In the U.S. of Voronoi, which state do you live in?

[jimrtex]

https://www.jasondavies.com/maps/voronoi/us-capitals/



It looks like I'm just within the Arizona border, since Phoenix is slightly closer to me than Sacramento. My grandparents are still living in California though.

World Capitals

Would most of those states even exist?   Note you can rotate the globe by dragging, and zoom with a scroll wheel.

Alaska would be administered from Tokyo, Reykjavik, and Ottawa.  Brazil would lose much territory since Brasilia is hardly inland at all.  Scotland, Wales, and Cornwall would come under Dublin dominion.

Antarctica would be ruled from Canberra, Wellington, Cape Town, Montevideo, and Santiago, with a tiny sliver from Buenos Aires.  Athens regains its Aegean coast of Asia Minor.

[jimrtex]

MSAs:

One could use this process to draw legislative or congressional districts, through an iterative process.

Michigan has around 6600 election precincts, and each could be considered a potential candidate for one of Michigan's 14 congressional districts.  Each iteration roughly 20% of the candidates would be eliminated.  We can determined the number of iterations by solving for x in the following equation.

6600 0.8^x = 14

x = (ln 14/6600) / ln .8

x = 27.59, which is rounded to the nearest integer, n, which is 28.

We can then solve for the actual decay rate, d.

6600 dⁿ = 14

ln d = (ln 14/6600) / n

d = e ^{(ln 14/6600) / n}

d = .80264

We can easily calculate the target number, t, for iteration i

t_i= nint (6600 dⁱ)

t₁ = 5297
t₂ = 4252
...
t₁₀ = 732
...
t₈₁ = 81
...
t₂₈ = 14

We can reduce each of our 6600 proto-districts to a single point, their center of population, and then calculate a Voronoi diagram, and determine the center of population and total population of each.

We keep the t₁ (5297) most populous proto-districts, and eliminate the 6600 - 5297 (1303) least populous proto-districts, and repeat the process.

We can visualize the elimination of these smallest districts as somewhat analogous to the popping of soap bubbles, and the surviving neighboring bubbles expand to fill the space left by the eliminated soap bubble (proto-districts).

Let's say that the Salt Lake district in the above map is the least populous and were eliminated.  The Denver, Las Vegas, Sacramento, Portland and Seattle districts would expand to fill the void.  Note that we would have used the center of population for the districts, so the centers of the districts would have been somewhat different.  The centers for Seattle, Portland, and Sacramento districts would have been pulled inward to begin with, and even further after Salt Lake were eliminated.

After 28 iterations we would have 14 districts of roughly equal population.

We can then adjust these districts by weighting the distance measure, so that less populous districts would expand towards more populous districts until the populations were equalized.

The survivng proto-districts in less densely populated areas will expand quickly, while in urban areas it will be slower.