0

For practical reasons I would like to know how many dots you can fit in an $7 \times 8$ box with no two dots closer than 2 metres from each other.

The simplest arrangement has 4 dots along each row and 5 rows. Can you do any better?

One method is to expand the box to $9 \times 10$ and fit circles of radius $1$. We can fit $23$ with:

enter image description here

But is it possible to fit more?

  • 1
    Does this answer your question? How many circles of a given radius can be packed into a given rectangular box? – JMoravitz May 22 '20 at 14:14
  • @JMoravitz I think that's a slightly different question. It would be like saying the centres of the circles have to be in the rectangle but the rest doesn't. Oh maybe you can just expand the box to $9 \times 10$? –  May 22 '20 at 14:15
  • so? Extend the rectangle with a 1meter margin on each side. Now the circles fit entirely within the margins. – JMoravitz May 22 '20 at 14:16
  • @JMoravitz Yes I noticed that too. But does the link give me a method to actually solve my specific problem? –  May 22 '20 at 14:17
  • You can expand the box with a $1$ meter margin on each side, becoming $6 \times 7$, because that is how far you are allowing the circles to stick out. If you take a look at http://www.packomania.com/ you will see that these problems are hard. – Ross Millikan May 22 '20 at 14:18
  • @RossMillikan Is there some way to get a better answer than 20 for my specific question? –  May 22 '20 at 14:19
  • There might be. I haven't tried. Likely it would involve a region of hexagonal packing in the middle, which is denser than square packing. The closest rectangle shown is $1 \times 0.8$ while yours is $1 \times \frac 67$. It shows the $4 \times 5$ grid to be the best known. – Ross Millikan May 22 '20 at 14:23
  • Oops, I got the dimensions wrong. The expanded box is $9 \times 10$. packomania still doesn't have $1 \times 0.9$ – Ross Millikan May 22 '20 at 14:25
  • @RossMillikan Maybe something like https://int-e.eu/~bf3/tmp/21.png ? –  May 22 '20 at 14:29

2 Answers2

1

You can do at least 23, since $7/4=1.75 > \sqrt{3}$:

enter image description here

Vasily Mitch
  • 10,414
0

Yes, in this example fitting 23 circles of radius=1 into an 9x10 rectangle is optimal.

This hexagonal packing arrangement plus the close cropped edges yields the highest packing efficiency for circles into a rectangle.

Packing efficiency for 23 circles = 23xPi/(9x10) = 80.3%

Packing efficiency for 24 circles = 24xPi/(9x10) = 83.7%

Referring to this graph: https://i.sstatic.net/ziRBe.jpg it would appear that 83.7% is not possible. (Note: this graph allowed for a slight spacing between circles, I could recalculate the graph for zero spacing as per the original question but I think you get the idea)

For further details on similar problems: How many circles of a given radius can be packed into a given rectangular box?

Paul McErlean
  • 41