Sure enough, they were the same character. This lead to further research and comparisons, and soon I had a series of principles and sub-principles that drew a solid link between the two styles. The English translations people used can vary, but the character is found to be the same for each style. Below is a work in progress but it is far enough along that I can share it.
The history of Tai Chi, correctly called Tai Ji Quan, disseminated to the masses, is often a mythical story that involves an art form thousands of years old with Taoist immortals, monks, and fairies. Commonly it is propagated that a non-existent type of magical energy, will heal the practitioners body and/or throw opponents without ever touching them. This is a fictional portrayal that in the West we call a fairy tale and in the East they call wu xia.