There exists a long history of movement and exercise systems which are associated with Taoism. In some sense one can see elements of all of these as contributing to the climate from which Tai Chi emerged.
There are some interesting inspirations for the movement philosophy of Tai Chi within the writings of Chuang Tzu, for example:
"The pure man of old slept without dreams and woke without anxiety. He ate without indulging in sweet tastes and breathed deep breaths. The pure man draws breaths from the depths of his heels, the multitude only from their throats."
The sage would not lean forward or backward to accomodate things. This is called tranquility on disturbance, (which means) that it is especially in the midst of disturbance that tranquility becomes perfect."
This approach is reflected in the entire movement philosophy of Tai Chi Chuan. There is, moreover, a long tradition of Taoist monks practicing exercises. Some of these were referred to as tai-yin or Taoist Breathing. Exactly what these were and what their origins were is obscure but they are mentioned in Chinese chronicles as early as 122 B.C.
Then in the sixth century A.D. Bodihdharma (called Ta Mo in Chinese) came to the Shao-Lin Monastery and, seeing that the monks were in poor physical condition from too much meditation and too little exercise, introduced his Eighteen Form Lohan Exercise. This approach gave rise to the Wei Chia or 'outer-extrinsic' forms of exercise.
Later in the fifteenth century A.D. the purported founder of Tai Chi Chuan, the monk Chang San-feng, was honored by the Emperor Ying- tsung with the title of chen-jen, or 'spiritual man who has attained the Tao and is no longer ruled by what he sees, hears or feels.' This indicates that already at this time there was a close association between the philosophy of Taoism and the practice of Tai Chi.
In the Ming dynasty (14th to 17th centuries), Wang Yang-ming a leading philosopher preached a philosophy which was a mixture of Taoism and Ch'an Buddhism which had certain associations with movement systems.
In any event the principles of yielding, softness, centeredness, slowness, balance, suppleness and rootedness are all elements of Taoist philosophy that Tai Chi has drawn upon in its understanding of movement, both in relation to health and also in its martial applications. One can see these influences (of softness and effortlessness) in the names of certain movements in the Tai Chi Form, such as:
Cloud Hands
Wind Rolls the Lotus Leaves
Brush Dust Against the Wind
Push the Boat with the Current
Winds Sweeps the Plum Blossoms
Moreover the contemplation and appreciation nature, which are central features of Taoist thought seem to have been reflected in the genesis of many Tai Chi movements such as:
White Crane Spreads Wings
Snake Creeps Down
Repulse Monkey
Embrace Tiger, Return to Mountain
White Snake Sticks Out its Tongue
Grasp Sparrow's Tail
Golden Cock Sands on One Leg
Swallow Skims the Water
Bird Flies into Forest
Lion Shakes it's Head
Tiger Hugs its Head
Wild Horse Leaps the Ravine
White Ape Devotes Fruit
Yellow Bee Returns to Nest

The story comes to us that Chang San-feng watched a fight between a bird and a snake and in this event saw how the soft and yielding could overcome the hard and inflexible. Particularly significant here is the reference to the White Crane (The Manchurian Crane, Grus japonensis), with its red crest an important symbol for Taoist alchemists.
Certain features of Taoist alchemy and talismanic symbolism have also penetrated the Tai Chi forms. As part of their contemplation of nature the Taoists observed the heavens and were keen students of astronomy and astrology. Movements of the Tai Chi Form such as :
Step Up to Seven Stars
Embrace the Moon
Biggest Star in the Great Dipper
Encase the Moon in Three Rings
The Smallest Star in the Big Dipper
Meteor Runs After Moon
Heavenly Steed Soars Across the Sky
Meditating Under the Protection of the Big Dipper.
Reflect this Taoist astrological concern.
Symbolism was a potent force in Taoist thinking. Taoist magic diagrams were regarded as potent talismans having great command over spiritual forces. They invoked the harmonizing influence of yin-yang and Eternal Change; the Divine Order of Heaven, Earth and Mankind; and the workings of the Universe through the principal of the Five Elements. These were symbolized by the Five Sacred Mountains (Taishan, Hengshan [Hunan], Songshan, Huashan and Hengshan Hopei, central places of Taoist development and pilgrimage.
Thus it is no surprise to find that the symbolism of names has, in important ways, infiltrated the forms of Tai Chi. There was a numerological component to this symbolism as well. The number '5' has a special mystical significance to Taoists (and to Chinese in general). There are the symbolic five mountains, five elements, five colours, five planets, five virtues, five emotions, five directions, etc. all of which have a mystic significance. Hence we see five Repulse Monkeys or Five Cloud Hands in the Tai Chi form. There are many instances where the numbers '1', '3', '5' and '7' figure prominently in the structure of Tai Chi.