The geometrical transformations of dissections rely on the splitting of a shape in smaller, similar shapes,
followed by a rearangement of these smaler shapes by rotations and symetries.
"Splitting" a shape in similar shapes is described very easily using the IFS formalism.
Take the simple example of a rectangle, split in 4 new rectangles in such a way that each edge of the parent rectangle is replaced by two parallel edges of child rectangles.
The corresponding IFS has 4 transformations.
These transformations can be fully parameterized by 2 positive values and 4 configurations chosen among a set of 8 possibilities.
The 2 positive values a and b control the ratio of surfaces between parent and child rectangles.
There are 8 different symetry/rotation configurations for the children :
The IFS transformation matrices for this kind of dissections are made from columns which are the barycentric positions
of child vertices in the space associated to the parent vertices (corners of the parent rectangle). These columns are :
[1 0 0 0], [a 1-a 0 0], [b 0 1-b 0], [ab (1-a)b a(1-b) (1-a)(1-b)]
See this page for further information on these matrices.
Basic examples
Below are a few basic examples of dissections. For each image, the following parameters are given :
a, b, the 4 configurations chosen for the 4 transformations, and the number of itterations.
[0.55 0.60] [00 00 00 00] 6
[0.55 0.60] [01 00 00 00] 6
[0.55 0.60] [10 00 00 00] 6
[0.55 0.60] [10 03 12 10] 6
Other examples
Many variations can be made from the previous examples. The z axis can be used in the IFS, as can the color of rectangles.
The children volumes need not fill the entire space of the parent volume. The following examples demonstrate this.
Applications ?
Here is an example of an application in a concrete application : the federation square building in Melbourn.
The use of dissections here is purely for esthetical purposes.
