Optimal Spacing for 16 spheres in a round pan.
November 27, 2024 8:55 PM Subscribe
I have always made my holiday dinner rolls in the format of 12 rolls at 60grams in a 10", tall cake pan and over the years have gradually figured out the placement to make them all fairly close to the same circumference. This year I need to adjust that to 16 rolls at 45 grams in the same 10" cake ban and my brain has broken when it comes to proper spacing. I'm guessing there's a formula or algorithm that would illustrate placement but that's not how my brain works. Any ideas or illustrations would be very welcome.
Best answer: You might take some inspiration from the diagrams in this Wikipedia article: Circle Packing in a Circle
posted by moonmilk at 9:50 PM on November 27 [37 favorites]
posted by moonmilk at 9:50 PM on November 27 [37 favorites]
Response by poster: Goddamn I love you folks. Pentagram rather than triangle in the center, gracias!
posted by nenequesadilla at 10:30 PM on November 27 [1 favorite]
posted by nenequesadilla at 10:30 PM on November 27 [1 favorite]
It's hard to actually say without knowing how big around your 12 vs 16 rolls are
To answer that part: if the 45g rolls are the same thickness as the 60g rolls which we define to have an area A60 and a radius r60=sqrt(A60/π), their area will be 3/4*A60 and their radius r45=sqrt(3*A60/4*π); this can be reduced to r45 = r60*0.87.
Circumference is 2*π*radius, or in simpler terms linear with radius, so c45 = c60*0.87.
posted by Stoneshop at 9:18 AM on November 28 [1 favorite]
To answer that part: if the 45g rolls are the same thickness as the 60g rolls which we define to have an area A60 and a radius r60=sqrt(A60/π), their area will be 3/4*A60 and their radius r45=sqrt(3*A60/4*π); this can be reduced to r45 = r60*0.87.
Circumference is 2*π*radius, or in simpler terms linear with radius, so c45 = c60*0.87.
posted by Stoneshop at 9:18 AM on November 28 [1 favorite]
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12 rolls in a circle is 3 rolls per quarter of a circle probably approximately in a triangle shape with one near the center of the pan and two on the edges. Repeat for each quadrant and you have 4 rolls around the center and 8 rolls around the edges. Or nearly so as approximations allow and leaving enough of a gap to let the rolls spread a little.
For 16 rolls (4 more than 12), the obvious thing to do is to make room for them in the middle of each quarter of the pan's triangle of rolls. Move the center 4 rolls closer to the center, move the 8 other rolls closer to the edge and fit the 4 new rolls in the middle of each quarter of the pan.
If you think about the line from the center of the pan to a center of a roll... from the 12 roll version move the 4 inner rolls closer to the center along that radius, move the outer 8 closer to the edge along that radius... until you have enough room for that extra roll in the middle.
It's hard to actually say without knowing how big around your 12 vs 16 rolls are, how much space you currently leave between them and how you pack the pan with the 12 rolls for your desired perfect roll.
posted by zengargoyle at 9:45 PM on November 27 [1 favorite]