Software that makes placemats

[jdaw1]

Software and manual updated.

Comments, complaints, praise, bug reports: all welcomed.

[jdaw1]

Googling the problem of packing circles reveals much learned comment, and lists of best-known packings at Packomania.com. But, as posted in sci.math, the placemat software knows some to be better, most obviously that for 1607 circles on a square page. If using Bunghole-standard glasses, it would have to be a large page.

[RAYC]

Very nice.

For smaller tastings, how about an A3 mat, guillotined lengthways (i.e 420 x 15mm), that can be placed near centre table (leaving uncluttered space for notes/meals etc.) and contain ports in an easy access row?

[jdaw1]

For smaller tastings, how about an A3 mat, guillotined lengthways (i.e 420 x 15mm), that can be placed near centre table (leaving uncluttered space for notes/meals etc.) and contain ports in an easy access row?

There are also three simple designs, /TopRow, /MiddleRow, /BottomRow, each having everything in one row, with obvious vertical position. There is also /Sides, with the obvious meaning.
For more than a few glasses these are too cramped.

Might /BottomRow do the job?

[PhilW]

(i.e 420 x 15mm)

That would certainly leave the table uncluttered.

[jdaw1]

Packomania.com is for geeky mathematicians, and has lots of examples of packing circles of maximal radius in various containers, including 7×10 rectangles. That is close to the proportions of A4, less a fixed margin.

Observe Packomania’s best for for 7 circles and 10 on a 7×10 rectangle:
 

I’ll implement this as two on the right (/Landscape) or top (/Portrait), reversed for /Mirror, of course, and three-row (three-column) /Diamonds for the rest. Packomania’s 10-glass solution is an epsilon better than that, but generalising the asymmetry would be too complicated. Compared to plain seven-glass /Diamonds, that adds ≈5% to the radius. PW would have wanted that for Warre versus Fonseca tasting.

What should it be called?

[djewesbury]

Are you thinking architectural, prosaic, allusive..? I can't see anything in the pattern that would provide a short metaphorical name at present. I'm guessing it'll probably end up being something like /DiamondsPlusTwo or /DiamondsAsymmetricalSeven or something like that.

[jdaw1]

/DiamondsPlusTwo

Current winner is ‟/DiamondsPlus”, which allows a little generalisation to something like Packomania’s 29-glass best:

[djewesbury]

I wouldn't like to be reaching for glass 16 on that placemat...

[jdaw1]

I wouldn't like to be reaching for glass 16 on that placemat...

It’s printed on A0. Each glass can hold most of a bottle. By the time you reach glass 16, it would indeed be difficult.

[Glenn E.]

Current winner is ‟/DiamondsPlus”

I can think of nothing better.

[PhilW]

Current winner is ‟/DiamondsPlus”

I can think of nothing better.

Ditto. Some possible insect outlines came to mind, but ‟/DiamondsPlus” is probably simpler and clearer, and therefore preferred.

[djewesbury]

Current winner is ‟/DiamondsPlus”

I can think of nothing better.

Ditto. Some possible insect outlines came to mind, but ‟/DiamondsPlus” is probably simpler and clearer, and therefore preferred.

I quickly discarded /BeetleCarapace

[jdaw1]

Step one: write code to solve a quartic equation. Worrying, I think that I have devised an algorithm as good as Brent’s Method, but simpler, and not needing a pre-chosen x-step.

Assume root bounded by LowerX and UpperX, with matching y values LowerY and UpperY. Interpolation would make the next x value be LowerX + (UpperX”“LowerX) × LowerY/(LowerY”“UpperY). This can fail for some shapes (e.g., y = x^4 ”“ c), as the interpolated value is always on the same side of the root, so only one side (say, LowerX) ever gets moved.

So instead make the next x value be LowerX + (UpperX”“LowerX) × Max[0.143, Min[0.857, LowerY/(LowerY”“UpperY) ]]

Repeat until UpperX”“LowerX ≤ Tolerance, that constant being pre-determined and small, at which time return the interpolated value (without the bounds).

When LowerX and UpperX are roughly even around the root, it interpolates. When one side is much closer, it brings in the other, moving it by a factor of 1÷0.143 ≈ 7.

FYI, the ‟0.143” constant came from a small experiment done in Excel. I do not know whether it should be precisely 1/7, or some other value. But a small non-exactitude in this would add only a tiny extra to the algorithm’s average time.

Indeed, this can be seen as a compromise between the slow robustness of interval bisection (‟! Max[0.5, Min[0.5, ! ]]”) and interpolation (‟! Max[0, Min[1, ! ]]”).

[Glenn E.]

Current winner is ‟/DiamondsPlus”

I can think of nothing better.

Ditto. Some possible insect outlines came to mind, but ‟/DiamondsPlus” is probably simpler and clearer, and therefore preferred.

I quickly discarded /BeetleCarapace

Hmm... it does look a bit like a top down view of a Star Trek shuttlecraft... though /Galileo would probably be too esoteric of a reference.

[jdaw1]

Hmm... it does look a bit like a top down view of a Star Trek shuttlecraft... though /Galileo would probably be too esoteric of a reference.

Not sure to which of the many possible uses of /Galileo this refers, but, as you say, too esoteric.

[djewesbury]

[jdaw1]

too esoteric of a reference.

[jdaw1]

Different question. In the arrangement thread for the 1966 horizontal on 27th June, with each update of the placemats pictures have been posted in the thread.

E.g.,

Updated draft of the placemats.

• +BMHR and his Croft.

Is this worth the effort?

[djewesbury]

yes

[DRT]

with each update of the placemats pictures have been posted in the thread.
!
Is this worth the effort?

yes

Agreed.

[PhilW]

Different question. In the arrangement thread for the 1966 horizontal on 27th June, with each update of the placemats pictures have been posted in the thread.

Is this worth the effort?

In the general case I'd say no; It's useful in the case you refer to because the first post is not currently being regularly updated, so the image posts are providing a useful update of current attendees and ports; if the first post were being regularly updated, I think the additional effort of pdf -> jpg -> image hosting -> post for each placemat iteration would be excessive.

[PhilW]

Step one: write code to solve a quartic equation. Worrying, I think that I have devised an algorithm as good as Brent’s Method, but simpler, and not needing a pre-chosen x-step.

Assume root bounded by LowerX and UpperX, with matching y values LowerY and UpperY. Interpolation would make the next x value be LowerX + (UpperX”“LowerX) × LowerY/(LowerY”“UpperY). This can fail for some shapes (e.g., y = x^4 ”“ c), as the interpolated value is always on the same side of the root, so only one side (say, LowerX) ever gets moved.

So instead make the next x value be LowerX + (UpperX”“LowerX) × Max[0.143, Min[0.857, LowerY/(LowerY”“UpperY) ]]

Repeat until UpperX”“LowerX ≤ Tolerance, that constant being pre-determined and small, at which time return the interpolated value (without the bounds).

When LowerX and UpperX are roughly even around the root, it interpolates. When one side is much closer, it brings in the other, moving it by a factor of 1÷0.143 ≈ 7.

FYI, the ‟0.143” constant came from a small experiment done in Excel. I do not know whether it should be precisely 1/7, or some other value. But a small non-exactitude in this would add only a tiny extra to the algorithm’s average time.

Indeed, this can be seen as a compromise between the slow robustness of interval bisection (‟! Max[0.5, Min[0.5, ! ]]”) and interpolation (‟! Max[0, Min[1, ! ]]”).

Presumably this also depends on any assumptions regarding the nature of the quartic to be solved, i.e. all-real roots (or at least a real root between the specified starting points), no discontinuities (no matching pole-zero root pairs) etc. In which case whether bisection, interpolation or your alternate specified scheme would be quicker in the general case would presumably depend on the nature of the group of potential curves across which the technique would be used? An alternative to the factor changed used to avoid never reaching the root could be to add a small proportion of the step delta determined from interpolation (deliberate over-adjust), though potentially decreasing the over-step with time to avoid oscillation; similar to techniques used to avoid getting stuck in local minima.

[jdaw1]

Presumably this also depends on any assumptions regarding the nature of the quartic to be solved, i.e. all-real roots (or at least a real root between the specified starting points), no discontinuities (no matching pole-zero root pairs) etc. In which case whether bisection, interpolation or your alternate specified scheme would be quicker in the general case would presumably depend on the nature of the group of potential curves across which the technique would be used? An alternative to the factor changed used to avoid never reaching the root could be to add a small proportion of the step delta determined from interpolation (deliberate over-adjust), though potentially decreasing the over-step with time to avoid oscillation; similar to techniques used to avoid getting stuck in local minima.

Assumptions: ∈ ℝ; continuous; starting points either side. No more. Desiderata include robustness, speed, and simplicity of code.

Don’t need to reach the root, only for the bounds either side to be closer than xTolerance, when do a final unconstrained linear interpolation.

[PhilW]

Assumptions: ∈ ℝ; continuous; starting points either side. No more. Desiderata include robustness, speed, and simplicity of code.

Don’t need to reach the root, only for the bounds either side to be closer than xTolerance, when do a final unconstrained linear interpolation.

Any possibility of multiple roots between initial bounds (excluding duplicate root)? (I.e. can we either exclude the possibility of multiple roots being present, and If not then do we care? i.e. are all roots required, or any root). Could there be any bounding of the relative ratio of xTolerance to initial delta between upper and lower bounds?

n.b. I assume direct calculation would not be appropriate?