The names are many but the truth is one;
- T’ai Chi
- Taichi
- Tai chi
- Taiji
- Tàijíquán
- T’ai Chi C’uan
- T’ai Chi Ch’uan
- Taiji Quan (spelled T’ai Chi Ch’uan in the Wade–Giles system of romanization)
See the wiki page “Tai chi” for a deeper dive into the etymology.