Wikipedia:Reference desk/Archives/Mathematics/2021 August 17

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August 17

Packing of partial disks.

I'm wondering about the packing density of partial disks (single pizza slices. :) )(all identical) for various fractions of the entire disk. While a unit fraction of a disk, such as 1/5, the packing is of course no worse than it is for the entire disk, if the fraction is small enough (say 1/16 or 2/31) a greater density can be achieved by alternating the direction and building bars for which the bumpy sides would be smooth enough to be at a higher density. My question is what fraction of a full disk has the *worst* packing density, I'm thinking about 85% where the empty slice can not be easily filled by the edge of the empty slice and outside and pushing two together doesn't gain much density. Ideas?Naraht (talk) 02:42, 17 August 2021 (UTC)

Let α stand for the angle taken out of the slice. I have examined the range 0 ≤ α ≤ 5π⁄12 = 75°. If my calculations are not off, a minimum is reached when α = 0.44556 radians = 25.529° (leaving 92.909% of the full disk), at which point the density is 0.87320. As α increases, the Lagrangian density of π⁄6√3 ≈ 0.90690 is reached and surpassed at α = 0.83120 radians = 47.624°. It then reaches a maximum at α = 1.07224 radians = 61.435°, at which point the density is 0.92277. (Then it decreases, but later it will rise again to approach density 1 arbitrarily closely as α tends to 2π.)  --Lambiam 16:59, 18 August 2021 (UTC) (edited) 17:54, 18 August 2021 (UTC)

Lambian Is the minimum at .44556 radians generated experiementally or are there two functions where that point represents the least value for the larger function. And any chance for a graph?Naraht (talk) 18:03, 19 August 2021 (UTC)

In the packing a unit is formed by two partial disks that share a maximal length of one of their two boundary rays. This unit is replicated in a tight lattice pattern. There is a minor change in the geometric configuration at α = π⁄4, which is reflected in a C¹ transition between two (analytic) density formulas, but there is no extremum there. The minimum occurs at a point where the derivative of the density with respect to α equals 0. If you can generate plots yourself, I can give you the formulas.  --Lambiam 18:40, 19 August 2021 (UTC)

A curious question (mathematical analysis of Spinoza's work)

Is there a purely mathematical analysis of Spinoza's work Ethics?--82.82.234.67 (talk) 21:46, 17 August 2021 (UTC)

On Academia.edu there is a paper entitled "A Mathematical Interpretation of Spinoza's Ethics: Short preliminary remarks".^[1] A purely mathematical analysis inspired by an aspect of his philosophy can be found in the article "Some Settheoretical Partition Theorems Suggested by the Structure of Spinoza's God".^[2]  --Lambiam 06:05, 18 August 2021 (UTC)