click on the slider and use the arrow keys to see the polygons frame-by-frame. question: some of the polygons look more “complicated” than others. which ones? is there a pattern? why?
how many points does the 3.5-gon have? what about the 3-gon (a/k/a the triangle)? what about the 3.25-gon? what about the 3.01-gon? (you might not want to count that last one, but can you guess?)
well, understanding why some of these designs are more complex than others requires understanding what these shapes are. dan meyer, who inspired it, explains it far better than me. but the idea is this: in ninth grade or whatever, you learn lots of facts about regular polygons (shapes where all the sides are the same length and all the angles are the same, like squares, stop signs &c.). some of the facts you learn are what the interior angles are: equilateral triangles have 60° angles, squares have 90° angles, pentagons have 108°, etc... eventually, you learn a formula for the interior angle of any regular polygon: an n-sided regular polygon has interior angles of 180 * (n-2)/n. so you memorize this formula, and maybe you have an exam where you have to calculate the interior angle of a 20-sided polygon, which works out to 180*(20-2)/20 = 162°, and then you forget it.
but there's still this formula: just sitting there. waiting to digest any number that comes its way. and what if—what if the number that comes its way happens to be not a natural number, like 3 or 4 or 5 or 20 or 3048, but what if—what if it's got decimal places? what if you plug, say, 3.42 into this equation? well, the equation tells you that a 3.42-gon should have internal angles of about 74.73°. but what does that mean? how can you have a shape with 3.42 sides? no ontology, no matter how sophisticated, can possible accept the existence of such a contradiction! but what if you do this: one way of making, say, a square, is to draw a line, then make a 90° angle, then draw another line of the same length, then make another 90° angle, and continue until you've joined up with the original line. so what if you decide to make your 3.42-gon by drawing a line, then drawing a 74.73° angle, and continuing until you've reach the original shape? what will happen? will it take 3.42 lines until you reach the original point? will you go around in a circle multiple times? (hint: see above.)
because these beautiful polygons are all vector graphics, you can zoom in to see more detail, or print them out at as much resolution as your printer can handle! the mathematics hasn't been perverted by pixels!