Help with math and science of tri-wiring?

I would think that they all need to be connected to the same tap. This is because:
each "ohm" tap is to allow for the same amount of power to match the paticular speaker. For example the 8 ohm tap will supply more voltage but less current than the 4 ohm tap. Gain of an amp = output Volts / input Volts. If you connected the woofer to a different tap then the midrange, it would see a different signal gain and the sound level would be off by that amount.

Yes Cford, but on some speakers the impedance variation between the mid/tweeter and the woofer is quite different. The "nominal impedance" is an average of the impedance curve. My speakers can be tri-wired. Admittedly mine are a special case - tweeter's impedance is 600 Ohm (!!), Midrange horn 8 Ohm, and woofer about 4 Ohm. Which made me suddenly wonder what would happen if I connected them to different output taps ...

Amfibius,
The impedance of the individual sections of your speakers was set by the designer to have the correct volume when the same voltage levels are going to each section as when they are not bi or tri-wired. If you want to bi or tri-wire then you still need to feed each with the same voltage level, which will only happen if you connect them all to the same tap.

I can't help you with the math or science, but I have tri-wireable speakers (Alon IV) and mono block ASL Hurricanes. The Canes have 3 output pots (4, 8, 16 ohm) and 1 negative pot per amp. I have bi-wired the mid and tweeter speaker posts to the 8 ohm pots and single wired the woofers to the 4 ohm pots. All negatives run to the negative pot on each amp.

In my system, there was a clearly audible improvement in the realism of the music. There was more body and dimension to the instruments as compared to the previous setup of tri-wiring from a single pot (I tried both 4 ohm and 8 ohm).

I would suggest that you try it and see if it helps your system. Of course YMMV.