Tai Chi, Taiji, Tai Chi Chuan, Taijiquan

Wikipedia is a good starting place to sort out what’s what:

Tai chi (simplified Chinese: 太极; traditional Chinese: 太極; pinyin: Tàijí), short for T’ai chi ch’üan or Tàijí quán (太極拳), sometimes colloquially known as “Shadowboxing,” is…