You mention flexibility in the context of yoga or dance and immediately most people envision some contortionist type person doing inexplicable types of movements with their limbs. While it is true that people turn to yoga for flexility of the body, long time practitioners know that the real gift of any mind body fitness regime is the mental flexibility developed. At its core, flexibility requires a great deal of strength. The ability to feel the subtle nuance between how much to stretch and what opposing muscles should be engaged. n fact, one might visualize strength and flexibility as the muscular yin and yang. Years of practice, a habit of visualizing how the body works and developing muscle memory culminate in the peak poses that are something to behold.
The process of this push and pull of strength and ease in mind body fitness pursuits, almost inevitably develops a second more important flexibility- the flexibility of the mind. Not only are there many paths to achieving some of the physical feats that wow, there is a profound trust that develops between the body and mind. The mind decides exactly which muscles to engage, which to stretch and to what extent. The body responds in turn to validate what feels correct and what doesn't. Not surprisingly, there is no short cut to developing this mind body partnership, to finding and feeling the sweet spot. It is discovered through many hours of practice, trial and error.
Applied to the bigger picture, flexible thinking, a feeling for nuance and an understanding that there are many ways to achieve and end goal are critical habits in the development of wisdom. Mind/body fitness then, is not only good physically, it makes a habit of the wisdom and muscle memory to find the 'sweet spot'. Not expending any unnecessary energy, being in the moment, feeling exactly when to engage and how. This mental training proves powerful beyond the gym or mat and hey, you may end up being able to do the splits too.