4 ms·

Erich's Packing Center

161 points by amadeuspagel 5y ago

madcaptenor 5y ago
Here's how to fit twelve circular muffins on a circular plate: https://gottwurfelt.com/2020/12/27/applied-circle-packing-2/ https://gottwurfelt.com/2020/12/27/applied-circle-packing-2/
ptero 5y ago
This is a fascinating collection (especially in-depth plots of each type), thank you for posting (and many, many thanks to the author for building this)!
tshaddox 5y ago
I’ve encountered this lovely site several times over the years, but this is the first time I’ve noticed the animated section. These animations of circles are quite satisfying: https://erich-friedman.github.io/packing/ciranima/ https://erich-friedman.github.io/packing/ciranima/
twic 5y ago
Who is Maurizio Morandi? He seems to have come up with a significant number of the most devious packings.
eezurr 5y ago
This is lovely and made me smile. Reminds me of the (incredibly fun) times I spent on attempting to solve problems that wont necessarily have an impact on the real world (e.g. Collatz conjecture)
pierrec 5y ago
I'm probably not the first one, but I wonder if there's a recognizable sequence in the "squares in triangles" packings: https://erich-friedman.github.io/packing/squintri/ https://erich-friedman.github.io/packing/squintri/ You'll notice that some of the packings have a threefold rotational symmetry, but only for certain numbers of triangles: 3, 6, 9, 15, 27, 36... A quick search in OEIS yields cryptic results, probably a red herring.
borepop 5y ago
My stepfather (a retired professor) is endlessly fascinated by these kinds of questions. If you ever ask him at dinner "so what are you working on?" the result is often a half-hour-long explanation of how many polygons could conceivably fit into some other polygon, if it's not about how many numbers could be derived from some other set of numbers. I've never been able to understand any of it, but it makes him happy.
etaioinshrdlu 5y ago
I love this kind of stuff but am always searching for a good engineering application of it, and I can't think of too much. Anyone have some ideas?
- kwhitefoot 5y ago
  It's not exactly the same but cutting shapes out of pieces of material in ways that waste as little as possible is surely related. The paper industry is very interested in a specialised subset of this kind of thing. So are factories that make transformer cores. The cores are made by stacking many (sometimes hundreds) of thin sheets of steel cut from rolls. Good quality core steel is expensive so it is important to waste as little as possible by making the cuts in the proper places.
  - sidpatil 5y ago
    These are specifically known as cutting stock problems [1]. Example of a possible application: I used to work at a home improvement retailer, and one of my responsibilities was to cut plywood boards for customers, to the dimensions they specified. I'd often wonder if the cuts I was making were optimal. The constraints were that any cuts I made had to go all the way through the entire piece, and that the blade itself was 1/8" thick. [1] https://en.m.wikipedia.org/wiki/Cutting_stock_problem https://en.m.wikipedia.org/wiki/Cutting_stock_problem
    - madcaptenor 5y ago
      A similar problem shows up in quilting - fabric comes in rolls of about 44 inches wide (called "WOF" for "width of fabric" in some patterns). (Occasionally there are other pieces - for example you can buy "fat quarters" which are 18 by 22 inches, i. e. half a yard by half WOF.) You may need many small pieces, say rectangles or triangles a few inches on a side. And so it's natural to ask what the best way to cut them is - either with minimal waste, or with making the smallest number of cuts.
- sidpatil 5y ago
  Which field(s) of engineering did you have in mind? One obvious application would be to find the minimum dimensions for a box to contain a number of identically-shaped physical parts. This would be a three-dimensional packing problem. Another application could be to find out how to run the maximum number of VMs on a finite quantity of physical machines. This would be a multidimensional packing problem, possibly irregular since the VMs could have different memory, CPU, disk storage, etc. requirements.

161 points by amadeuspagel 5y ago

Here's how to fit twelve circular muffins on a circular plate: https://gottwurfelt.com/2020/12/27/applied-circle-packing-2/ https://gottwurfelt.com/2020/12/27/applied-circle-packing-2/

ptero 5y ago

This is a fascinating collection (especially in-depth plots of each type), thank you for posting (and many, many thanks to the author for building this)!

tshaddox 5y ago

I’ve encountered this lovely site several times over the years, but this is the first time I’ve noticed the animated section. These animations of circles are quite satisfying: https://erich-friedman.github.io/packing/ciranima/ https://erich-friedman.github.io/packing/ciranima/

twic 5y ago

Who is Maurizio Morandi? He seems to have come up with a significant number of the most devious packings.

eezurr 5y ago

This is lovely and made me smile. Reminds me of the (incredibly fun) times I spent on attempting to solve problems that wont necessarily have an impact on the real world (e.g. Collatz conjecture)

pierrec 5y ago

I'm probably not the first one, but I wonder if there's a recognizable sequence in the "squares in triangles" packings: https://erich-friedman.github.io/packing/squintri/ https://erich-friedman.github.io/packing/squintri/ You'll notice that some of the packings have a threefold rotational symmetry, but only for certain numbers of triangles: 3, 6, 9, 15, 27, 36... A quick search in OEIS yields cryptic results, probably a red herring.

borepop 5y ago

My stepfather (a retired professor) is endlessly fascinated by these kinds of questions. If you ever ask him at dinner "so what are you working on?" the result is often a half-hour-long explanation of how many polygons could conceivably fit into some other polygon, if it's not about how many numbers could be derived from some other set of numbers. I've never been able to understand any of it, but it makes him happy.

etaioinshrdlu 5y ago

I love this kind of stuff but am always searching for a good engineering application of it, and I can't think of too much. Anyone have some ideas?

kwhitefoot 5y ago
It's not exactly the same but cutting shapes out of pieces of material in ways that waste as little as possible is surely related. The paper industry is very interested in a specialised subset of this kind of thing. So are factories that make transformer cores. The cores are made by stacking many (sometimes hundreds) of thin sheets of steel cut from rolls. Good quality core steel is expensive so it is important to waste as little as possible by making the cuts in the proper places.
- sidpatil 5y ago
  These are specifically known as cutting stock problems [1]. Example of a possible application: I used to work at a home improvement retailer, and one of my responsibilities was to cut plywood boards for customers, to the dimensions they specified. I'd often wonder if the cuts I was making were optimal. The constraints were that any cuts I made had to go all the way through the entire piece, and that the blade itself was 1/8" thick. [1] https://en.m.wikipedia.org/wiki/Cutting_stock_problem https://en.m.wikipedia.org/wiki/Cutting_stock_problem
  - madcaptenor 5y ago
    A similar problem shows up in quilting - fabric comes in rolls of about 44 inches wide (called "WOF" for "width of fabric" in some patterns). (Occasionally there are other pieces - for example you can buy "fat quarters" which are 18 by 22 inches, i. e. half a yard by half WOF.) You may need many small pieces, say rectangles or triangles a few inches on a side. And so it's natural to ask what the best way to cut them is - either with minimal waste, or with making the smallest number of cuts.
sidpatil 5y ago
Which field(s) of engineering did you have in mind? One obvious application would be to find the minimum dimensions for a box to contain a number of identically-shaped physical parts. This would be a three-dimensional packing problem. Another application could be to find out how to run the maximum number of VMs on a finite quantity of physical machines. This would be a multidimensional packing problem, possibly irregular since the VMs could have different memory, CPU, disk storage, etc. requirements.