it. So the coordinates of the first set by itself are D(1,0,0,0,0,0.....,0) and the coordinates of a_1 intersect a_2 are D(1,1,0,0,0,0,0,....,0). Naturally, the coordinates of no sets is D(0,0,0,.....,0). Exercises left to the reader: Suppose there existed a dimension that contained all natural numbered dimensions (R^n for all n in N) and call it R^infinity. Do there exist diagrams of dimension R^infinity? If so, how many are there? If not, why not? If there was a dot on the real number line for each coordinate in R^infinity , what is the thickness of all these dots (e.g., the "thickness" of intervals [1,3] or [2,4] is 3-1=4-2=2 units)?
This video opened my eyes 👀. Because i study IT, there i learn about numeral technology, and this is very, but very related! Thanks 🙏 Liked, commented, subbed!
Im pretty sure you can make it work with four circles. First you draw your two big circles for your to set. Then you draw your third circle in the middle and bellow that has to cover at least slightly more than half of the middle region. Then you draw your fourth circle directly above the third circle with the exact same dimensions that will cut slightly more than 50% of the middle region as well. For reference, this will look something like a flower.
The solution is obvious, you need n dimensions to draw a n+1 region venn diagram if you use circles (or whatever their equivalent is in n space). ex: a 4 region venn diagram must be composed of spheres. Construction is simple: start with a regular n-simplex (triangle, triangular pyramid, etc.) with a side length of 1, place an n-ball at each vertex with a radius of 1. Remove the n-simplex, and each ball will form a region of an n+1 region venn diagram.
Expect we cant draw in n dimensions in a way that is easy to understand And thus such a solution conflicts with the unspoken premise that most people will have making it imo a less obviously solution
@@wwellthemage8426 Sure, it is inconvenient to use, but I think it's the most logical extension of venn diagrams to more sections. Obviously it's more useful to ditch the circles and use something else, but then it isn't really a venn diagram anymore.
@@haph2087 I personally would consider using a sphere or hypersphere more fundamentally different from a venn diagram than an oval Yes a sphere is perfectly symmetrical within it's own respective dimension but one cant compare things from different dimensions (a good explanation of this is in a vidoe titled something along the lines of "can you paint an object with infinite area")
@@wwellthemage8426 I think you took the wrong message from that video. You can't compare things with different dimensions in the sense that you can't say one is greater than the other, but that doesn't mean they don't have relationships, nor does it mean there can't be similarities between things in different dimensions. Anyways, I do understand how one could say that a that the 2d ness matters more than it being an n-ball. I just think that giving up the 2d-ness in exchange for keeping the symmetries is the more elegant than the alternative shown in the video, and I like that it generalizes to any number of regions without losing any more symmetries.
Another way to think of the how many regions for n sets question: If we have n sets, obviously we need 1 region to contain 0 sets. We need n regions to contain 2 sets, 1 region per set. How many regions to contain 2 sets? Simple! n choose 2 does the trick! So the number of regions is simply the sum of (n choose k) for all k from 0 to n. Those with a background in combinatorics will know that this always sums to 2^n
