I wonder if the guy that invented top bar hives had trouble with geometry? 
I knew the guy who invented the top bar hive..... actually he never invented them but worked with a world development organization on a project in Kenya.
After studying the African bee the challenges faced both from working with the bees, the pests and predators, and the resources the native people had at their disposal. And looking at all the different types of hives available and that have been used thru out the world, the top bar hive was selected and promoted. It was chosen for it's simple design, ease of construction, and the availability of material to build it with. This is where
the guy that invented top bar hives had trouble with geometry? Not really they were constructing hives with rough sawed lumber that was random in width so the geometry changed to fit the lumber used on any particular hive.
John Corner Kenya Top Bar Hive
I had the privilege to have known John Corner, to hear him speak, watch his slide show presentations, to work and talk with him, He was the head of the Provincial Apiary Branch Department of B.C.
He also did some extension work in 3rd world country's. In the early 1980's he took on a project of introducing managed beekeeping in Kenya as up till then most honey was retrieved by honey hunting and robbing the colonies. the task wasn't as simple as taking plans for a Langstroth hive and saying build this hive.
European and North American hive sit on the ground This set up of a hives would be to easy for predator to attack. The equipment needed to make all the various pieces of the Langstroth hive was not readily available to beekeepers in the small villages, or the power to run the equipment. The beekeepers didn't have the equipment to extract the frames of honey or the finances to perches extracting equipment. The top bar hive was selected as the hive that would best allow for managed beekeeping in the area.
The top bar hive could he hung from trees keeping the predators from getting to the hives. It allowed for hive manipulation for easy access for removing honey and moving brood to make more colonies. The design of the top bar hive allowed for the construction of the hive with only needing to cross cut the boards to length, a task that could be done easily with a hand saw. The honey harvested with the wax, the wax being more valuable than the honey to the local villagers. They also took designs for veils, protective clothing, and the construction of smokers and hive tools. Once they had the tools to keep bees and become beekeepers rather than honey robbers. The new beekeepers had to be taught how to manage the colonies for honey production, for swarm prevention and to make increases in the number of colonies by splitting, and having colonies raise new queens.
John had a picture of a news clipping of 2 nuns being caught and arrested while flying in from Europe, for trying to smuggle 2 packages of bees into the country under their dress. The Kenya agriculture branch were concerned that the introduction of European bees would compromise the native African bees ability to survive in the harsh environment and with their predators.
While head of the Provincial Apiary Branch he put together a book on building Langstroth hives. It has been updated a little and published as PDF on the web but it is close to the same as was originally published.
Here is a link to it.
http://www.agf.gov.bc.ca/apiculture/construction.pdf