So I decided to build a hive that was the same length as it was in width. Inquiring at Beesource, I discovered that this idea was not new and that there are square hives in other countries. At some point I discovered while building it in Google Sketchup, that three mediums is about 19 7/8" tall. Presto, three medium hive is a cube hive. Coincidentally, it's also approximately equal to three deeps depending on how you calculate the comb area.
Some benefits I'm expecting: more efficient use of wood, shorter hive, more spherical cluster, bigger hives, and the ability to place successive boxes at 90 degrees to one another.
Here's a picture of a deep I made to help make the transition from deeps to mediums. 12 standard frames fit comfortably.
Finally, as I mentioned above, three mediums forms a cube. It's not precisely a cube on the inside due to the thickness of the wood, but that won't stop me from calling it the Parker Cube Hive.
Hopefully I'll get the opportunity to utilize them fully this year.