A graphics system that utilizes the equilateral triangle instead of the common square pixel, instead of using squares to represent the image, use a grid of equilateral triangles, I have constructed two grids of the same size, one using the triangles and one using the squares of the same size, the equilateral triangle has the smoother and sharper image, you have noticed that the smaller the pixel is the smoother the and finer the image, well you can make a smaller shape with an equilateral triangle. And seeing how the triangle is only shape with the least amount of sides, (and the shape pyramid starts with triangles and ends in circles). I hope you can see where I am going with this.
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It's more logical to address the pixels in a grid as squares.
I can't understand why you think the triangles would make a 'sharper' picture-though I don't know if it's true. You can only 'make a smaller shape' if you have more pixels. If more triangles fit in the same space that means they're smaller. Either type of pixel will be 'sharper' with smaller elements.
The logic could work. One square pixel could be divided into two oblong horizontal pixels, orginized as such:
r1 g2 b2g1 b1 r2
Display and image sensor manufacture would need to be changed significantly though, and things would need to be changed between old sensors and displays and the new one.
Maybe you mean "hexagonal" instead of triangular. Put a penny on the table and surround it with six others to see what I mean. If you have a "Lite-Brite" toy you can see a hexagonal layout. There are three dimensions (say a, b and c) for each location on the flat hexagonal layout, a coordinate for each of three directions out from a center point, though you only need to know two of the three to find a location on the flat surface.
If you try it in "3D" you start describing a tetrahedral coordinate system, so it is really 4D with an additional coordinate directly up.
I think designers of computers should take this idea seriously. I can imagine a stop-sign-shape display on a do-everything gadget worn around my neck. You would also want to put the pixels in a camera's CCD in the same hexagonal layout (I don't think this exist now --2008).