Matrix & Octaves in development

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

Matrix

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.

Octaves

A piece based on the concepts developed in Matrix, displayed as part of the*Instrumental* show at the Gallery Project in Ann Arbor, MI

*Detroit artist Sambuddha Saha's three-part photo print "Octaves: 2nHz, 4nHz, 8nHz" pulses with an energy that flows from the collision of math, op-art and the synaptic "a-ha!" moments that come when a complex concept like sound frequencies are rendered in visual metaphor. Deceptively simple, "Octave" is a sequence of three 2-foot-by-2-foot prints with solid dots of varying shades of gray on white. Sit with it for a while, and the idea reaches out of the static* – Chris Handyside, *Metro Times, April 11 2007*

Matrix

The expansion can be calculated by:

initially

2 |

1 |

4 | 2 |

2 | 1 |

16 | 8 | 8 | 4 |

8 | 4 | 4 | 2 |

8 | 4 | 4 | 2 |

4 | 2 | 2 | 1 |

initially

1 |

0 |

2 | 1 |

1 | 0 |

4 | 3 | 3 | 2 |

3 | 2 | 2 | 1 |

3 | 2 | 2 | 1 |

2 | 1 | 1 | 0 |

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.

Octaves

A piece based on the concepts developed in Matrix, displayed as part of the