## Wednesday, December 31, 2014

### Hexagonal-grid city life

Based on a board game, I started thinking about what putting a city on a hexagonal grid would look like:

The streets (in blue) run north-south around the hexagonal blocks. I numbered them, although obviously, the indexing is off and 1st Street should be at the far west end of the city.

The avenues (in red) run northeast-southwest. I gave them Latin letters. The boulevards (in green) run northwest-southeast. I gave them Greek letters.

Two things to note:

1. intersections are three-way (three roads come together)
2. two different roads "intersect" for a whole block

The east-west (horizontal) blocks are on both an avenue and a boulevard. The northeast-southwest blocks (diagonal bottom left to upper right) are on both a street and an avenue. The northwest-southeast blocks (diagonal upper left to bottom right) are on both a street and a boulevard. Can you locate A Avenue and Gamma Boulevard?

Addresses could be given this way: "22 A Ave. and Gamma Blvd." I think address numbers would reset for each block; there would be a "22 A Ave. and Beta Blvd."

### How hard would it be to navigate?

Let's say you lived at 1st Street and A Ave. and wanted to travel to 4th Street and Alpha Blvd. You know that you need to go east (to higher numbered streets) and not sure whether you need to travel north or south overall. But you don't need to worry. Just head northeast on A Ave. Either you'll intersect 4th Street first or Alpha Blvd. first.

How would the road signs look? Let's work our example in detail. Again, you want to travel east (from 1st to 4th), so you would take A Ave. northeast. (Taking 1st Street north would get you there but not necessarily in the most direct way.)

Let's look at the points (intersections) on the grid you would travel through:

At point U, the signs would say:

LEFT - N on 1st Street, NW on Gamma Blvd.
RIGHT - NE on A Ave., SE on Gamma Blvd.

At point V, the signs would say:

LEFT - N on 2nd Street, NE on A Ave.
RIGHT - S on 2nd Street, SE Gamma Blvd.

At point W, the signs would say:

LEFT - N on 2nd Street, NW on Beta Blvd.
RIGHT - NE on A Ave., SE on Beta Blvd.

At point X, the signs would say:

LEFT - N on 3rd Street, NE on A Ave.
RIGHT - S on 3rd Street, SE on Beta Blvd.

At all four points, you would choose to continue driving northeast on A Ave. The numbered streets are increasing, which is correct, and the Greek letters are decreasing, which is also correct. You know you're going the right way.

At point Y, the signs would say:

LEFT - N on 3rd Street, NW on Alpha Blvd.
RIGHT - NE on A Ave., SE on Alpha Blvd.

You've reached Alpha Blvd, so you're going to turn on it. Which way? Northwest or southeast? You're at 3rd Street. Not far enough east, so turn southeast on Alpha Blvd.

At point Z, the signs would say:

LEFT - NE on A Ave., N on 4th Street
RIGHT - SE on Alpha Blvd., S on 4th Street

You've almost arrived. A right turn puts you onto 4th Street and Alpha Blvd., the block you are looking for.

It would take a little getting used to, but I don't believe this kind of navigating in insurmountably complex.

### Would it take longer to travel on a hexagonal grid?

If one is traveling in a straight line on the rectangular grid, the hexagonal grid is 15% longer.

But if one is traveling in a "straight" line on the hexagonal grid (at an oblique angle), the rectangular grid is about 18% longer.

I tested out a bunch of random points; it more or less seems to even out. Sometimes the rectangular grid is shorter. Sometimes the hexagonal grid is shorter. Here's the results of 10,000 random trials:

Averaging them all together came up with that the hexagonal grid is about 0.3% longer, but that really might just be variability of taking a random sample. I'd say it just about evens out. So, are travel times the same?

One thing to consider is that the intersections on the hexagonal grid are faster, three-way intersections as opposed to the slower, four-way intersections in a rectangular grid. In a three-way intersection, there must be three cycles. Each right turn is allowed in 2/3 cycles; each left turn is allowed in 1/3 cycles. In a four-way intersection, there must be four cycles. Each right turn is allowed in 2/4 cycles (or 3/4 cycles if right-on-red is allowed); each left turn and each straight is allowed in 1/4 cycles. Comparing the two types of intersections, left turns are 33% faster in the three-way intersection; assuming no rights-on-red, right turns are also 33% faster in the three-way intersection. That means that the three-way intersections are 33% faster overall.

Because the travel distance are just about equal, the hexagonal grid is actually going to be faster due to the faster intersections.

Maybe bees are onto something.

As one more thought, it'd be quite easy to make park blocks at a ratio of 1/7 blocks.

No location would be more than 2 blocks from a park.

1. If you make all the streets one-way then you never need to stop or cross traffic. It's possible that every intersection either merges with another street or divides into two.

2. You're blowing my mind.

3. all the intersections should be round-a-bouts :)

4. This is a very good thought, there are various capacity and safety advantages for 3 way intersections. A further thought is that a Cairo tessellation would allow a straighter route for the trunk roads and might provide a bit more flexibility.

5. That's an interesting suggestions Tom, but Cairo tessellation still has four-way intersections. I wonder if there are any other irregular tessellations with advantageous properties?

6. How did you compare the hexagonal vs square plots? Do they have the same area? When you compare the time it takes to travel from one point to another do you also count the number of intersections crossed? My intuition is that you would have to cross more in a hexagonal grid than in a square one but I have not tested this.

7. I don't believe it makes a difference what size the plots are, though I might be wrong. The "straight" line on the hexagonal grid is 15% longer (blue diagram), whether the hexagons are large or small. Likewise, the corners in the rectangular grid add about 18% (red diagram), whatever the block size. It's really about whether the locations line up on grid lines or not, which is basically a wash.

However, you make a really good point -- the size of the blocks will make a big difference about how many intersections one has to go through. I would imagine the hexagonal blocks have to be bigger to keep intersections at the same frequency.

8. Square grids where all roads are one-way streets that alternate will only have 2 traffic cycles.

9. I have a hex pattern with 1 traffic cycle (no lights, no traffic crossing).

Best case: 33.33% longer. Worst case: 53.96% longer.
Worst case is at least 33% less frequent on hex grids [vs. square].
Only lane-changes are necessary. Each intersection consists, clockwise, of these segments: dual lanes in, dual lanes out, two-way road.
Rather than using a traffic light to obstruct 50% of routes at a time, the slightly longer travel distances can handle dense traffic faster (without requiring any stoppage).

10. Nice one Tiki.

Why would you make parks to be a whole block? Much better to make the blocks much bigger and put a park in the middle of each block.