# Looking for references about a tessellation of a regular polygon by rhombuses.

A regular polygon with an even number of vertices can be tessellated by rhombuses (or rhombii, or lozenges), all with the same sidelength. The three following figures display a common structure.

I had already seen this kind of tessellation, and I met it again in a recent question on this site (Tiling of regular polygon by rhombuses).

The main feature of this tessellation is as follows. Let the polygon be $n$-sided with $n$ even. The starlike pattern of rhombuses issued from the rightmost point, that we will call the source can be structured into patterns of similar rhombuses. The first pattern $R_1$ of rhombuses (there are precisely $m$ of them where $m:=\dfrac{n}{2}-1$) with the most acute angles, then moving away from the source, a second pattern $R_2$ with $m-1$ rhombuses, etc. with a grand total of $\dfrac{m(m+1)}{2}$ rhombuses.

It is not difficult to show that rhombuses in $R_p$ are characterized by angles $p\dfrac{\pi}{m+1};$

Consider now (figure 4) the regular polygons that are obtained by successive rotations with angle $\dfrac{\pi}{m+1}$ around the “source” point of the reference regular polygon. It is clear that the rhombus pattern described above results from this larger structure.

The figures displayed have been produced by different Matlab programs.

The program generating the 2nd figure is given at the bottom of this text.

• where can I find some references?

• are there known properties/applications?

Matlab script for the second figure:

 hold on;axis equal
m=9;n=2*m+2;
i=complex(0,1);pri=exp(2*i*pi/n);
v=pri.^(0:(n-1));
for k=0:m-1
col=rand(1,3);
z=1-(pri^k)*(1-v(1:m+2-k));
plot([z,NaN,conj(z)],'color',col,'linewidth',5);
end;