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seen Mar 27 at 12:35

I like higher dimensional spaces.


Feb
28
awarded  Teacher
Feb
27
answered Fourier transform
Jun
7
awarded  Scholar
Jun
7
comment For an $n$-dimensional object, how many types of holes are possible?
NKS, thanks. I'll need to read up a lot obviously, but your answer is phrased in a way that seems very familiar to the way I was analyzing the issue. Betti numbers were a tougher link. You mentioned knot theory, which is also nice. I had wondered whether to bring that up, since as best I can tell knots tend to fall apart as you embed them in higher spaces.
Jun
7
accepted For an $n$-dimensional object, how many types of holes are possible?
Jun
7
awarded  Commentator
Jun
7
comment For an $n$-dimensional object, how many types of holes are possible?
NKS, thanks! Keywords like that are very useful to me. I don't mind doing the homework, but good starting threads help immensely.
Jun
7
revised For an $n$-dimensional object, how many types of holes are possible?
An update on my efforts to answer my own question
Jun
6
comment For an $n$-dimensional object, how many types of holes are possible?
Ah! The $b_0$ Betti numbers are best understood by dropping the "solid/empty" distinction and thinking more in terms of two viscous but immiscible fluids $A$ and $B$. Assume $A$ the dominant fluid. Self-cohesive (connected) droplets of $B$ within dominant $A$ then become "holes" in terms of the $A$ fluid, and that is what the $b_0$ term is counting: A connected object is a type of hole, a void or cavity in everyday terms, in the "substance" of the vacuum. You then get trees of alternating $A$ and $B$. Other quick notes: Signed cycles? Please! 2D centric. Gradients fields make more sense.
Jun
5
revised For an $n$-dimensional object, how many types of holes are possible?
Adding a figure that shows generalized holes up through n=4.
Jun
3
comment For an $n$-dimensional object, how many types of holes are possible?
@LeeMosher also: I think the equivalence requires that there be only one non-zero Betti number in the part of the sequence $b_1,b_2,\ldots$, since more than one non-zero number seems to implies multiple holes as I have defined them. I do not know how $b_0$ fits in.
Jun
3
comment For an $n$-dimensional object, how many types of holes are possible?
@LeeMosher, if you can put that as an answer I would be delighted to check it for you -- you nailed it, thanks! I even looked at Betti numbers at one time, but most of the descriptions are so different from the way I was approaching the holes issue that I did not see the connection. The $m$ columns of my table are not Betti numbers, but the diagonals of that same table are very close, since they describe "voids" if increasing dimensionalities -- the interiors of $n$-spheres. So again, thanks! This is a lead I can work with. So far, I've seen nothing related to object interactions, though.
Jun
2
revised For an $n$-dimensional object, how many types of holes are possible?
typo
Jun
1
revised For an $n$-dimensional object, how many types of holes are possible?
typos
Jun
1
revised For an $n$-dimensional object, how many types of holes are possible?
typos
Jun
1
revised For an $n$-dimensional object, how many types of holes are possible?
some typos, minor edits
Jun
1
revised For an $n$-dimensional object, how many types of holes are possible?
Large-scale typo fixes (deleting junk remains of duplicated paragraphs), some clarifications
Jun
1
revised For an $n$-dimensional object, how many types of holes are possible?
Large-scale typo fixes (deleting junk remains of duplicated paragraphs), some clarifications
Jun
1
revised For an $n$-dimensional object, how many types of holes are possible?
Added an extended explanation of why different m holes have different properties.
Jun
1
revised For an $n$-dimensional object, how many types of holes are possible?
Clarified the axis expansion procedure to avoid the "drilling" interpretation.