5d Basis Vectors of Penrose's Tilings

Question

I have been writing some software to display/render Penrose tilings. I was hoping to use the approach of projecting a 5-dimensional lattice into 2d and apply some coloring based on regions etc. I was successful at doing this, but I am getting shapes like the following:

which have some of the features of a correct outcome, such as $D_5$ symmetry and bounded output values. I am fairly certain that I need a different 5d->2d projection (right now I am mapping the 5d unit cell to a pentagon).

I was hoping to find a reference for what the basis is to get Penrose's tilings, or a principled way to calculate it myself.

Answer 1

This reference clarified everything for me. The grid $j$'th gridline is given by $$\vec{x} \cdot \vec{\epsilon}_j + \gamma_j = N,$$

for some integer $N$.

Where the grid directions are given by: $$\epsilon_j = \left(\cos\left(\frac{2\pi j}{5}\right), \ \sin\left(\frac{2\pi j}{5}\right) \right) ,$$ for j = (0, 1, 2, 3, 4) and the shift of the each of the gradation from the origin must obey:

$$ \sum_{j=0}^4 \gamma_j = 0$$

Thanks all.