Addenda (January 24, 2006)

In our study of the dense set of cusps on the arc, we found that it was relatively easy to generate large
numbers of cusps and represent them in a picture of a cloud of points, but we had to work harder to
find out what  order they occurred in the arc, that is, how the points in the cloud should be joined up.  
Between ourselves, we called this "the Calvin-and-Hobbes problem"  because of the highly appropriate
Calvin and Hobbes comic strip of October 6, 1990,  and October 1, 1994.  There,  Calvin
complains that he has connected the  dots on the page of a puzzle book expecting to get a picture but
has only got a mess; Hobbes explains that the dots are supposed to be connected in the order that they
are numbered;  Calvin then grumbles about everything having to have rules.

The Calvin-and-Hobbes problem has been posed and solved, independently, elsewhere, although
without proof; see page 327 of the delightful book:

David Mumford, Caroline Series and David Wright,  Indra's pearls, the vision of Felix Klein, CUP,
Cambridge, 2002.

                                                         Errata (September 28, 2007)

p.142 line 13.     "千里之行, 始于足下." is a famous sentence from Chapter 64 of  Dao De Jing that
is cited in our article in the form "A journey of a thousand miles began with a single step.", which was very
appropriate because the article considers the extent to which the limit of an infinite sequence of steps is
determined by the first step.  However, "千里之行, 始于足下." does not mention first steps, and, moreover,
is in the present tense.  The excellent  Zhongwen.com web site  gives interpretations of pictograms, allowing
us to translate the characters in order as "thousand li of walk, begins from foot below.". In the order corresponding
to English, the list is "walk of thousand li, begins from below foot.". Over the centuries,  the li-to-mile ratio has
varied in the interval from  0.30 to 0.41.  The sentence could be rendered as: "A four-hundred mile walk begins
beneath one's feet." with the sense "Even the longest journey will start from where you stand.". These forms are
not readily applicable to anything considered in our article.  It is a consolation that the Zhongwen.com web site
accepts tradition and gives the translation of the sentence in question as "A journey of a thousand miles begins
with a single step.".

p.148 line 3.  Both occurrences of "\langle A,B,C\rangle" should be changed to  "\langle ABC \rangle".

p.164 line -15.   "(a-c  b-d)"  should be  "(a-c   a-c+b-d)"
                          (   c    d  )                      (  c        c+d   ).

p.164 line -10.  "= a + b" should be  "= 2a + b".

p.164 line - 8.  "\le a + b - 2" should be "\le 2a + b - 2".

p.165 line - 5.  "\le 2a + c - 1" should be "\le 2a + c - 2".

p.166 line 5.      G = C'FC should be G = C' \overline F C'.

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