A Curious Note About the Geomantic Joys

I was pondering the Joys of the Geomantic Figures and realized something interesting: Mercury and Venus cannot rejoice together. This is because the first mother's head must be latent if Mercury rejoices, the first daughter's head must be active if Venus rejoices, and these cannot occur simultaneously. (Weren't violetcabra and I just talking about this yesterday?)

I was curious if such a relationship held for any other pair of planets, and so I wrote a computer program to examine every geomantic chart and see which pairs of planets either always or never rejoice together. It turns out there is exactly one other possibility: the Sun and Jupiter also cannot rejoice together.

The analysis for this is a little more involved. Let's name the boolean variables that make up the mothers as follows: A B C D for the first mother, E F G H for the second mother, etc.; the daughters are the transpose of these: A E I M for the first daughter, B F J N for the second daughter, etc.; and finally, the nieces are the XOR of their parents: A⊕E B⊕F C⊕G D⊕H, etc.; as in the following shield chart:

D H L P	C G K O	B F J N	A E I M	M N O P	I J K L	E F G H	A B C D
C⊕D G⊕H K⊕L O⊕P		A⊕B E⊕F I⊕J M⊕N		I⊕M J⊕N K⊕O L⊕P		A⊕E B⊕F C⊕G D⊕H
⋯				⋯
⋯

We will focus on the head and neck lines of the ninth house (where the Sun rejoices) and the eleventh house (where Jupiter rejoices). First, please note that the Solar figures Fortuna Major and Fortuna Minor always have identical head and neck lines, while the Jovial figures Acquisitio and Laetitia always have different head and neck lines. Therefore, if there is a Solar figure in the ninth house and a Jovial figure in the eleventh house, then A⊕E=B⊕F [eq. 1] and A⊕B≠E⊕F [eq. 2]. However, if we add B⊕E to both sides of eq. 1 and cancel, we get A⊕B=E⊕F [eq. 3]. Eqs. 2 and 3 cannot both be satisfied, therefore the Sun and Jupiter cannot rejoice together. ∎