I had assumed that a 10x10 grid was best, but it turns out it's not.

## Guide to The Optimal Square Grid

### Introduction

I always figured that if I was going to build a nice, regular, square grid that of course it should be 10×10: That way each of the blocks is fully zonable and the road length highlight and road guideline circles make it easy to draw.

But it turns out we can do better.

### Time for Mathematics

For the 10×10 grid, the calculation is simple: we get 8² zonable tiles in a 10² tile area, for a density of exactly 64%. (Here, and in the rest of the section, we assume a small—2u—road).

For the general case of a square of side 𝓁, the zoneable area is (𝓁-2)²-max(0, 𝓁-10)² -- the area inside the road minus the area in the middle where the zoning doesn't reach -- and the total area is 𝓁².

It's hard for me to grok that quotient in my head, so let's just graph it and see what happens:

That shows that the "obvious" 10×10 grid is actually only as good as a 15×15 grid:

It also looks like 12u is the best. Can we prove it? Sure, let's go back to Calculus I:

- 0 = d/d𝓁 (𝓁-2)²-(𝓁-10)² / 𝓁²
- 0 = d/d𝓁 ( (𝓁-2)+(𝓁-10) )*( (𝓁-2)-(𝓁-10) ) / 𝓁²
- 0 = d/d𝓁 (2𝓁-12)*8 / 𝓁²
- 0 = d/d𝓁 (16𝓁-96) / 𝓁²
- 0 = ( (16)*𝓁² - (16𝓁-96)*2𝓁 ) / 𝓁⁴
- 0 = (16𝓁 - 32𝓁 + 192) / 𝓁³
- 0 = -16𝓁 + 192
- 𝓁 = 192/16
- 𝓁 = 12

So yes, it's a block length of exactly 12 that gives the best density.

Note: Coincidentally, that's also the maximum segment length for an axis-aligned road in CSL.

How much better? ⅔ of the area (66.66…%) instead of just 64%.

Now, I admit that only about 4% more zoning doesn't sound that exciting. But not only is it more dense, but it's also cheaper.

How's that? Well, you need to pay for the roads. And for the 10×10 blocks, those 64 tiles of zoning need 40u of roads (1.6tiles/u), but with a 12×12 block you get 96 tiles of zoning out of only 48u of roads (2tiles/u). So you're also paying 16% less on the roads to grid the same area with more stuff.

For some concrete numbers, let's compare a 60×60 area (as it divides evenly in a bunch of ways) using a variety of block sizes and the basic small two-lane road:

So if you make a 10×10 grid, you'll spend 50% more than you need to for the area you'll get.

Note: That table also shows that zonable area drops slower than road cost, but don't forget that lower density is also increased cost for any service with a radius of effect.

### Conclusion

If you want the densest square grid, make it 12×12 (using small roads).

You'll fit 1⁄24 more RICO in the same space but spend 1⁄6 less on the roads, compared to a 10×10 grid.

### Appendix A: Medium or Large Roads

What if you're using a 4u-wide road, like the Medium and Large Roads?

Then the optimal-density square grid is 16×16.

But you pay a heavy density price for the larger roads: only 50% of the area is usable for zoning.

### Appendix B: Rectangles

Rectangles, as they get longer and longer, can arbitrarily approach the limit† of 80% density.

The density formula for an α×β block is ((α-2)(β-2) - (α-10)(β-10))/(αβ), assuming both sides are 10-or-longer for simplicity (it's clear that if both are smaller than 10 it's worse than the square, and I don't care what happens for silly things like a 5×50 block).

So there's a surprising amount of choice if you just want to do better than the 10×10 block:

But your choices are far more limited if you want to beat a 12×12 block:

Indeed, if you restrict yourself to integers (so you can draw it with Snap to Road Length), then the only things that beat the best square block are α ∋ {10, 11}, β ∋ {13, 14, 15, ...}.

How much better? Well, 10×16 has a density of 70%, an extra 5% more than the 12-square. Getting up to 75% density takes 10×32, which is more oblong than I personally like for a grid.

Is the extra density worth being less regular? I'll leave that up to you.

The limit is parallel lines that are infinitely long and never connected, as that way there's never any potentially-zonable area taken up by a cross street. A quick look at the cross-section of such a road shows 4u zoneable on each side and a 2u road, for (4+4)/(4+2+4)=0.8 density.

### Appendix C: Real Cities

These are measured from satellite maps, so don't always match the "canonical" answers. For example, the Portland Bureau of Transportation says its blocks are 200ft (≈7.6u).