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.    The following is a famous sentence from Chapter 64 of  Dao De Jing:

            "A journey of a thousand miles began with a single step."

This form is very appropriate because the article considers the extent to which the limit of an infinite sequence
of steps is determined by the first one.  However, the Chinese quotation 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 at foot below

In the order corresponding to English, the list is

          walk of thousand li begins at 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."

or

          " Even the longest journey must 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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