Timeline for All the regular $n$-gons are nested tightly around a unit circle. How to order them to minimize the outer radius, and what is that minimum radius?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| May 15, 2023 at 0:06 | comment | added | LSpice | Re, what you want is to impose some order on the natural numbers greater than $2$ that does not necessarily have the same order type as the existing order. I guess you have to decide what properties you require of this order—almost certainly it should be linear, but, for example, do you want it to be well ordered? Anyway, you'd need to decide on that, but, even once you do, I'm not sure of any existing terminology. | |
| May 14, 2023 at 22:06 | comment | added | Dan | @LSpice Yes, I mean to allow this situation. Is there a word, instead of permutation, that allows this? | |
| May 14, 2023 at 15:22 | comment | added | LSpice | Re, I understand what you mean geometrically, but $s(4), s(3), s(5), \dotsc$ is not a permutation of the integers greater than $2$ (because, for example, $3$ has infinitely many predecessors), and similarly for $3, s(4), s(6), s(5), \dotsc$. Do you mean to allow this situation? | |
| May 14, 2023 at 14:23 | comment | added | Dan | @LSpice Yes, I do. If we assume the polygons have a finite number of sides, then any permutation would have some coprime consecutive polygons, in which case minimizing the outer one would (I think) mean that they are not concentric. On the other hand, in the following permutation the polygons are all minimized and concentric: $s(4),s(3),s(5),s(7),s(9),...$ where $s(k)$ means $k, 2k, 2^2k, ..., 2^\infty k$ but it can be shown that this permutation is not as economical as say $3,s(4),s(6)$ then $s(5),s(7),s(9),...$. | |
| May 14, 2023 at 10:58 | comment | added | LSpice | Do you expect the minimum to occur with non-concentric polygons? | |
| May 14, 2023 at 3:01 | comment | added | Dan | @GerryMyerson By radius I mean the distance from the centre to a vertex. I've added a link to clarify. | |
| May 14, 2023 at 2:54 | history | edited | Dan | CC BY-SA 4.0 |
added 60 characters in body; edited title
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| May 14, 2023 at 2:48 | history | edited | Dan | CC BY-SA 4.0 |
edited title; added 60 characters in body
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| May 14, 2023 at 2:44 | comment | added | Gerry Myerson | What is meant by the "radius" of a polygon? | |
| May 14, 2023 at 2:27 | history | asked | Dan | CC BY-SA 4.0 |