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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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]