Hex Grid: Fill & Download for Free

I am presuming you mean "why are computer pixels laid out in a rectangular grid?" (since they aren't necessarily square per se, and some, such as original IBM PC with CGI graphics, are not even aspect ratio 1:1) Note that other people interpreted your question as "why do computer displays have a pixel aspect ratio of 1:1?", which I don't really think is such an interesting question because it seems to me that, given a rectangular grid, it is fairly obvious why 1:1 would be the optimum aspect ratio.I can think of one reasonable alternative to a rectangular grid, which would be a hex grid, like a honeycomb. Each pixel shares an edge with 6 other pixels (rather than sharing an edge with 4 pixels and a corner with 4 more pixels).Low res image with hex shaped pixelsSame image with square pixelsIt could be done, it's certainly been thought of, and if you ignore certain practical issues, I would argue that it is "better." Bees would probably agree.In fact, when you get down to the subpixel level, some displays actually are approximating this, and it has advantages due to needing to have a red, green and blue component: squares don't divide up into thirds quite as well as hexagons:One negative is that rectangular images displayed on a hex grid would have ragged edges. In theory we could get rid of the idea of rectangular images, and have all image borders at 30, -30, or 90 degrees (rather than 0 and 90), but that is probably a tough sell. Anyway, with super high definition displays like on the lastest iPhones and iPads, the ragged edges might not matter so much.The other big negative is that the math is harder. Not that much harder, but it certainly complicates things for programmers to have to think in hex-grid terms. So higher level software will probably stay thinking of graphics as having a 1:1 rectangular grid (i.e. "square pixels") while the display hardware then sorts things out when mapping things to a hex-grid screen.

Count the # of Hamiltonian tours in a hex grid with obstacles (joking (well, sort of (cows be warned) ) )My favorite by far is this problem from CTSC 09 authored by ahyangyiYou have 3 pegs labelled 1~3, and n chips marked 1~3, all initially sitting on peg 1.In each move, you can take the top chip off some peg, and put it on top of another peg. (basically the moves that you can make in towers of Hanoi, except there are no restrictions on chips being on top of each other)Find the minimum number of moves to move all the 1s to peg 1, 2s to peg 2, 3s to peg 3.Intended solution is a [math]O(n^2)[/math] time DP, 1 person got it during the contest. Even more ridiculous fact: the more that you think about this problem, the less code you'll write. At the end of the day there is a 40 line, O(n) time DP solution for this.

YesVariants exist for triangular, pentagonal and hexagonal grids.Here is a website with links to exampleshttp://kaytdek.trevorshp.com/projects/computer/neuralNetworks/gameOfLife2.htmA paper explaining triangular, pentagonal, hexagonal and irregular pentangular and hexangular variations. More resources in the citations.http://www.complex-systems.com/pdf/15-3-4.pdfCarter Bays is on the USC (So. Carolina) faculty, but the website cited in the paper and the site linked by the University are unavailable as I type this.A Google search for this yielded many examples and variations.conway game of life hex gridThe standard form is 2D. 1D and 3D variants have also been found.