I have become obsessed with drawing heptagons recently
While regular heptagons are impossible to construct using a straightedge and compass, they are easily approximated using only units, halves, and thirds.
Start with a 5x5 square. Five of the seven corners will end up touching it.
The bottom two corners of the heptagon are each 1⅓ units from the corner of the square
The next two corners are 1⅔ units up the sides
Along the top edge of the square, the next corner falls on the midpoint 2½ units along.
From there, mark off the final two corners one unit down and 2 units across towards the sides of the square. They are exactly ½ a unit from either edge, and 4 units from each other.
The resulting heptagon has 4 different side lengths ranging from 2.134 to 2.386, close enough for government work.
w = 2.33333...
x = 2.13437 (square root of 41/9)
y = 2.38630 (square root of 205/36)
z = 2.23607 (square root of 5)
A regular heptagon has interior angles of exactly 900/7 degrees (128 4/7°, 128.571428...°). I'll spare you the rest of the trigonometry, but this approximate heptagon has one angle of 126.87°, four angles of 128.66°, and two angles of 129.245°
Regular of approximate, the heptagon does not perfectly tile the plane. The densest packing has concave octagonal gaps in between groups of six heptagons. Looking at the grid, the gaps can be thought of as two irregular pentagons side by side.
Using only halves and thirds, I was able to fill an entire page with heptagons. There's no point to any of this, I just thought it was fun to do. It's calming to me. Simple but non-trivial. A good time filler (not a time waster, mind you)
