Timeline for The "stained glass window problem": Draw many random chords in a circle; which kind of polygon ($3$-gon, $4$-gon, etc.) occupies the most total area?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Oct 28 at 7:52 | comment | added | Timothy Budd | No, see the same paragraph of the paper Hilhorst & Calka. The Crofton cell is biased by the area, compared to the typical cell. | |
| Oct 27 at 14:25 | comment | added | Dan | @TimothyBudd Is the side distribution of the Crofton cell the same as the side distribution of a typical cell? | |
| Oct 27 at 10:22 | comment | added | Timothy Budd | Wouldn't the fraction of the area covered by $d$-gons converge as $n\to\infty$ to the side distribution $p_d$ of the Crofton cell of the Poisson line process? See the summary of Hilhorst & Calka in Section 1.2 of arXiv:0802.1869, which claims that $p_d$ has mean $\pi^2/2=4.93\ldots$ and peaks at pentagons $d=5$, citing Matheron. | |
| Oct 26 at 22:57 | history | edited | Dan | CC BY-SA 4.0 |
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| Oct 25 at 12:09 | comment | added | Peter Taylor | Empirically it seems to be pentagons. Out of $1000$ trials of $5m$ chords for $m \in [1, 20]$ pentagons took the lead at $20$ chords with $416$ wins (vs $395$ for quadrilaterals), and by $100$ chords they were winning more than $80\%$ of trials, with quadrilaterals picking up the rest. | |
| Oct 23 at 22:23 | history | edited | Dan | CC BY-SA 4.0 |
added 226 characters in body
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| Oct 22 at 9:44 | history | asked | Dan | CC BY-SA 4.0 |