The function pack is designed to iteratively pack a specified number of non-overlapping disks within a given region while maximizing the radius of each disk. Here’s a detailed breakdown of its components and behavior:

1. Function Definition and Parameters:

•

pack[reg_, n_]

: The function takes two parameters: reg, which represents the region within which the disks are to be packed, and n, which is the number of disks to be packed.

2. Local Variables:

• disk: Represents a disk with center

{x, y}

and radius r.

•

diski[i_]

: Represents the i-th disk with center

{x[i], y[i]}

and radius

r[i]

•

circi[i_]

: Represents the boundary of the i-th disk.

• cons: A list of constraints ensuring that a disk is within the specified region.

•

consDJ[i_]

: A function that returns a constraint ensuring that the i-th disk is disjoint from the main disk.

3. Optimization Loop:

• The function uses a loop (

loop) to place each disk iteratively.

• For each disk i, it maximizes the radius r subject to the constraints that the disk must be within the region and disjoint from all previously placed disks.

•

NArgMax

is used to find the maximum radius and corresponding center coordinates for each disk.

• After finding the optimal disk parameters, cons is updated to include a new constraint ensuring that the newly placed disk is disjoint from the main disk.

4. Performance Output:

• The function prints the total optimization time taken to place all disks.

5. Packing Fraction:

• The packing fraction is calculated as the sum of the areas of all packed disks divided by the area of the region.

• This value is printed to indicate how efficiently the disks have been packed within the region.

6. Visualization:

• The function uses

Show

to create a visualization of the region and the packed disks.

• The region is displayed with a light brown color and thick black edges.

• Each disk is displayed with a distinct color, and their boundaries are outlined in black.

Overall, the function aims to demonstrate an optimization process for packing disks within a region, calculating the efficiency of the packing, and providing a visual representation of the result.

Packing Code

pack[reg_,n_]:=Block[{disk,diski,circi,cons,consDJ,x,y,r,time,ds,cs,pf},disk=Disk[{x,y},r];(*Definetheinitialdisk*)diski[i_]=Disk[{x[i],y[i]},r[i]];(*Definethediskifunctiontocreatediskswithvaryingradii*)circi[i_]=RegionBoundary[diski[i]];(*Definethecircifunctiontogettheboundaryofthedisks*)cons=List@@(RegionWithin[reg,disk]);consDJ[i_]:=RegionDisjoint[diski[i],disk];(*Performtheoptimizationandmeasurethetimetaken*)time=AbsoluteTiming@Do[{Thread[{x[i],y[i],r[i]}=NArgMax[{r,cons},{x,y,r}]],cons=Join[cons,{consDJ[i]}]},{i,n}];(*Calculateandprintthepackingfraction*)Print["optimization time = ",time[[1]]];pf=Sum[Area@diski[i],{i,n}]/Area[reg];Print["packing fraction =",pf];(*Displaythepackeddiskswithintheregion*)Show[Region[Style[reg,LightBrown,EdgeForm[{Thick,Black}]]],Graphics@Table[{Hue[0.8*(i-1)/n],diski[i]},{i,n}],Graphics@{Thick,Black,Table[circi[i],{i,n}]},Frame->True]]