# A Cube, Decorated with Icosagons and Their Diagonals

This icosagon, with its diagonals showing, first appeared here. I then used Stella 4d to make the cube. You can try Stella for yourself at this website.

# Quadrilaterals with Perpendicular Diagonals

I just learned these things are officially called orthodiagonal quadrilaterals. I’ve been calling them Qw⊥Ds (pronounced “quids”) for years, have studied their properties, and have even tested students’ knowledge of Qw⊥D esoterica.

Obviously, on grounds of symmetry alone, it is easy to determine that Qw⊥Ds include all squares. With congruent triangles, it is also possible to prove that all rhombi, kites, and darts are Qw⊥Ds.

As for other parallelograms, such as the rectangle, they are Qw⊥Ds iff they are also rhombi. No non-rhomboidal parallelograms have perpendicular diagonals.

With no parallel sides, altering darts and kites to make their diagonals off, slightly, from being perpendicular would be easy. In the process, though, the figure would lose its “dartness” or “kiteness.”

With exactly one pair of parallel sides — what most Americans call “trapezoids” (that word has multiple, troublesome definitions) — things get more messy. A non-isosceles trapezoid (lower left) can either have perpendicular diagonals (red) or not (yellow). As can be seen at the lower right, the same is true of isosceles trapezoids.

## Tetrakaiicosagon with Diagonals

### Image

How many diagonals does a polygon with 24 sides have?

First, consider that there are 24 vertices for diagonals to come from, and they each have 21 places to go, since they can’t go to themselves, or to the adjacent vertices. (24)(21) therefore equals twice the number of diagonals, since I just counted each one twice (once per endpoint). There are therefore (24)(21)/2 = 252 diagonals.