Geomancy and Mathematics
The four binary elements of each figure allow for 16 different combinations, each called a tableau. As each chart is generated from the four Mothers, there are a total number of 164, or 65536, possible charts. Due to the mathematics of the chart, only figures that have an even number of points total can become Judges; each of the eight Judges then has 8192 charts associated with it. Traditional practitioners of geomancy use this knowledge as a type of parity check on the chart to ensure that no mistakes have been made while computing the figures.
In each chart, if all sixteen figures are observed (the four Mothers, the four Daughters, the four Nieces, the Witnesses, Judge, and Reconciler), at least two of the figures must be the same. However, as the Reconciler is usually termed an optional figure, 16 combinations of Mother figures can yield a chart where the Mothers, Daughters, Nieces, Witnesses, and Judge are all unique. Notably, Populus cannot appear in these charts, since mathematically it either requires two figures to be the same in order to be formed, or produces a duplicate figure when added to another figure. In such charts, the Judge will always be one of Conjunctio, Amissio, Carcer, or Acquisitio. The sixteen combinations of Mothers, in order from the First to the Fourth Mother, are
- Puer, Caput Draconis, Tristitia, Albus
- Conjunctio, Puella, Fortuna Major, Tristitia
- Puella, Puer, Tristitia, Albus
- Puella, Cauda Draconis, Tristitia, Albus
- Rubeus, Laetitia, Puella, Puer
- Rubeus, Laetitia, Cauda Draconis, Puella
- Rubeus, Laetitia, Cauda Draconis, Caput Draconis
- Rubeus, Laetitia, Caput Draconis, Puer
- Acquisitio, Puella, Albus, Fortuna Major
- Laetitia, Fortuna Minor, Puer, Conjunctio
- Laetitia, Fortuna Minor, Acquisitio, Cauda Draconis
- Cauda Draconis, Caput Draconis, Tristitia, Albus
- Caput Draconis, Amissio, Fortuna Major, Tristitia
- Caput Draconis, Carcer, Albus, Fortuna Major
- Fortuna Minor, Rubeus, Puer, Amissio
- Fortuna Minor, Rubeus, Carcer, Cauda Draconis
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