/DiamondsAndAlternate, for 14 on /A3

[jdaw1]

When fitting fourteen glasses on /A3, the mathematical best (a pattern provisionally known as /Bespoke14) is bigger than the software’s best (a pattern known as /DiamondsAndRectangular). /Bespoke14’s diameter is larger than /DiamondsAndRectangular’s by ≈ 0.39mm ≈ 0.46%. In issue #151 it was decided that /Bespoke14 will not be coded, Won’t Fix, as the mathematical best is ugly, and its order-18 polynomial breaks PostScript’s single-precision arithmetic.

But there is a compromise, a pattern provisionally called /DiamondsAndAlternate. Its diameter is better than /DiamondsAndRectangular’s by 0.32mm ≈ 0.37%, and hence worse than /Bespoke14’s by only −0.07mm ≈ −0.084%. That is, it captures ≈ 81.4% of the possible improvement /DiamondsAndRectangular→/Bespoke14, for a loss of symmetry that is partial but not total.

The following GIF has three frames.

1. Diameter ≈ 86.41mm, /DiamondsAndRectangular, requiring solving a quadratic polynomial. For all glasses, one or two other glasses have the same x. And for twelve of the fourteen glasses, three others have the same y. /DiamondsAndRectangular has been in the code since July 2013 (e.g., 21 Oct 2014, 21 Apr 2016, 15 June 2016, 08 June 2017, 10 Oct 2017, 10 Apr 2018, 08 May 2018, 05 June 2018, 01 May 2019, 11 June 2019, 23 Oct 2019, 21 Nov 2019, 20 Oct 2021).

2. Diameter ≈ 86.73mm, /DiamondsAndAlternate, requiring solving a quartic polynomial. All glasses are in proper columns, so one or two other glasses have the same x. But the ‘rows’ are messy, with only one non-trivial pair of matching y’s (i.e., excepting glasses on the bottom and top edges). /DiamondsAndAlternate is doable, and is the subject of this issue.

3. Diameter ≈ 86.80mm, /Bespoke14, requiring solving an order-18 polynomial, which was the subject of issue /Bespoke14 for /A3 #151. There is only one non-trivial pair of matching x’s, and no non-trivial matching y’s. /Bespoke14 has been rejected: Won’t Fix.