Below is a shape that is the union of three smaller copies of itself (which can be colored separately by clicking the hint button). Each copy is half as big as the original, and can either be a rotation of the original by a multiple of $90^\circ$ (Up, Right, Down, Left for $0^\circ, 90^\circ, 180^\circ, 270^\circ$ degrees clockwise, respectively), or obtained by first reflecting the original about a vertical line, then doing one of the four rotations (called -Up, -Right, -Down, -Left). Can you determine which three transformations correspond to the three similar shapes in the figure?
Note that if $S$ is the figure, and $S_0$, $S_1$, and $S_2$ are the three copies (which are always congruent to each other and similar to the original), then each copy determines a unique affine map $W_0$, $W_1$, and $W_2$, respectively, such that $$S=W_0(S)\cup W_1(S)\cup W_2(S)$$ Furthermore, $S$ is the unique closed bounded figure that satisfies that equation. So we are essentially given the solution to an equation in geometry, and guessing what equation it solves. See the Lecture Notes for Math 320 for details.