Thelma and I have been implementing C++ maps between different
versions of a Lorentz transformation.  Going around a loop of these
mappings and thereby returning to where you started---except for
welcome rounding error which proves that you actually did more than
copy the original---helps a lot in fixing mistakes.  The different
versions are represented by different C++ types.  That guarantees that
we aren't fooling ourselves in thinking that some particular version
truly contains complete information, and on how it connects to the
other versions.  We call this the roundrobin diagram.  The few
paragraphs here do not tell you how to click around anything---they
just seek to give you the flavor of our little game.

     THE DIAGRAM IS A CIRCLE, with infinitesimal stuff on top and
finite stuff on bottom, real on the left and complex on the right.

                   16               17                18
        ._____.____->___._______.___->___.________.___->____._____.
        |Fld^^|         | (E,B) |        |  E+iB  |         |Sigma|
        |_____|____<-___|_______|___<-___|________|___<-____|_____|
        |     |    15               14                13    |     |
       gl    eg                                            ln    exp
       /|\   \|/                   +or-                    /|\   \|/
        |     |    1      2          5         6       7    |     |
        ._____.____->.___.->.______.->._______.->.____.->___._____.
        | Ltz |      |Can|  |Report|  |Creport|  |Ccan|     | Spn |
        |_____|____<-|___|<-|______|<-|_______|<-|____|<-___|_____|
           |       4      3         10         9       8       |
           |                                                   |
           |________________________<-_________________________|
                                   L(S)


     Ltz's are the usual 4by4 real matrices.  Can  gives the ew
(Eigenwerten---i.e. eigenvalues) and ev (Eigenvektoren---eigenvectors)
[why German? just for the tight  ew ev  abbreviation]
which started by using a canned Maple program, and retains our
sneering label Can though we have distanced ourselves from that, as Lorentz
transformations have their own special properties which makes generic
algebra have the quality of overkill.   Report  tells it all very
visually---the two ends of the axis, and two more data, a
``homerapidity'' and a ``homeangle'';  axis  is the line connecting the
eccentric poles in an example shown in our online
``Sinus Figure''; that you *can* click on:

Seeing all Six Dimensions of the Lorentz Group of Special Relativity  

     Logarithmic ambiguity comes from going from the finite lower level to
the infinitesimal upper level.  So our major early testing, with loops
confined to the lower level, were log-ambiguity-immune.
We did use the upper level, but only to start there, go low; and we
did not come back.  That was done merely to generate an Ltz by exponentiating a
random antisymmetric Fld^^ (exponentiating in the eg = exp o g__ sense).
     The  +or-  ambiguity in step 5 is wiped out by step 10, in our Ltz
to Ltz loop.  It is a loop which begins on the right, that may close on
the negative: of course, an unfancy ``5'' selects a sign, according to
some more or less arbitrary rule, and we did do selection, rather than
waste effort on following two merely sign-opposite branches.
     We are currently adding ambiguity-resolving windingnumber fields,
which should do a Riemann-surface-like lift on all of that.