r digrams in order of decreasing frequency out of a 10 000-letter count. The corresponding probability distribution is obtained by normalizing this histogram by 10 000 (from data in Fig. 9.5 with c(y|x) > 100). into single characters, calling them, for instance, T, , or A, respectively. 15 Ancient scriptures include lots of such symbolic digram contractions, also called ligatures, which helped the early writers, sculptors, or printers to produce their works more effectively or artistically. 16 Figure 9.7 shows a logarithmic-scale plot of the 676 digrams in decreas- ing order of frequency. The distribution is seen to be very nearly exponential, as the numerical fit indicates. The digram counts that were found to be zero in the reference were changed to 0.5 and 0.1, to allow and to optimize this exponential fit. The fact that the distribution is exponential suggests that Shannon-Fano coding should be a good approximation of an optimal (Huffman) code, considering the relatively large size of the source. The digram-source entropy is calculated to be H(X(2))=7.748 = bit/digram. Interest- ingly, it is smaller than H" = 2 X H (X () = 8.368, where H(X ) = 4.185 bit/character is the entropy of the English-language source. This fact is a positive indication that the These are not to be confused with phonetics, which uses other super-alphabet codes in an attempt to emulate and classify foreign-language sounds and vocal spelling. A surviving contemporary example of digram ligatures is the symbol character &, called amper- sand (or esperluette in French). It is in all computer keyboards, and is now used internationally as the word contraction of "and"; see for reference: http://en.wikipedia.org/wiki/Ligature_(typography), www.adobe.fr/type/topics/theampersand.html. The graphics of the ampersand character actually come from the Latin word et. While the French use the same word as Latin, et, the Italian and Spanish use the shorter words e and y, respectively, which do not call for symbol contraction, in contrast with most other