The size (number of rows and columns) of each matrix is equal to the population of the preceding matrix, i.e. also square of the size of the preceding matrix. It is the series 2^(2^n) = 2; 4; 16; 256; 65,536 ... The population of the matrices is the series 2^(2^(n+1)) = 4; 16; 256; 65,536; 4,294,967,296 ...

Through the visualization of this process of progressive elaboration of the simplest originating opposition – 1 versus 0 – we are able to observe subtle patterns of repetition at different scales within the structure of the binary series.

The expansion can be calculated by:

**multiplying literal values:**

initially

2 |

(2^1), and

1 |

(2^0), to generate:

4 | 2 |

2 | 1 |

,

16 | 8 | 8 | 4 |

8 | 4 | 4 | 2 |

8 | 4 | 4 | 2 |

4 | 2 | 2 | 1 |

, etc.

**adding powers of 2:**

initially

1 |

and

0 |

, to generate:

2 | 1 |

1 | 0 |

,

4 | 3 | 3 | 2 |

3 | 2 | 2 | 1 |

3 | 2 | 2 | 1 |

2 | 1 | 1 | 0 |

, etc.

**concatenating binary figures:**

or simply populating the cells of the matrix with a ascending arithmetic series starting at 0,

initially

1 |

and

0 |

, to generate:

11 | 10 |

01 | 00 |

,

1111 | 1110 | 1101 | 1100 |

1011 | 1010 | 1001 | 1000 |

0111 | 0110 | 0101 | 0100 |

0011 | 0010 | 0001 | 0000 |

, 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.

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