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 have absolutely no meaning). 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. In 4 cases (on a total of 6) we provide a plot in the 2-dimensional case. We keep a large Black bullet at the origin, that is the position of the parent. Here is the nomenclature :

• BLACK
  • 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

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

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

• LIME

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

• BROWN

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

• WHITE
  • dimension = 3 : (a, a, b) associated to N = 24 particles

2 examples

• PURPLE pattern

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

  • dimension = 3. The pattern is (a, a, 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)

• BROWN pattern

  • dimension 2. The pattern is (a, b) 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 : PINK
• N = 3 : BLACK + PINK (exact for p = 1)
• N = 4 : PINK + PINK (exact for p = 2)
• N = 5 : BLACK + PINK + PINK (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 : PURPLE
• N = 5 : BLACK + PURPLE for for p = 1, BLACK + PINK for p=2 and p=3
• N = 8 : PINK + PURPLE
• N = 9 : BLACK + PINK + PURPLE (exact for p = 1)
• N = 16 : PURPLE + BROWN + PURPLE (exact for p = 2)
• N = 25 : BLACK + PINK + PURPLE + PINK + BROWN + PURPLE (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 : PINK
• N = 7 : BLACK + PINK
• N = 8 : PURPLE
• N = 8 : BLACK + PURPLE
• N = 12 : LIME
• N = 13 : BLACK + LIME
• N = 14 : PINK + PURPLE
• N = 15 : BLACK + PINK + PURPLE
• N = 18 : PINK + LIME
• N = 19 : BLACK + PINK + LIME
• N = 20 : PURPLE + LIME
• N = 21 : BLACK + PURPLE + LIME
• N = 26 : PINK + PURPLE + LIME
• N = 27 : BLACK + PINK + PURPLE + LIME (exact for p = 1)
• N = 64 : LIME + WHITE + WHITE + LIME + (exact for p = 2)
• N = 125 : BLACK + PINK + PURPLE + PINK + WHITE + PURPLE + LIME + WHITE + WHITE + LIME (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.