2

Imagine you have a ring and you want to fill as much of this ring as possible with one type of shape (triangle, square, hexagon, must be the entire shape repeated within the ring no overlaps). To be efficient filling up the ring you want the total number of shapes to be as low as possible. So which shape will have the best ratio of area filled up per number of shapes used (A/n). linked are some simple examples using squares (area of the circle = 1).

Edit: n>1, only straight edge shapes Edit 2: shape must tile the plane

Jl137
  • 29
  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community May 18 '22 at 19:51
  • 1
    Are you fixing the area of the shape? If not, you can choose the shape to be the circle itself, which would give an optimal ratio. – Jair Taylor May 18 '22 at 21:02
  • For any number n you can divide the disc into n sectors, i.e. pizza slices with 100% efficiency. You'll have to be a bit more clear about what you're asking for. – Jaap Scherphuis May 19 '22 at 14:49
  • I’ll add that the total number of shapes must be greater than 1. Also no curved edges (boring). As you decrease the size of the shape and are able to fill more voids the area filled in the circle will increase but so will the total number of shapes, infinite isosceles triangles with 100% area (1.0/infinity) doesn’t tell me that much, if more triangles means a better ratio always would be better. For each shape either theirs an optimal number of shapes used, ie 7 with hexagons or the ratio tends to become better towards infinity, in which case look at growth rate or best for a given n – Jl137 May 27 '22 at 01:59
  • If you're really measuring by the number of shapes, then you'll just want $1$ regular polygon with lots and lots of sides. – Greg Martin May 27 '22 at 02:11
  • 4
    You want $n>1$? Take a 10000-gon and cut it in halves. – Ivan Neretin May 27 '22 at 09:53
  • Ill say the shape has to tile the plane since I cant come up with a better (more general) way to think of this optimization problem that is essentially using a variable Number of polygons to approximate a circle to an arbitrary degree. These are clever answers but they work too well I’m only interested in (the best of the) bad approximations for whatever reason. I’m very curious how I’ll be outsmarted now. – Jl137 Jun 02 '22 at 22:41

0 Answers0