Magical Medieval Mathematical Models

I basically crammed in as many math related ideas as possible.

King and Queen's Castle- they like to show their power so a model of the equation y=x^2 (power of 2) is built into the castle.

Princess's Tower- The princess is very beautiful so she fittingly has a tower that has many parts that follow the golden ratio. This ratio is approximately 1.618 and has been used in architecture since ancient times as it is said to make things look beautiful. This ratio is determined by a segment A having the same ratio to segment B as B does to A + B.

Prince's House- The prince is a pretty cool dude but he's stuck in the 90's and still uses the word radical to say awesome. His house matches this and is shaped like a radical (square root symbol). There are also some numbers and symbols on top to easily stamp into the sand.

Wall/Moat- The wall is designed after a sine wave(or cosine depending on your view) from 0 to pi. It can also be flipped into a moat with tessellated crocodiles to pr