# Generation and Animation of Intertwinings, Larsen Effects and more

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#### CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France

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Abstract: How to build and animate intertwinings? How to generate Larsen effects?

Keywords: Picture Synthesis, Entrelacs, Intertwinings.

# 1-A LIBRARY OF ELEMENTARY SYMBOLS:

One simple way to build bidimensional intertwinings is to define a library of elementary symbols by means of pictures (obviously, the most interesting ones are X and Y periodical); for example:

By the way, these elementary symbols 'PA' can be themselves intertwinings (recursive definition); for example:

# 2-DEFINITION OF TRANSFORMATIONS:

Then, these elementary symbols 'PA' can be transformed using arbitrary functions as follows:

• PR[x,y] = PA[FX[x,y],FY[x,y]]

where 'PR' denotes the resulting intertwining (a picture). Moreover, 'FX' and 'FY' denote respectively two arbitrary functions defined as follows (bidimensional case):
• ```FX[x,y] = A11.PX[x,y] + A12.PY[x,y]
```
• ```FY[x,y] = A21.PX[x,y] + A22.PY[x,y]
```
or again as (pseudo-tridimensional case):
• ```FX[x,y] = A11.PX[x,y] + A12.PY[x,y] + A13.PZ[x,y]
```
• ```FY[x,y] = A21.PX[x,y] + A22.PY[x,y] + A23.PZ[x,y]
```
where 'PX', 'PY' and 'PZ' denote respectively three pictures; 'Aij' are six arbitrary coefficients.

This process can be recursively iterated using:
• ```PX[PX[x,y],PY[x,y]]
```
• ```PY[PX[x,y],PY[x,y]]
```
for example...

This process can be extended using "picture albums" -a set set of pictures- (tridimensional case):

• PR[x,y,z] = PA[FX[x,y,z],FY[x,y,z],FZ[x,y,z]]

where 'PA' and 'PR' denotes respectively two picture albums: the 'elementary' symbol and the resulting intertwining. Moreover, 'FX', 'FY' and 'FZ' denote respectively three arbitrary functions defined as follows:
• ```FX[x,y,z] = A11.PX[x,y,z] + A12.PY[x,y,z] + A13.PZ[x,y,z]
```
• ```FY[x,y,z] = A21.PX[x,y,z] + A22.PY[x,y,z] + A23.PZ[x,y,z]
```
• ```FZ[x,y,z] = A31.PX[x,y,z] + A32.PY[x,y,z] + A33.PZ[x,y,z]
```
where 'PX', 'PY' and 'PZ' denote respectively three picture albums; 'Aij' are nine arbitrary coefficients.

# 3-"BIDIMENSIONAL" INTERTWININGS:

Here are some resulting intertwinings (please note that the colors of the pictures 'PA' and 'PR' might differ due to specific color post-processings):

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=) and then a transposition.
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=TapestryEffect(CenterOf()))
```
• ```(PR=) = (PX=,PY=)(PA=TapestryEffect(CenterOf()))
```

• ```(PR=) = (PX=,PY=)(PA=TapestryEffect(CenterOf()))
```
• ```(PR=) = (PX=,PY=)(PA=TapestryEffect(CenterOf()))
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = MountainEffect((PX=,PY=)(PA=))
```

• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

# 4-"PSEUDO-TRIDIMENSIONAL" INTERTWININGS:

• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```

• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```

• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```

• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```

• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) =TapestryEffect((PX=,PY=,PZ=)(PA=))
```
• ```(PR=) =MountainEffect(TapestryEffect((PX=,PY=,PZ=)(PA=)))
```

Here is a summary of some associations between surfaces and bidimensional intertwinings:

 A plane -zero curvature-.

 A sphere -positive curvature-.

 The Bonan-Jeener-Klein triple bottle.

 The Boy surface.

# 5-"BIDIMENSIONAL" DYNAMICAL INTERTWININGS:

At last, here are some resulting dynamical intertwinings (please note that the colors of the pictures 'PA' and 'PR' might differ due to specific color post-processings):

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```

# 6-"PSEUDO-TRIDIMENSIONAL" DYNAMICAL INTERTWININGS:

• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```

• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```

• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```

# 7-"TRIDIMENSIONAL" INTERTWININGS:

• ```(PR=) = (PX=,PY=,PZ=)(PA=) with a double mirror effect.
```

• ```(PR=) = (PX=,PY=,PZ=)(PA=).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```
• ```(PR=) = (PX=,PY=,PZ=)(PA==).
```

Here is a summary of some associations between surfaces and trodimensional intertwinings:

 A sphere -positive curvature-.

 The Bonan-Jeener-Klein triple bottle.

# 8-"BIDIMENSIONAL" LARSEN EFFECTS:

When iterating that kind of transformation and adding simultaneously some rotations, Larsen effects can be obtained:

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

# 9-MISCELLANEOUS EFFECTS:

• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```

• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```

• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```

• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```

• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```

• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```
• ```(PR=) = (PX=,PY=,PZ=)(PA=)
```

• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```
• ```(PR=) = (PX=,PY=)(PA=)
```