When I'm feeling stressed and like everything is meaningless. I just watch the PBS videos(space time, infinite series, etc). Although I don't exactly understand every term exactly. It calms me down to hear science and math. I don't know why. But its like my happy place. Thankful for these videos I only half understand lol
6:43 The explanation of one dimensional holes could be clearer. Two points that I think could be clarified: (a) these circles are holes in the Betti sense because they have an inside and an outside, while the "donut hole" in the middle of the torus doesn't count here. (b) the reason these two circles imply that there are two holes is that if you draw any third circle on the torus, it can be either continuously deformed into one of those circles or shrunk to a point (and so it's not a hole), but you can't deform the two circles into each other, thus there are exactly two one-dimensional holes.
i would say the "contents of the donut" explanation of two-dimensional holes is even worse. two-dimensional holes are interior hollow spaces of three dimensional hollow objects. in my opinion, before going to the torus, they should add a circle and a hollow sphere as simpler examples.
You can actually draw other circles on the torus that can't be deformed into either of them, but in a sense it'll be the "sum" of those two rather than a new one
Yes, you are correct. You can draw a curvee that circles both the "donut hole" and the "donut arm", which I assume is what you meant. You can also draw a curve that (say) just circles the "donut hole" twice, but it must cross itself.
Hi Kelsey, this episode felt really rushed. For example, some graphics were on the screen for less than a second, and these are pretty tough concepts. I like most of your other videos, but this one was particularly hard to follow
absolutely agree. it is the nature of the subject that it requires long explanations for even the most rudimentary understanding. it is already a three video series, and it makes sense to not want to make it ridiculously long, but it may be necessary
I agree. I don't understand why the torus has 2 one dimensional holes. But I feel like it will be dumbly obvious once I know. The shape clearly has two curvatures like you drew, but why are they holes?
Answer to "Why are they holes:" If you draw any loop on a sphere, it's always possible to shrink it to a point. In other words, you can't find any holes on the surface of a sphere. However, it's possible to draw two types of loops on the torus (they're different because they can't be rotated and slid along the torus to overlap) such that they can't be shrunk to a point. Thus, in this sense, the torus has 2 1d holes (1d because the loops are 1d).
You know, when the formula was shown for the euler characteristic in higher dimensions with its alternating pattern, I couldn't help but think whether there is a comnection between that and the formula for probability of the union of sets, derived from set theory. Perhaps the characteristic is a sort of measure associated with the union of all the points embodied in the polyhedron, and is measured in a way analogous to probability by counting the measure of the individual points, then subtracting number of 1d lines (analogous to all pairwise intersections between points), adding number of 2d faces (all triple intersections between points), etc.
Yes there is a connection between combinations of sets and simplicial complexes, in topology we can take a "covering" of a space by taking a bunch of regions, and we might care about all the intersections of these covering regions pairwise, in triples, etc. This data is described simplicially by the "nerve" of the covering.
Topology is one of the most difficult subjects in mathematics I’ve ever learned through elementary school to university. The difficulty ranking will be: Topology > manifolds > functional analysis > Lebesgue integral > set theory.
that's interesting. For me it is spectral theory> dynamical systems> measure theory>point set topology >homologies >galois theory/polynomial rings. The experience of every mathematical student will vary wildly. I've come to love the fact that there is no one steadfast rule on which math is the hardest. Each branch can radically resonate differently with different people.
Knowing a bit about neuroscience, I can see this being very useful for analysis of the electrical activity of neurons (assuming you have some way to model changes in the activation threshold of a neuron as a consequence of firing... once a neuron fires, it needs to recover for a bit and doesn't immediately return to the same quiescent state it was in before firing... repeated firing can also cause it to form new dendrites...) but I get the feeling that it will hit a wall as soon as neurotransmitters are considered. Those work on a vastly different timescale, and the chemistry in the gap and on the surface of the neuron becomes very important and very time-dependent.
Sometimes you explain things fairly well, but then you just tell us that the Euler characteristic as you increase the number of holes decreases by 2 (with no reason) and that the 1st Betti number of a torus is 4 (with no reason)
5:17 That "giant awkward shape without a name" is a truncated-icosehedron. Of course, the Euler Characteristic is true for this, as with all 3D convex polyhedra. Vertices: 60 Edges: 90 Faces: 32 60-90+32=2 Apparently I'm not the first to say this, but I probably won't be the last either.
Neat video! I remember in my early twenties, I was reading about neural networks, & I was fiddling around with the idea of a number of points; each point would represent a computer CPU. I wondered if there was a way to figure out how many lines it would take to connect every CPU (point) to every other. Eventually, I discovered [as opposed to invented!!!] that the formula was: x = (n(n-1))/2. [where 'n' is the number of points, and 'x' is the number of connections] I was pretty proud of that. But I had no idea what it meant, or what it might relate to. Imagine my surprise when, 20 years later, I discovered (while playing Q-Bert) that the number of squares he's hopping around on can be derived by knowing the length of one side of the triangle [playing board], & using: x = (n(n+1))/2. Weird! Thanks to your wonderful video, I now have something to 'connect' all of this TO! It's fascinating that all of these things: triangles, tetrahedra, neural networks, complete graphs, etc., are indeed CONNECTED! This sort of thing really gets my brain's wiring firing! Thanks again. :D Rikki Tikki.
2:12 - "a 4-simplex is difficult to imagine" Yes, it is! But let's try... As Kelsey says 0:45 - all vertices of a simplex are connected pairwise to form the edges. Just like a 2D object has a 1D boundary; a 3D object has a 2D boundary; a 4D object - and therefore the 4-simplex - has a 3 dimensional boundary. We also know that because everything is connected pairwise, the boundary must be made of tetrahedra. Tetrahedra have 4 vertices. We can do the math to see how many Cells (3 dimensional boundary pieces) compose the 3D boundary of the 4-simplex. So, we want all the ways that we can form a tetrahedron from these 5 vertices that make up the 4-simplex, In other words, we can evalaute (5 choose 4) = 5. So there are 5 tetrahedron cells (Ah! No wonder the 4-simplex is also called the "5 cell"!) Next we can define 5 unique vertices in 4D space (x,y,z,w). Then we can find the intersection of the w=0 hyperplane with each of the 5 tetrahedron cell boundaries to find the boundary of the "3D slice" of 4-simplex. Finally, by rotating and translating the 4-simplex through the w=0 hyperplane (which is 3D xyz space that we live in) We can visualize the geometry of the 4-simplex. Furthermore, as Kelsey says 2:53 - "we can stick higher dimensional simplices together along higher dimensional faces.." Let's go ahead and try that! Since we are dealing with the 4-simplex, 2 connected 4-simpices are interfaced along an entire 3D Cell. So, let's pick any 4 vertices of our first 4 simplex and define a 2nd 4-simplex using these 4 vertices and then a 5th unique one. Using the same simulation, we can visualize this 4 dimensional simplicial complex!! I've programmed renderings of this so we can all see the geometry, and I just emailed a video showing the demo of the 4-simplex and 4D simplical complex. I give permission to share the video on this channel if you want to.
This video is much better and comprehensible for non advanced mathematicians at speed 0.75. Try it. Incidentally, Kelsey looks and sounds high at this speed : )
I don't get what "holes" of different dimensions mean. Why are there two 1d holes in the torus, and why is there one 2d hole in both the torus and the two-hole torus?
What about the 1st Betti number for a 2-holed torus? There are two holes in each of the handles, one in the bit which joins them together, and then one longitudinally around each donut hole, and then one which goes longitudinally around both holes. I know you can construct paths which look like circles but can be decomposed into a mix of two on a torus, is that what is happening here?
So I looked into this and it looks like those "holes" are a different concept called genus. Ignoring that, from the definition in (b) of Tehom's comment, I'd think there are 6 1d holes. (Around left handle, around middle part, around right handle, around left donut hole, around right donut hole, around both donut holes.) I know that's wrong, so maybe two of them are somehow combinations of the other four? I still don't know how this works, but I think I'm getting closer.
Alex Knauth That was what I was thinking. But I remember seeing something about being able to put two circles together to make a third. So in the case of a donut, you can mix a longitudinal circle with a minor circle to get a path which travels through the centre 'hole' once and once around the entire ring.
Every Betty number indicates how many of "spaces" of dimension n+1 are between a n dimensional object and "itself". Which is literally the definition of a hole. Imagine a bowl with no thickness. One might at first think that the B2 of the bowl is 1, as there is 3d space between the 2d angled face, but it isn't enclosed, so it's not a hole. You can notice this because topologically a bowl is the same as a disc, but simply bent. A disc has B2=0. Draw any topological shape of 2 dimensions, so it's not a solid. It could be said a "3d without thickness". Now, take any point on the surface of that shape and make a line that goes out of that point into any direction. That line should either touch another point of the same surface or it won't. If you can do this for every point in a surface, into every direction, you've got yourself a hole. In the case of the torus the "hole" is between the inner face of the shape and itself. in the case of the outer face you can easily see that if you do a line from every point, into every direction (except 90 degrees, which i don't know if that is "allowed") you will not reach that same face. Now do this same thing for the Betty 1. Take your shape, draw your 1 dimensional object (in this case a straight line). Pick any point on that line and fire off into any direction on top of the 2d surface. If you reach that same line, you've got a hole. Remember that when you draw a straight line on a shape it might look "bent" from our perspective. Think of the classic ant on the surface. For the ant, it is a straight path. The B1 holes in the torus are made by making the line that looks like "a ring" put around the torus (that which you wouldn't be able to take apart if they were 2 solid objects). In this case you can take any point on that line and go into any direction in a straight path and you will eventually reach that same line, no matter how long you take (say you pick 45 degrees of a point and your path will look like a corkscrew). The other hole is more confusing. It looks like it's "in" the torus, but it's actually a straight line on the surface. If you do you your straight path on what would be the "outer" or "inner" side of a doughnut you would create a line that, topologically, would be equivalent to the line presented on the figure in the video. In topology you can stretch and "transpose" at will without affecting the shape. A doughnut is a mug. (by transposing I mean like if the surface were magnetic fabric over a surface and you can move this fabric around and it will keep the shape of the object "within" it) This was a long answer but this is how I understood Betty numbers. Please correct me if I'm wrong for I'm not a math student so I don't know if this is how it actually is.
Oliver: Your intuition is good. Unfortunately, the rigorous definition of Betti numbers is very technical - even the vast majority of mathematics undergraduates will never see it, and it is often only taught in masters courses. (The advantage of this technical definition, of course, is that it allows you to calculate Betti numbers for weird shapes which are impossible to properly visualise e.g. the Klein bottle.) One of the technicalities involves, as Alex and Alastair have discussed, deciding when one hole is different to other holes. There are several comments which described how intuitively it seems you can find five or six B1-holes in the 2-genus torus, but the first Betti number of the 2-genus torus is 4. Out of curiosity, I'd be interested to know what you think is the Betti-1 number of a sphere? (By sphere, I mean the 2D surface with empty inside, like a balloon).
Stating that a neural connections (connectome) graph is a tetrahedral complex, seems the same as stating that it is a graph that contains one or more cliques. Cliques being fully connected subgraphs. I'm less sure about this, but the number of faces is the number of subgraphs of said clique that contain 3 nodes, the length being 2 for edges, 4 for volumes, etc...
Daniel Bamberger my maths knowledge is mostly in calculus (being engineer and all) and I followed what a k-simplex is, how to build simplexes together to form complexes, and how to characterise them with euler's characteristic. only thing I didn't manage to see is betti's numbers. I just don't see the 2 one-dimensional holes she's speaking about and got a bit lost.
I literally saw the thumbnail for the part 3 and clicked it without having seen the other 2 parts. About 3 seconds passed and I was already searching the part 1 to understand the 3 seconds of talking she did. Alas, here I am, ready to embark once again into the quest, hopefully this time prepared. As for the difficulty I think it's definitely a bit rushed and complex, requiring knowledge, concentration and sometimes pausing the video to ponder. But I am glad it is like that. There are enough science channels on youtube that explain certainly interesting but not so deep stuff because they can't go very deep. Only when you assume your audience has the capacity can you start explaining more advanced stuff.
OMFG. I just wrote about 10 paragraphs explaining my take on betty numbers and when I hit "reply" it only sent those 2 paragraphs.... god damnit youtube.
[05:32] How is the torus EC = 0 with 1 face − 0 edge + 0 vertex = 1, and approximated its superpolyhedron would = 2, and too it's not a 'hull' here to meet his definition... And why leave out Euler on 2D-which would = 0 if strictly-2D....
Well noticed. Yes it is (at least in some circumstances)! en.wikipedia.org/wiki/Euler_characteristic#Inclusion.E2.80.93exclusion_principle For what is relavant to this video, the Euler characteristic does follow a version of the inclusion-exclusion principle.
finding a k-simplex in a simplicial complex is equivalent to finding the (k+1)-cliques in the underlying graph. that is not a task that requires algebraic topology, and therefore if that is the goal, it is of no use building a simplicial complex out of the graph. we only build simplicial complexes when algebraic topology becomes useful for a goal
Knot theory probably could play a real nice role in the deep blue project as you can probably replace those vertex with holes and get a better understanding of the riemannian topology and use that to model the actual landscape of the folds as a pseudo-metric as the neurotransmitters are made in different areas of the brain and involve multidimensional processes like impulsion by catecholamines, and maybe see how the fit in the receptor protein and with what electron density. Although thats a lot of data points to follow
I agree that studying the wiring isn't going to be sufficient as it sounds like the chemical nature matters a lot too - an interesting parallel could be made with epigenetics, where it turns out proteins are more important than we initially thought.
My own bet'd be modelling the wires as phonon networks to account for (tiny) quantum 'acoustical' effects, as well as the classical action potentials - but the ass of the problem is indeed the neurojunk. I suppose one could assign a cross section for acetylcholine, etc. hitting a detector, put everything in a nice density matrix, and see what happens. But still I fail to see what neurotransmitters have to do with Riemannian geometry -_-'
thstroyur yes those phononic effects are most definitely an avenue! An fmri can only do so much, but a eulerian amplification coupled with some tensor flow could do a lot! As for the neurotransmitters and reimannian topology, imo, i think the electron density at the neuroreceptor, is what influences the overall direction the synapse will take (as the electrovolt potential is locked at certain threshholds (75 millivolts or something) but the actual release of the chemically mediated process is dependent on the cells overall configuration and therefore the protein ribbons are the overall "gate keepers". So to me, the reimannian topology comes into play at the displacement of energy of the amine (like how atp is a release of energy as its broken by the enzymes and we use that as energy) so the topology comes in (bear with me im ranting now) with the time dependent schrodinger equation. This is correlated with the topology of the electron angle of the amine molecule, and therefore consistent with directed model of the particular motivation that the amine (or hormone!) has had on the evolution of our dna. So the info contained in the codons exons and introns, are the actual mathematics explained in this video represented in a statistical way. Rant over Its tough to combine so many areas and not be challenged without proof so i accept any attacks of such wild claims
So lemme see if I understand: 1) electron density at neuroterminals influences synapse direction, 2) said electron density is dependent on biomolecular configurations, 3) said configurations are altered by energy-releasing processes but molded after 4) the electron density in nucleotides and neurotransmitters. But even so, I still see no deep connection with Riemann theory (i.e., metric manifolds), even if you wish to visualize the 'topology' of molecules, as you call it, as a potential-surface-like landscape wherein time-dependent/nonequilibrium evolution takes place. As for proofless rants, I'm understanding; recently I showed a (simple) formal solution for the TDSE to a researcher, and I got a sound scolding >_
thstroyur ok here is my proposal... as points close to each other diverge polynomially(at least in a unipotent geodesic flow as well as in the horocycles) and closed sets are not supported by piecewise linear manifolds, hyperbolic surfaces allow a horocycle flow, keeping unipotent flows nondivergent, showing holonomy of ergodicity supported by affine submanifolds. Using polynomial shearing to diverge invokes vector growth as stable foiliatons on linear subspaces using cantor sets, keep the polynomial shearing divergence as an exponential drift, showing that inverse measures are closed orbits only if the holonomy can be stable by identifying the unit tangent bundle as an ergodic measure of the invariance being equidistributed. So as any horosphere inverse measure is geometric, and we are using polynomial drift to equidistribute, i would allow a lebesgue measure framed in a fourier series. So the electric part of an action potential can be measured ergodically by ablelian differentials, and cant be additive like the chemical releases can Can you summarize your schrodinger take?
you didn't really explain the betti numbers very well. I don't see how a circular hole is 1-dimensional, and the things you're calling holes don't seem like holes.
I hope this helps: a) A circular hole is best described as being 2-dimensional object, rather than 1-dimensional. This may have been a mistake on their part, or, an un-intuitive convention in mathematics. b) If you get a piece of paper, and cut circular holes in it, then the number of holes in that paper is the first Betti number of that paper. c) If you have a block of glass with bubbles in it, then the number of bubbles is the second Betti number of that block. d) If you have a sponge, the number of holes in the sponge would be equal to its second Betti number. e) A balloon's second betti number is 1. It is possible the word hole is not the best choice of word in all cases, but hopefully these examples illustrate what some of the intuition.
@@TheManxLoiner So is that what u mean that a one dimensional hole is that which can be filled by a two dimensional plane and a two dimensional hole means which can be filled with a 3-dimensional space? Can we indirectly say this as a definition? And hence follows the rest of the holes? Is that mean the betti number of the universe is infinity?
@@ponaki6891 This is not my expertise so I do not know for sure. 'one dimensional hole can be filled by a 2-dimensional place' - yes, that is the intuiion. 'can we say this as a definition' - Probably not - the formal definition is very technical, and I assume there's a reason there is not a simpler definition. 'betti number of the universe is infinity'. First, which betti number? Two, you have to be more specific about what you mean by 'the universe'. Do you mean all the objects in the universe? Do you mean space-time? Lastly, the universe might be finite anyway.
I've just realised, or maybe she mentioned it in the video and I didn't get it. The number of nodes in an N dimensional simplex is (1 chosen from N), the number of edges is (2 chosen from N), faces is 3, volumes is 4, four dimensional hyper volumes is 5, and so on.
Yes I see the Mastro-Master reference in the 7 pointed star Game of Thrones uses. Look up Lagarde 7's where she talks about this at the esoteric level which most people & she knew obviously wouldn't have any real understanding of. 7 is also G which is a JAR stand in hieroglyph which is why Lucas used Jar Jar Binks as the reference he did, 77 is also 49 which looks like XLIX in roman numeral another esoteric tell as LI means powerful like what my name says but it says that I burn or light or expose them which obviously I do lol.
What if the "learning process" of the brain is just creating or destroying holes and those holes are like "pebbles in rivers that change the flow so the result is different"?
Is there a way to add weight to more important vertices, edges, faces, volumes, etc? I'm pretty sure the connections in the brain aren't just on or off, some are used more and are more important than others. Maybe a connection is more important the more vertices, edges, faces, volumes it shares? idk, cool shit tho.
Should we drill a square shaped hole through a square shaped face of a cube, we would still have one face. Thus the drilled-through cube (that is, central shaft removed) has 16 vertices, 24 edges and 10 faces. Then we have 16-24+10 = 2 opposed to the video where it is stated that this value should be zero. Any thoughts on this?
It will come to zero if you assume there are 8 additional edges connecting each outer vertices with corresponding inner vertices. It will split both front and back faces into 4 pieces each. In this case it will be 16 vertices, 32 edges and 16 faces. 16 - 32 +16 =0. May be that is the correct way of counting.
I was aware of this trick. In fact you need to add only one such an additional edge for each of the two faces to change the result. So you will have 16-26+10 = 0, but the problem with it is that, such additions seem quite arbitrary and how do you know the number of required additional edges in more complicated cases?
I'm struggling a bit with the Euler characteristic of a torus... I can see one face (its surface), but no edges or vertices - which gets me to an Euler characteristic of 1 (0-0+1). I assume my definition of vertices and edges must be off. I'm defining them as discontinuities in the surface function in one dimension (edge) or multiple dimensions (vertex). So what am I doing wrong here?
I knew pretty much all of this but I learned it from reading and rereading my way through wikipedia articles over months and months, not from a nice, well-explained 8 minute video. This channel is such an amazing resource keep up the good work!
If this had been like other videos in the series, she would have taken much more time to walk through the way Euler characteristics and Betti numbers work. I think she was also speaking significantly faster than int previous episodes. As a result I didn't have time to get even a surface understanding of an idea before we were jetting into implications of it. And with all that, I never had a sense of what the topics discussed mean to the brain model. "It's important" is fine, butting might have helped my understanding if I knew how categorizing topologies is important to modelling a brain. On previous videos I have rarely had to pause and hit Wikipedia to figure out what's being discussed. This time I have watched it twice, paused many times, and spent as much time reading the wiki as on the video. This video also gets a thumbs up from me. I just wasn't expecting a significantly more difficult deep dive for this miniseries.
Wonderful Video. I do wonder if there is any sort of dynamical tool that could be applied to the Simplicial Complexes so that order that the edges are traversed by a signal could be taken into account. So various sub Simplicial Complexes could be thought of as comprising something like logic elements and the order that they are traversed could correspond to a brain function that would be understood on a time-like basis.
