Thanks again! Let me know how you like homology. It is awesome, but a bit unintuitive. So there is probably not a canonical way to explain it. (Not that there is for any other topic, but certainly one sometimes has "more canonical" ways, whatever that means.)
That is a good question - I would really like to know that as well! Applied topology has quite a few things one can do, but they sometimes go under a different header such as persistent homology. See e.g. github.com/appliedtopology/javaplex Maybe that is helpful?
@@navneetsinha451 Simplicial complexes are special cases of hypergraphs, see for example www.reddit.com/r/math/comments/8vhjre/difference_between_hypergraph_and_simplical/
I do not deserve your praise: I try to explain things they way it works best for me. Really very selfish ;-) Nevertheless, I hope you enjoyed the video, and it will help you with topological data analysis (which is one of the coolest fields of modern topology).
I really appreciate this video and how you try to make it simple! I am trying to understand what the 4 dimensional simplicial complex would look like :) I liked the two definitions of a simplicial complex "smallest convex set with n vertices" and that "any subset of points of a simplicial complex also makes a simplicial complex". I have been thinking, it seems that each simplicial complex is made up of a couple of the n-1 simplicial complex in different basis glued together. I am a bit confused by what you meant at time 6:50 when you talked about gluing c and d together and what it means for an intersection to be a "face of both" simpliical complexes?
You are trying to visualize 4D? Oh, you are brave ;-) There are essentially three ways trying to do this, say for the 4D tetrahedron which is the prototypical 4D simplicial complex: First, via a projection. Maybe you have seen the picture of the 4D cube? See the animation here en.wikipedia.org/wiki/Tesseract This is the most popular form to illustrate 4D, but I am not sure whether that works for me. I still find that pretty much mysterious. Second, by adding a non-space dimension, e.g. via a movie where time is the fourth dimension. www.dtubbenhauer.com/slides/my-favorite-theorems/48-volume-balls.pdf www.researchgate.net/publication/342697434/figure/fig1/AS:960092123975683@1605915208302/Examples-of-mathematical-drawings-to-communicate-geometry-and-topology-a-Foxs.png Not a new idea, but it is still hard to find good illustrations that are freely available. Not sure why this is not well-known. I am guessing right now that during the history of computer-animation someone decided that projections are nicer, and then everyone just copied that. That what humans do, I guess. So most pictures you find are projections, but at least for me movies work much better. To illustrate the 4D tetrahedron is a fun exercise ;-) Finally, abstractly. Here is the tetrahedron: {[0]}, {[0],[1],[0,1]}, {[0],[1],[2],[0,1],[1,2],[0,2],[0,1,2]} etc. The last one is the triangle with [0,1,2] being the face, [0,1] being the edge from 0 to 1 etc. The 4D tetrahedron now has [0,1,2,3,4] and all of its “subsimplices”. This of course is not very visual, but easily works in any dimension (or is what your computer would use). Oh, that was a lot of waffle! Sorry for that. To come to your question, proably the confusion comes from what “face” means. This is 3D terminology, so it should be the (n-1)-dimensional part of an n-simplex. Let me illustrate that using the abstract notation, which is the third above. Let us do a small example. Say you have two “volumes” [a,b,c,d] and [e,f,g,h], and they intersect. The intersection should be a face of both, e.g. [a,b,c]=[e,f,g] if a=e, b=f, and c=g. Does that make sense?
@@VisualMath I see what you mean, basically the simplex has to look nice :). Only part of an edge can't be shared or there can't be random simplexes going through other simplexes. I thought perhaps c + d meant you were adding the two pictures together in some way haha! I think it would have been better if you added a comma like "c, d" or do "c and d". Just curious, how do we call the addition of two n simplexes? I think the 4D hypercube is the explanation that has made the most sense to me. I feel like if you connected a 2D hypercube to a another 2D hypercube with aluminum wires the length of a side, they would automatically go into the 3rd dimension. There's no way do to that for a 3D hypercube, unless... Watch out for my research paper!!! XD
@@kaushikdr I see - now I understand the question ;-) Yes, the “+” should be an “and”, which always worked for me but maybe not for everyone. Thanks for the tip - I try to be more careful in the future. Not sure what addition of n-simplexes actually is; a priori you can't add them. I guess it doesn't have a name
Sorry, I am unsure what you mean. What is it you want, maybe you can elaborate and I will see what can be done? I think I never used any fancy visualization techniques in this video.
This explanation was exactly what i was looking for when i was stuck in my script. Thanks 😊
Thanks for the feedback; I am glad that the video was helpful! I hope you enjoy your topology journey!
Thank you for your lecture that is very helpful for me to understand how to form the basic energy of universe.
I am glad that the video was helpful, thank you so much for the feedback. I hope you will enjoy AT!
Awsome presentation as always
Thanks again!
Let me know how you like homology. It is awesome, but a bit unintuitive. So there is probably not a canonical way to explain it. (Not that there is for any other topic, but certainly one sometimes has "more canonical" ways, whatever that means.)
Thanks for adding complexity , I would agree.
Thanks for the feedback! I hope the video was helpful and enjoyable.
thank you! helped me understand a lot
I am glad that the video was helpful. Thanks for the feedback!
great explanation thanks!
You're welcome! I hope you enjoy your AT journey!
How can we draw Simplicial Complexes using Python or Matlab?
That is a good question - I would really like to know that as well!
Applied topology has quite a few things one can do, but they sometimes go under a different header such as persistent homology. See e.g.
github.com/appliedtopology/javaplex
Maybe that is helpful?
@@VisualMath Is there any relations between Hypergraphs and Simplicial Complexes?
@@navneetsinha451 Simplicial complexes are special cases of hypergraphs, see for example
www.reddit.com/r/math/comments/8vhjre/difference_between_hypergraph_and_simplical/
@@VisualMath Thanks sir, it was really helpful
@@navneetsinha451 Welcome.
P.S.: Sorry for being annoying, but I go by "they/them" so "sir" is not the correct way of addressing me.
You are amazing. Thank you.
I am learning about Topological Data Analysis algorithms.
I do not deserve your praise: I try to explain things they way it works best for me. Really very selfish ;-)
Nevertheless, I hope you enjoyed the video, and it will help you with topological data analysis (which is one of the coolest fields of modern topology).
Thank you!
Thanks you are welcome, I hope you enjoyed the video!
Can you share an book about it?
Usually I recommend pi.math.cornell.edu/~hatcher/AT/AT.pdf for topology
I appreciate
@@LisantinaNhangave Welcome, friend!
Great video thanks!!
Glad that you liked it, you are welcome! I hope you will enjoy AT as much as I do ;-)
I really appreciate this video and how you try to make it simple! I am trying to understand what the 4 dimensional simplicial complex would look like :)
I liked the two definitions of a simplicial complex "smallest convex set with n vertices" and that "any subset of points of a simplicial complex also makes a simplicial complex". I have been thinking, it seems that each simplicial complex is made up of a couple of the n-1 simplicial complex in different basis glued together.
I am a bit confused by what you meant at time 6:50 when you talked about gluing c and d together and what it means for an intersection to be a "face of both" simpliical complexes?
You are trying to visualize 4D? Oh, you are brave ;-) There are essentially three ways trying to do this, say for the 4D tetrahedron which is the prototypical 4D simplicial complex:
First, via a projection. Maybe you have seen the picture of the 4D cube? See the animation here
en.wikipedia.org/wiki/Tesseract
This is the most popular form to illustrate 4D, but I am not sure whether that works for me. I still find that pretty much mysterious.
Second, by adding a non-space dimension, e.g. via a movie where time is the fourth dimension.
www.dtubbenhauer.com/slides/my-favorite-theorems/48-volume-balls.pdf
www.researchgate.net/publication/342697434/figure/fig1/AS:960092123975683@1605915208302/Examples-of-mathematical-drawings-to-communicate-geometry-and-topology-a-Foxs.png
Not a new idea, but it is still hard to find good illustrations that are freely available. Not sure why this is not well-known. I am guessing right now that during the history of computer-animation someone decided that projections are nicer, and then everyone just copied that. That what humans do, I guess. So most pictures you find are projections, but at least for me movies work much better. To illustrate the 4D tetrahedron is a fun exercise ;-)
Finally, abstractly. Here is the tetrahedron: {[0]}, {[0],[1],[0,1]}, {[0],[1],[2],[0,1],[1,2],[0,2],[0,1,2]} etc. The last one is the triangle with [0,1,2] being the face, [0,1] being the edge from 0 to 1 etc. The 4D tetrahedron now has [0,1,2,3,4] and all of its “subsimplices”. This of course is not very visual, but easily works in any dimension (or is what your computer would use).
Oh, that was a lot of waffle! Sorry for that. To come to your question, proably the confusion comes from what “face” means. This is 3D terminology, so it should be the (n-1)-dimensional part of an n-simplex. Let me illustrate that using the abstract notation, which is the third above. Let us do a small example. Say you have two “volumes” [a,b,c,d] and [e,f,g,h], and they intersect. The intersection should be a face of both, e.g. [a,b,c]=[e,f,g] if a=e, b=f, and c=g. Does that make sense?
@@VisualMath I see what you mean, basically the simplex has to look nice :). Only part of an edge can't be shared or there can't be random simplexes going through other simplexes.
I thought perhaps c + d meant you were adding the two pictures together in some way haha! I think it would have been better if you added a comma like "c, d" or do "c and d".
Just curious, how do we call the addition of two n simplexes?
I think the 4D hypercube is the explanation that has made the most sense to me. I feel like if you connected a 2D hypercube to a another 2D hypercube with aluminum wires the length of a side, they would automatically go into the 3rd dimension. There's no way do to that for a 3D hypercube, unless... Watch out for my research paper!!! XD
@@kaushikdr I see - now I understand the question ;-) Yes, the “+” should be an “and”, which always worked for me but maybe not for everyone. Thanks for the tip - I try to be more careful in the future.
Not sure what addition of n-simplexes actually is; a priori you can't add them. I guess it doesn't have a name
"Hatcher’s book (I sometimes steal some pictures from there)." 😂😂😂 hatcher the homology god 💜
Indeed ;-)
Can I have the software for visualizing simplicial complex from point cloud data?
Sorry, I am unsure what you mean. What is it you want, maybe you can elaborate and I will see what can be done? I think I never used any fancy visualization techniques in this video.
Do you know about TDA?
One of my favorites ☺
This is a nice (fairly modern) summary: arxiv.org/pdf/1710.04019
@@VisualMath amazing💪
I love it! Thank you so much for this treasures!
@@LisantinaNhangave Welcome ☺
Are you married
Let me not answer this one ;-)
Thank you for this awesome lesson please i need your support as the beginner of this field
😂❤@@VisualMath