Terry Bollinger
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 Sep24 awarded Autobiographer Feb28 awarded Teacher Feb27 answered Fourier transform Jun7 awarded Scholar Jun7 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. Jun7 accepted For an $n$-dimensional object, how many types of holes are possible? Jun7 awarded Commentator Jun7 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. Jun7 revised For an $n$-dimensional object, how many types of holes are possible? An update on my efforts to answer my own question Jun6 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. Jun5 revised For an $n$-dimensional object, how many types of holes are possible? Adding a figure that shows generalized holes up through n=4. Jun3 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. Jun3 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. Jun2 revised For an $n$-dimensional object, how many types of holes are possible? typo Jun1 revised For an $n$-dimensional object, how many types of holes are possible? typos Jun1 revised For an $n$-dimensional object, how many types of holes are possible? typos Jun1 revised For an $n$-dimensional object, how many types of holes are possible? some typos, minor edits Jun1 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 Jun1 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 Jun1 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.