Saturday, September 10, 2016

[egvimqga] Triangle in square

Consider an isosceles triangle with base 1 and height 1 inscribed in a unit square.  There are 4 different ways of doing it with the base coincident to an edge of the square.

Stack two such squares on top of each other: 16 possibilities, an encoding of hexadecimal.  4 such squares arranged in a big square has 256 possibilities, encoding a byte.

Consider stretching the tip of the isosceles triangle beyond the edge of the square so that it touches the triangle in the adjacent square.  This helps the set look like one connected character.  One tricky case is 4 triangles each "pointing" to the next one around the big square.

Update: For the tricky case, the height of the isosceles triangle is 0.5 + sqrt 0.5, and the apex angle is 45 degrees, making it surprisingly a wedge of an octagon.

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