Matrix
in development
discipline:
computational art
theme:
binary/geometry
A graphic exploration of self-similitude and self differentiation through the squaring expansion of a binary pair as a series of matrices. Each matrix is the product of the multiplication of its parent matrix with itself. The resulting square matrices populated with powers of 2 reveal the patterns of repetition and difference inherent in the arithmetic integer series.
The expansion can be calculated by:
multiplying literal values:
initially
(2^1), and
(2^0), to generate:
,
, etc.
adding powers of 2:
initially
and
, to generate:
,
, etc.
concatenating binary figures:
or simply populating the cells of the matrix with a ascending arithmetic series starting at 0,
initially
and
, to generate:
,
, etc.
The series can also be understood geometrically by visualising the superimposition of the differently scaled units which multiply to produce each subsequent generation of the matrix.
One orientation/scale of the matrix,
superimposed over the other, at 50% opacity
produces,
which, carried over to the next column at both scales,
At any one level of the matrix, each cell is multipiled by the entire matrix in order to produce the next generation of cells in that fraction of the matrix.
related projects: Octaves: 2nHz, 4nHz, 8nHz, Infinity > Infinity + 1, Squared
A graphic exploration of self-similitude and self differentiation through the squaring expansion of a binary pair as a series of matrices. Each matrix is the product of the multiplication of its parent matrix with itself. The resulting square matrices populated with powers of 2 reveal the patterns of repetition and difference inherent in the arithmetic integer series.
The expansion can be calculated by:
multiplying literal values:
initially
| 2 |
| 1 |
| 4 | 2 |
| 2 | 1 |
| 16 | 8 | 8 | 4 |
| 8 | 4 | 4 | 2 |
| 8 | 4 | 4 | 2 |
| 4 | 2 | 2 | 1 |
adding powers of 2:
initially
| 1 |
| 0 |
| 2 | 1 |
| 1 | 0 |
| 4 | 3 | 3 | 2 |
| 3 | 2 | 2 | 1 |
| 3 | 2 | 2 | 1 |
| 2 | 1 | 1 | 0 |
concatenating binary figures:
or simply populating the cells of the matrix with a ascending arithmetic series starting at 0,
initially
| 1 |
| 0 |
| 11 | 10 |
| 01 | 00 |
| 1111 | 1110 | 1101 | 1100 |
| 1011 | 1010 | 1001 | 1000 |
| 0111 | 0110 | 0101 | 0100 |
| 0011 | 0010 | 0001 | 0000 |
The series can also be understood geometrically by visualising the superimposition of the differently scaled units which multiply to produce each subsequent generation of the matrix.
One orientation/scale of the matrix,
superimposed over the other, at 50% opacity
produces,
which, carried over to the next column at both scales,
At any one level of the matrix, each cell is multipiled by the entire matrix in order to produce the next generation of cells in that fraction of the matrix.
related projects: Octaves: 2nHz, 4nHz, 8nHz, Infinity > Infinity + 1, Squared