The most obvious feature of Takemiya's go is the consistent use of large-scale frameworks (omoyo). This cuts across the development of a territorial style which is more reliable in terms of results, in top players from Rin Kaiho and Ishida Yoshio to Kobayashi Koichi and Cho Chikun, and these days outstandingly in Yi Ch'ang-ho. Takemiya's way is more artistic, in the sense that would be understood in Japanese tradition, more personal, and also is much admired by amateur players for its apparent freedom.
These considerations resemble those under discussion at the time of the shinfuseki revolution. It would be superficial to equate the 'cosmic style' with a recycled shinfuseki, though. While there are some common features, such as the use of sanrensei, the details are different and give the impression of fresh research - Yamashita Keigo's style is rather closer to shinfuseki in some ways, but is surely not yet completely developed.
After one period near the top, Takemiya found he had to modify the cosmic style somewhat. With success, it has to be said, in that he still can produce outstanding games.
The other thing that ought to be pointed out is that he is a naturally strong player (except perhaps in the endgame) who has in his own way gone back to a style based on honte, a deep matter for a pro. The spectacular strategies are just the icing on the cake.
Ochanomizu (AGA 1 kyu): Nice, lucid bio. I have one minor point of contention: I feel it's a bit misleading to say that Takemiya's best period saw him 'near the top'. I'm not just trying to be pedantic here. When you consider that from the mid-to-late 1980's he held the Honinbo for numerous years in succession, and, more importantly, won the 1st and 2nd Fujitsu tournaments, I think it's more appropriate to say he reached the top. It wasn't the kind of domination that Cho achieved in the early eighties, or that which Kobayashi then produced for much of the period between 1985 through the early nineties. You could also point to Nie Weiping's dominance in the Japan-China exchanges and then Cho Hunhyun's impressive garnering of international titles in the early nineties. Maybe Takemiya's was a less conspicuous ascendency than these, but I think winning the Fujitsu in its first two years is VERY impressive, considering he had to play his way all the way back to the final the second year; and this, at a time when international stars like Nie, Ma, Cho Hunhyun and Yu Cheonghyuk were emerging with impressive force, and a fervor to prove the mettle of their respective countries' go strength, in the face of presumed Japanese chauvinism. All I'm saying is: the case can be made that, for a very brief period, Takemiya was the man to beat.
flyingfox (KGS 7d): One common misconception about cosmic style is that people think it's about building large moyos and surrounding a huge chunk of territory. The real essense of cosmic style, however, is to force the opponent to come into the moyo and attack the invading stones (not necessarily kill them). In fact, in many cases in which Takemiya was successful in enclosing his moyo (thus surrounding a big territory), he still lost. To quote Takemiya, "back when I was a student, my first teacher always told us not to win through gaining territory, but to win through fighting. It was due to my first teacher that I started to prefer influence over territory".