One major mistake was pointed out in my work on the Takeno line element from yesterday[6]. When working on a theory that describes the behavior of physical quantities with respect to velocity, (like special relativity), don't decompose the velocity into its components like distance and time, or in my case into angular displacement, and time. **Just leave omega, (angular velocity), as omega! **Since omega isn't one of the variables that the derivatives are being taken with respect to, the many terms due to chain ruling out the innards of the hyperbolic trig functions disappeared and the phi and time components of the line element as calculated by Takeno fell out pretty simply.

I also found out last night while reading through one of the Takeno referencing articles[5] I mentioned a few days ago that Takeno was scooped on his transform by about thirty years. As it turns out, Phillip Franklin[2], (picture 2), a then recent PhD from Princeton, beat Takeno to the punch in 1922[3, open access!]. Franklin was a mathematician by training and did his PhD thesis under Veblen on the four color problem. Veblen is also interesting because he wrote a book on projective geometry[4, open access], a topic which turns up more or less subtly all over relativity.

As it turns out Takeno wrote more about the relativity than just the Takeno transforms. Here's an open access example in which he wrote about spherically symmetric space-times[1]. He also helped to develop something called Wave Geometry. I haven't been able to get my hands on any of the sources that contain Takeno writing on wave geometry yet, but here's what Gibbons had to say about it[7]

"Conformal Killing spinors of course arise naturally in conformal supergravity [16]. As a further illustration of historical antecedents, it is interesting to recall that the existence of solutions to an equation of the form (2) was the basic assumption behind the theory of “Wave Geometry” which was extensively developed in Hiroshima in the ’30.s. The introduction to [38] describing the history of these ideas and the fate of those working on them seems to me to be one the most poignant in the physics literature."

**References:**

1. Takeno on space-times

http://projecteuclid.org/euclid.jmsj/1261734863

2. Franklin biography

http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Franklin.html

3. Franklin's transforms

http://www.jstor.org/stable/pdfplus/84534.pdf?acceptTC=true

4. Veblen's book on projective geometry

https://archive.org/details/117714799_001

5. Article referencing Takeno and Franklin

http://arxiv.org/pdf/1208.1913.pdf

6. Takeno line element recalculation begins

http://copaseticflow.blogspot.com/2014/03/re-calculating-takeno-line-element.html

7. Gibbons on Takeno and wave geometry

http://arxiv.org/pdf/1110.1206.pdf?origin=publication_detail

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