Strictly speaking, they are different. "Taiji", literally meaning "the grand ultimate", is the cosmos, whereas Tai Chi Chuan is a martial art. The concept of Taiji is found in many Chinese disciplines, such as in Taoism, philosophy, metaphysics, feng shui (the Chinese science of Geomancy) and military strategy.
The underlying principle of Taiji is yin-yang, which symbolizes the two complimentary yet opposing aspects of reality, which contain a seed of its opposite. Nevertheless, many people, especially in the West but also some Chinese themselves, shorten the term Tai Chi Chuan to Taiji. Almost always those who say they "play" Taiji are those who practice a debased form of Tai Chi Chuan without its martial dimension.
One may play Taiji, but not Tai Chi Chuan which is a serious martial art demanding great effort and endurance. More.
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