TITLE: A special tessellation
QUESTION [12 upvotes]: Let $P$ be a convex $n$-gon. Suppose that we have an infinite number of $P$s, and that each of them is colored either red or blue. Here, let us consider the following operations :
Operation 1 : Place a red $P$ on a plane.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
Operation 2 : Place $n$ blue $P$s around the red $P$ such that each of the blue $P$s and the red $P$ are laid symmetrically with each $E_i\ (i=1,2,\cdots,n)$ where $E_i$ is an edge of the red $P$.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
Operation 3 : Place $n$ red $P$s around every blue $P$ in the same way as operation 2.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
Operation 4 : Place $n$ blue $P$s around every red $P$ in the same way as operation 2.
Operation 5 : Repeat operation 3 and 4.
Here, let us consider the following conditions :
Condition 1 : These $P$s are plane tessellation figures.
Condition 2 : There exists no place where both red $P$ and blue $P$ are placed.
Note that a regular hexagon, for example, does not satisfy the condition 2. See the figure below. The blue $P$ on which a letter $P$ is written, for example, will be colored red by the next operation.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $
Then, here is the first question.

Question 1 : Is the following true?
$P$ satisfies these two conditions $\iff$ $P$ is either "a $45–45–90$ triangle", "a $30–60–90$ triangle", "an equilateral triangle" or "a rectangle".

I reached this conjecture by considering the inner angles of $P$. The followings are what I've thought : Every inner angle, say $\alpha$, of $P$ has to satisfy $2m\alpha=360^{\circ}$ where $m\ge 2\in\mathbb N$. Hence, $\alpha$ has to be any of the positive divisors of $180$ except $180$. This leads $n\le 4$ and so on.
Then, here is the second question.

Question 2 : Letting $P$ be a convex polyhedron, how about the three dimensional version of this question? Suppose that we consider the plane symmetry instead of the line symmetry.

In the two dimensional version, I think we can consider the inner angles of $P$. However, I don't have any good idea for the higher dimensional version. Can anyone help?

REPLY [2 votes]: This is in response to questions posed in the comments, but is too long for a comment.
The use of the reflection trick in mathematical billiards, and detailed results for regular billiards are contained in

S. Tabachnikov Geometry and billiards (AMS, 2005).

including the classification of regular 2D billiards already given in another answer here.  The connection between higher dimensional billiards and Coxeter groups was made in

A. Y. Plakhov and A. M. Stepin Scattering and Coxeter groups Russ. Math. Surv. 53 401-402 (1998).

Both conditions 1 and 2 are needed for regular polytope billiards.  The red and blue polytopes then correspond to regions where the mapping back to the original space is a translation or a reflection.  The condition that angles in a polygon are of the form $\pi/n$ is exactly what is needed for corners to be regular, in other words, parallel billiard paths that are close but pass on opposite sides of a corner remain close after reflections near the corner.  The example of a hexagon, which tesselates the plane but does not satisfy condition 2, has corners not of this form and so parallel billiard paths on opposite sides of the corner emerge at different angles.  I am not aware of results which distinguish this case from the very popular more general case of billiards with rational angles.  These are described by translation surfaces of higher genus, for example see

V. Delecroix, P. Hubert and S. Lelievre Diffusion for the periodic wind-tree model arxiv:1107.1810