Let $P$ be a convex $n$-gon. Here, let us consider the following operations :

Operation 1 : Place $P_0$ on a plane.

Operation 2 : For every $i\ (i=1,2,\cdots,n)$, place another $P$ such that $P$ and $P_0$ are laid symmetrically with $E_i$ where $E_i$ is an edge of $P_0$. Let these $P$s be $P_1$.

Operation 3 : For every $P_1$, place $n$ $P$s in the same way as operation 2. Let these $P$s be $P_0$.

Operation 4 : For every $P_0$, place $n$ $P$s in the same way as operation 2. Let these $P$s be $P_1$.

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 : Every point on a plane except on the edges of $P$ is on either $P_0$ or $P_1$.

(Note that a regular hexagon, for example, does not satisfy the condition 2.)

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 square".

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 $n\ge 4$ and so on.

Then, here is the second question.

Question : Letting $P$ be a convex polyhedron, how about the three dimensional version of this question?

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? [1]: http://en.wikipedia.org/wiki/Tessellation#Tessellations_in_higher_dimensions