Beniamin BOGOSEL

Theorem 17. (from our article) Among all infinite group 3D models we have the following classification in terms of angles:

• exactly one right angle
  • $200$ models in $G_6$
  • $837$ in $G_4$
  • $77667$ in $G_2$
  • $31005$ in $G_1$
• exactly two right angles
  • $57935$ models in $G_3$
  • $1552$ in $G_2$
  • $28893$ in $G_1$
• three right angles
  • $40$ models in $G_3$
  • $563$ in $G_1$
• three angles equal to $2\pi/3$ (Tetrahedral partition)
  • $2$ models in $G_4$
  • $3$ models in $G_1$
• triangles with three angles $\arccos(1/3) = 0.3918265520\pi$ (polar triangle of the equilateral triangle with angles $2\pi/3$)
  • $5$ triangles in $G_1$

• Model in $G_1$ corresponding to three right angles with "no drift" (sum of steps equal to the zero vector): critical point for the inventory $(1,1,1)$ $$M = \begin{pmatrix} -1 & -1 & -1& -1 & -1 & 0 & 0 & 1 & 1 & 1 & 1 & 1\\ -1 & -1 & 0 & 1 & 1 & -1 & 1 & -1& 0 & 0 & 0 & 1\\ -1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & -1 & 0 & 1 & -1 \end{pmatrix}$$
• Model in $G_1$ with triangle with three right angles and drift in each direction (sum of steps has all components non-zero): critical point for the inventory $(1.6180,1.6180,1.6180)$ $$ M = \begin{pmatrix} -1& -1& -1& -1& -1& 0& 0& 0& 1& 1\\ -1& -1& -1& 0& 1& -1& 0& 1& -1& 0\\ -1& 0& 1& -1& -1& -1& 1& 0& -1& 0 \end{pmatrix} $$
• Model in $G_1$ with triangle with three right angles and drift in each direction (sum of steps has all components non-zero): critical point for the inventory $(0.6180,0.6180,0.6180)$ $$ M = \begin{pmatrix} -1& -1& 0& 0& 0& 1& 1& 1& 1& 1\\ -1& 0& -1& 0& 1& -1& -1& -1& 0& 1\\ 1& 0& 1& -1& 0& -1& 0& 1& 1& 1 \end{pmatrix} $$

• There are $1394$ equilateral triangles corresponding to models in $G_1$. A full list can be found in the following document.

  The list of angles with tolerance $10^{-8}$ can be found in the following file: angles_equiG1.csv. In some case, explicit algebraic expressions can be found using symbolic computations. There are Matlab codes in the Numeric section below which can be run on the list of equilateral triangles given above in order to find the precise algebraic expression for these angles.

  Of course, among these there are triangles with other angles than $\pi/2$ or $2\pi/3$. In particular, the equilateral triangle with angles $\arccos(1/3) = 0.3918265520\pi$ is the polar to the Kreweras triangle with three angles $2\pi/3$. There are five such triangles in $G_1$. Two examples are given below: $$M = \begin{pmatrix} -1& -1& 0& 0& 1& 1\\ -1& 0& -1& 1& 0& 1\\ 1& 0& 1& -1& 1& 1 \end{pmatrix} $$ $$M = \begin{pmatrix} -1& -1& 0& 0& 1& 1\\ -1& 0& -1& 1& 1& 1\\ 1& 0& 1& -1& -1& 0 \end{pmatrix} $$