SplitPattern

Définition of a pattern

For a given number of refined particles, the splitting procedure will involve a given set of patterns. The patterns will be identified by a COLOR (which choice is absolutely random). There is thus a bijective relation between color-pattern. Some of these patterns are defined for all dimensions, and some others are not.

For a given pattern, all refined particles are sharing the same weight. They all have a different position (relative to the one of their parent) but this position rely on a given value(s) of delta(s). This position can be provided by a singleton/couple/triplet, depending on the dimension. Here is the nomenclature :

• RED

  • dimension = 1 : (0) associated to N = 1 particle
  • dimension = 2 : (0, 0) associated to N = 1 particle
  • dimension = 3 : (0, 0, 0) associated to N = 1 particle

• GREEN

  • dimension = 1 : (1) associated to N = 2 particles
  • dimension = 2 : (1, 0) associated to N = 4 particles
  • dimension = 3 : (1, 0, 0) associated to N = 6 particles

• BLUE

  • dimension = 2 : (1, 1) associated to N = 4 particles
  • dimension = 3 : (1, 1, 0) associated to N = 12 particles

• ORANGE

  • dimension = 3 : (1, 1, 1) associated to N = 8 particles

• PURPLE

  • dimension = 2 : (2, 1) associated to N = 8 particles
  • dimension = 3 : (2, 1, 0) associated to N = 24 particles

• CYAN

  • dimension = 3 : (2, 2, 1) associated to N = 24 particles

2 examples

• BLUE pattern

  • dimension = 2. The pattern is (1, 1), so there are 4 refined particles, defined for a single delta value that we call a. These refined particles are located at (-a, -a) (-a, +a) (+a, -a) (+a, +a)

  • dimension = 3. The pattern is (1, 1, 0) so there are 12 refined particles, also defined for a single delta value that we also call a, located at (-a, -a, 0) (-a, 0, -a) (0, -a, -a) (-a, 0, +a) (-a, +a, 0) (0, -a, +a) (+a, 0, -a) (+a, -a, 0) (0, +a, -a) (+a, +a, 0) (+a, 0, +a) (0, +a, +a)

• PURPLE pattern

  • dimension 2. The pattern is (2, 1) so there are 8 refined particles, defined for two delta values that we call a and b. These refined particles are located at (-a, -b) (-a, +b) (+a, -b) (+a, +b) (-b, -a) (-b, +a) (+b, -a) (+b, +a)

List of pattern at 1D

• N = 2 : GREEN
• N = 3 : RED + GREEN (exact for p = 1)
• N = 4 : GREEN + GREEN (exact for p = 2)
• N = 5 : RED + GREEN + GREEN (exact for p = 3)

The cases N = 4 only holds for interpolation order p = 2 and p = 3. The cases N = 5 only holds for interpolation order p = 3.

List of pattern at 2D

In some cases, the pattern color(s) depend(s) on the interpolation order p. In such cases, the interpolation order is indicated.

• N = 4 : BLUE
• N = 5 : RED + BLUE for for p = 1, RED + GREEN for p=2 and p=3
• N = 8 : GREEN + BLUE
• N = 9 : RED + GREEN + BLUE (exact for p = 1)
• N = 16 : BLUE + PURPLE + BLUE (exact for p = 2)
• N = 25 : RED + GREEN + BLUE + GREEN + PURPLE + BLUE (exact for p = 3)

The cases N = 16 only holds for interpolation order p = 2. The cases N = 25 only holds for interpolation order p = 3.

List of pattern at 3D

• N = 6 : GREEN
• N = 7 : RED + GREEN
• N = 8 : BLUE
• N = 8 : RED + BLUE
• N = 12 : ORANGE
• N = 13 : RED + ORANGE
• N = 14 : GREEN + BLUE
• N = 15 : RED + GREEN + BLUE
• N = 18 : GREEN + ORANGE
• N = 19 : RED + GREEN + ORANGE
• N = 20 : BLUE + ORANGE
• N = 21 : RED + BLUE + ORANGE
• N = 26 : GREEN + BLUE + ORANGE
• N = 27 : RED + GREEN + BLUE + ORANGE (exact for p = 1)
• N = 64 : ORANGE + CYAN + CYAN + ORANGE + (exact for p = 2)
• N = 125 : RED + GREEN + BLUE + GREEN + PURPLE + BLUE + ORANGE + CYAN + CYAN + ORANGE (exact for p = 3)

The cases N = 64 only holds for interpolation order p = 2. The cases N = 125 only holds for interpolation order p = 3.

JUNKYARD

correspondance between the 'colors' used in the pdf for dim=3, order=3, and the above color green -> RED orange -> GREEN cyan -> BLUE purple -> GREEN grey -> PURPLE red -> BLUE brown -> CYAN lime -> CYAN magenta -> ORANGE black -> ORANGE