New info on energy parks

[Chaos Theory]

Read what I posted and consider what happens if a mirror has less than 8 panels next to it, or a panel has less than 8 mirrors next to it. Everything I wrote still holds.

Look at it slightly differently. Take a field of panels and mirrors. For each panel, calculate how much extra energy it receives from mirrors (in that field). For each mirror, calculate how much extra energy it contributes to panels (in that field). Obviously, the sums of these sets must be equal, and each in isolation represents the actual energy gained over homogenous solar panels.

Swap every mirror and panel. As I explained in my previous post, for each tile, the bonus energy it contributes or receives must stay the same. Therefore, all of the bonuses in the mirror set are mapped into the panel set, and vice versa. Their sums remain the same and still represent the energy gained over simply using solar panels.

Nothing here depends in any way on the shape of the field, its elevations, solar flares, or any other qualities of the field. However, the following qualities may vary when you exchange mirrors and panels:

The variation in energy from tiles may vary. In other words, one field may be more desirable than the other if you don't have enough crawlers to cover the field.
Terraforming time will likely vary.
...can't think of anything else significant that varies right now.

[Skanky Burns]

That makes more sense now.

[Patashu]

Okay, so energy parks are most efficient when they're circular and are painted with stripes of solars and mirrors, right?

[Skanky Burns]

   e4e   		   3e3   		2e4e4e4e2
 e5e6e5e 		 2e5e5e2 		3e6e6e6e3
 e6e6e6e 		 4e6e6e4 		3e6e6e6e3
3e6e6e6e3		e5e6e6e5e		3e6e6e6e3
3e6e6e6e3		e6e6e6e6e		3e6e6e6e3
3e6e6e6e3		e5e6e6e5e		3e6e6e6e3
 e6e6e6e 		 4e6e6e4 		3e6e6e6e3
 e5e6e5e 		 2e5e5e2 		3e6e6e6e3
   e4e   		   3e3   		2e4e4e4e2

Tiles: 61		Tiles: 61		Tiles: 81
e: 32 s: 29		e: 29 s: 32		e: 36 s: 45
Ftime: 457		FTime: 328		FTime: 423
Energy: 148		Energy: 148		Energy: 200
e/t: 2.43		e/t: 2.43		e/t: 2.47
e/F: .324		e/F: .451		e/F: .473


  3e3			  e4e  			2e4e4e2
 e5e5e			 4e6e4 			3e6e6e3
3e6e6e3			e5e6e5e			3e6e6e3
3e6e6e3			e6e6e6e			3e6e6e3
3e6e6e3			e5e6e5e			3e6e6e3
 e5e5e			 4e6e4 			3e6e6e3
  3e3			  e4e  			2e4e4e2

Tiles: 37		Tiles: 37		Tiles: 49
e: 17 s: 20		e: 20 s: 17		e: 21 s: 28
Ftime: 196		FTime: 211		FTime: 252
Energy: 86		Energy: 86		Energy: 114
e/t: 2.32		e/t: 2.32		e/t: 2.33
e/F: .439		e/F: .408		e/F: .452


Key:
e = eschelon mirror
s= solar collector
FTime = Former time (8e + 3s)
e/t = energy per tile
e/F = energy per Former time

Regardless of the metric used to measure this, square energy parks seem superior to circular parks. They are built faster compared to the energy they produce, and they net you more energy per tile used.

EDIT: Fixed both middle figures.

[Skanky Burns]

           e4e4e             2e4e4e4e4e4e4e4e4e4e4e4e4e2
        3e5e6e6e5e3          3e6e6e6e6e6e6e6e6e6e6e6e6e3
      3e5e6e6e6e6e5e3        3e6e6e6e6e6e6e6e6e6e6e6e6e3
     e5e6e6e6e6e6e6e5e       3e6e6e6e6e6e6e6e6e6e6e6e6e3
    3e6e6e6e6e6e6e6e6e3      3e6e6e6e6e6e6e6e6e6e6e6e6e3
   e5e6e6e6e6e6e6e6e6e5e     3e6e6e6e6e6e6e6e6e6e6e6e6e3
  3e6e6e6e6e6e6e6e6e6e6e3    3e6e6e6e6e6e6e6e6e6e6e6e6e3
  4e6e6e6e6e6e6e6e6e6e6e4    3e6e6e6e6e6e6e6e6e6e6e6e6e3
 e5e6e6e6e6e6e6e6e6e6e6e5e   3e6e6e6e6e6e6e6e6e6e6e6e6e3
 e6e6e6e6e6e6e6e6e6e6e6e6e   3e6e6e6e6e6e6e6e6e6e6e6e6e3
 e6e6e6e6e6e6e6e6e6e6e6e6e   3e6e6e6e6e6e6e6e6e6e6e6e6e3
3e6e6e6e6e6e6e6e6e6e6e6e6e3  3e6e6e6e6e6e6e6e6e6e6e6e6e3
3e6e6e6e6e6e6e6e6e6e6e6e6e3  3e6e6e6e6e6e6e6e6e6e6e6e6e3
3e6e6e6e6e6e6e6e6e6e6e6e6e3  3e6e6e6e6e6e6e6e6e6e6e6e6e3
3e6e6e6e6e6e6e6e6e6e6e6e6e3  3e6e6e6e6e6e6e6e6e6e6e6e6e3
3e6e6e6e6e6e6e6e6e6e6e6e6e3  3e6e6e6e6e6e6e6e6e6e6e6e6e3
 e6e6e6e6e6e6e6e6e6e6e6e6e   3e6e6e6e6e6e6e6e6e6e6e6e6e3
 e6e6e6e6e6e6e6e6e6e6e6e6e   3e6e6e6e6e6e6e6e6e6e6e6e6e3
 e5e6e6e6e6e6e6e6e6e6e6e5e   3e6e6e6e6e6e6e6e6e6e6e6e6e3
  4e6e6e6e6e6e6e6e6e6e6e4    3e6e6e6e6e6e6e6e6e6e6e6e6e3
  3e6e6e6e6e6e6e6e6e6e6e3    3e6e6e6e6e6e6e6e6e6e6e6e6e3
   e5e6e6e6e6e6e6e6e6e5e     3e6e6e6e6e6e6e6e6e6e6e6e6e3
    3e6e6e6e6e6e6e6e6e3      3e6e6e6e6e6e6e6e6e6e6e6e6e3
     e5e6e6e6e6e6e6e5e       3e6e6e6e6e6e6e6e6e6e6e6e6e3
      3e5e6e6e6e6e5e3        3e6e6e6e6e6e6e6e6e6e6e6e6e3
        3e5e6e6e5e3          3e6e6e6e6e6e6e6e6e6e6e6e6e3
           e4e4e             2e4e4e4e4e4e4e4e4e4e4e4e4e2

Tiles: 553			Tiles: 729
e: 275 s: 278			e: 351 s: 378
Ftime: 3034			Ftime: 3942
Energy: 1554			Energy: 2054
e/t: 2.81			e/t: 2.82
e/F: .512			e/F: .521

EDIT: Still not as efficient as a square.

[Chaos Theory]

In your next to last post, you didn't count the energy properly in the second figure. The numbers you listed sum to 144, not 140, but the numbers themselves are slightly wrong, too. They should be:

   3e3
 2e5e5e2
 4e6e6e4
e5e6e6e5e
e6e6e6e6e
e5e6e6e5e
 4e6e6e4
 2e5e5e2
   3e3

Note where some 4s have changed to 5s. I noticed the problem because your second figure is just the first with all the panels and mirrors swapped, yet your energy totals differed. You made similar errors in your fifth figure.

I wonder if a circular energy park without all the wasted mirrors on the periphery (replace with panels) would be more efficient than a square park? For example, replace every mirror that contributes 3 or less bonus energy with a panel.

[Skanky Burns]

Doesn't seem so. For example, changing any of the mirrors on the left-most row of diagram in your post removes 3 energy from the grid, replacing it with either 2 or 1.

And thanks for checking the diagrams, fixed now.

[Santiago_Claus]

Damn, SB, I'm impressed! Hah! Not even I went to the trouble of making models of circular energy parks, but I'm glad somebody did.

Building things in a circle is ... umm, "impractical" I think is the word I'm looking for. Besides, I'm into parallel ridges (have I exceeded my quota for that phrase this month?)

However, in defense of the circle it bears mentioning that both circle and square efficiency is directly proportional to the overall size of the park, and that the lean, compact circle with a mere 553 tiles slightly tops the ponderous old square which tips the scales at 729.

Put another way, a square of 553 tiles would be less efficient; a circle of 729 tiles would be more.

Not that I would advocate building circular energy parks. Where would you fit the thing?

Hmm ... if you raised concentric rings centered on Mount Planet you could tell how old the park was by counting them ... ok, why is everybody looking at me like that?

[Chaos Theory]

I counted 1558 energy in your circle, for an energy/tile of 2.817. However, as I look at your circle, I see obvious improvements. Any empty tile next to 3 mirrors should be filled with a panel, since the average bonus/tile is less than 3. Likewise, any empty tile next to three panels should be filled with a mirror. That produces an octagon. After that, I add rows and columns until I no longer gain energy/tile. It turns out that adding a row of n tiles gives 3n - 1 energy, but adding a column of n tiles gives 3n energy. Therefore, the right and left sides of the octagon (now hexagon) become points, but the top and bottom remain flattened. Then, here is how I can improve the circle:

               3e4e4e3 
              e5e6e6e5e
             4e6e6e6e6e4
            e5e6e6e6e6e5e         
           4e6e6e6e6e6e6e4        
          e5e6e6e6e6e6e6e5e       
         4e6e6e6e6e6e6e6e6e4      
        e5e6e6e6e6e6e6e6e6e5e     
       4e6e6e6e6e6e6e6e6e6e6e4    
      e5e6e6e6e6e6e6e6e6e6e6e5e   
     4e6e6e6e6e6e6e6e6e6e6e6e6e4
    e5e6e6e6e6e6e6e6e6e6e6e6e6e5e
   4e6e6e6e6e6e6e6e6e6e6e6e6e6e6e4
  e5e6e6e6e6e6e6e6e6e6e6e6e6e6e6e5e
 4e6e6e6e6e6e6e6e6e6e6e6e6e6e6e6e6e4
e4e6e6e6e6e6e6e6e6e6e6e6e6e6e6e6e6e4e
 4e6e6e6e6e6e6e6e6e6e6e6e6e6e6e6e6e4
  e5e6e6e6e6e6e6e6e6e6e6e6e6e6e6e5e
   4e6e6e6e6e6e6e6e6e6e6e6e6e6e6e4
    e5e6e6e6e6e6e6e6e6e6e6e6e6e5e
     4e6e6e6e6e6e6e6e6e6e6e6e6e4
      e5e6e6e6e6e6e6e6e6e6e6e5e   
       4e6e6e6e6e6e6e6e6e6e6e4    
        e5e6e6e6e6e6e6e6e6e5e     
         4e6e6e6e6e6e6e6e6e4      
          e5e6e6e6e6e6e6e5e       
           4e6e6e6e6e6e6e4        
            e5e6e6e6e6e5e         
             4e6e6e6e6e4
              e5e6e6e5e
               3e4e4e3

Bonus: 1896 (hex) vs 1558 (circle) vs 2054 (square)
Tiles: 667 (hex) vs 553 (circle) vs 729 (square)
B/T = 2.84258 (hex) vs 2.81736 (circle) vs 2.81756 (square)
Mirrors: 333 (hex)
Panels: 334 (hex)
Ftime: 3666 (hex)
B/F: .51718 (hex)


To optimize for former efficiency, since F(panel) = 3, place panels such that they give more than B/F * F(panel) = 1.55 energy each. Plenty of empty tiles are adjacent to two mirrors. Furthermore, mirrors should be eliminated that give less than B/F * F(mirror) = 4.14 energy each. If this is done, the energy park looks like this:

                 2 2
               3e4e4e3 
             2e5e6e6e5e2
             4e6e6e6e6e4
           2e5e6e6e6e6e5e2        
           4e6e6e6e6e6e6e4        
         2e5e6e6e6e6e6e6e5e2      
         4e6e6e6e6e6e6e6e6e4      
       2e5e6e6e6e6e6e6e6e6e5e2    
       4e6e6e6e6e6e6e6e6e6e6e4    
     2e5e6e6e6e6e6e6e6e6e6e6e5e2  
     4e6e6e6e6e6e6e6e6e6e6e6e6e4
   2e5e6e6e6e6e6e6e6e6e6e6e6e6e5e2
   4e6e6e6e6e6e6e6e6e6e6e6e6e6e6e4
 2e5e6e6e6e6e6e6e6e6e6e6e6e6e6e6e5e2
 3e6e6e6e6e6e6e6e6e6e6e6e6e6e6e6e6e3
 3e6e6e6e6e6e6e6e6e6e6e6e6e6e6e6e6e3
 3e6e6e6e6e6e6e6e6e6e6e6e6e6e6e6e6e3
 2e5e6e6e6e6e6e6e6e6e6e6e6e6e6e6e5e2
   4e6e6e6e6e6e6e6e6e6e6e6e6e6e6e4
   2e5e6e6e6e6e6e6e6e6e6e6e6e6e5e2
     4e6e6e6e6e6e6e6e6e6e6e6e6e4
     2e5e6e6e6e6e6e6e6e6e6e6e5e2  
       4e6e6e6e6e6e6e6e6e6e6e4    
       2e5e6e6e6e6e6e6e6e6e5e2    
         4e6e6e6e6e6e6e6e6e4      
         2e5e6e6e6e6e6e6e5e2      
           4e6e6e6e6e6e6e4        
           2e5e6e6e6e6e5e2        
             4e6e6e6e6e4
             2e5e6e6e5e2
               3e4e4e3
                 2 2

Bonus: 1954
Mirrors: 331
Panels: 366
Ftime: 3746
B/F: .52162 vs .52106 (square)

[Skanky Burns]

   3e3				  3e3
  e5e5e				2e5e5e2
 4e6e6e4			3e6e6e3
e4e6e6e4e			3e6e6e3
 4e6e6e4			3e6e6e3
  e5e5e				2e5e5e2
   3e3				  3e3

Tiles: 39			Tiles: 41
e: 19 s: 20			e: 17 s: 24
Ftime: 212			Ftime: 208
Energy: 92			Energy: 94
e/t: 2.36			e/t: 2.29
e/F: .4340			e/F: .4388


				23333332
2333333333332			eeeeeeee
eeeeeeeeeeeee			46666664
2333333333332			eeeeeeee
				23333332

Tiles: 39			Tiles: 40
e: 13 s: 26			e: 16 s: 24
Ftime: 182			Ftime: 200
Energy: 74			Energy: 88
e/t: 1.90			e/t: 2.2
e/F: .4066			e/F: .44

Figure 1 features a standard hex park. Figure 2 is the optimised version, which also gives the highest total energy bonus. Strangely enough though, figure 4 gains you more energy per former time than other designs using a similar amount of tiles.

[Skanky Burns]

    2
   e4e
  4e6e4
 e5e6e5e
3e6e6e6e3
 e5e6e5e
  4e6e4
   e4e
    2

Tiles: 41
e: 20 s: 21
Ftime: 223
Energy: 96
e/t: 2.34
e/F: .4305

And here is figure 1 again with the ends all the way to a point. Solars and mirrors have been swapped, but this has no effect on bonus energy as proven by Chaos Theory's Theory. As can be seen, the two tips only add 2 bonus energy which drags the average bonus per tile down. If aiming for highest energy per tile, then figure 1 (previous post) is the one to go for.

OT: I have fixed the energy count and related stats for the giant circular energy park. I have also triple-checked my latest models, but no guarantees are made about their accuracy as it is late and I am drunk. As usual.

And I must also say: You are one smart cookie, Brian. Nice work on the optimisation of these energy parks, particularly regarding the equations to determine what tiles to add/delete.

[Chaos Theory]

It should be noted that these are still not necessarily the optimal shapes for a given number of tiles, since I focused on adding/removing only individual tiles, rows, and columns, and there could be more exotic operations that raise E/t (in fact, there are, since increasing the overall size of the park increases E/t towards the limit of 3).

[Santiago_Claus]

The question that comes to mind now is whether the optimal shape as area -> infinity is

1. A "trimmed circle"

2. An n-gon, where "n" is constant

3. An n-gon, where "n" increases as a function of increasing area (an increasing "n" makes this a lot like a "trimmed circle")

[Chaos Theory]

For optimal energy/tile, the optimal shape should be exactly a hexagon, with one diagonal perpendicular to the lines of mirrors, though I'm not sure how to prove this. I'm confident that simple polygons, such as octagons or squares, are inferior to hexagons of equal area, assuming alternating lines of mirrors and panels. The shapes themselves break down in the limit of small area, of course.

There is nothing to gain by having curvature in the energy park.