regular triangle discovery

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I made a discovery about regular triangles today. For a long time I've wondered, is it possible to create a regular triangle on a computer which has all of it's points exactly on an integer position. I finally succeeded in doing it with inkscape. I know almost nothing about the math behind angles. However, when creating lines with the path tool, inkscape displays the angle created by the current location relative to the first point of the line. For this I used a square grid. The ratio in pixels I found in the smallest triangle while one corner pointed straight up or down is. width/height 82/71 How I did this was first use inkscape's polygon tool to automatically create a regular triangle. Then I made a grid of squares with spacing of 1 pixel. Then I snapped one corner to a grid intersection. Then I tried to follow the triangle and see if it's sides ever crossed an intersection of the grid. And I was so amazed that it did, and since inkscape lets me automatically snap to the grid, I quickly made my own triangle path that looks like a really good regular triangle. I'm not sure if I can use this information for pixel art or not but it will be very useful for any vector editor with a square grid. I'm not sure if this discovery is correct according to geometry,algebra,trigonometry, or whatever other math I know nothing of, but it is at least a good way to approximate a regular triangle on a grid of squares.
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emekdemir said on 2011-08-19 18:14:54

Nice try and this is a very close approximation. However mathematically it is relatively easy to prove that you can't fit a regular triangle on a regular lattice (Pick's Theorem works).

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