Gödel, Escher, Bach An Eternal Golden Braid by Douglas R. Hofstadter
A collapsed version of the previous figure. 715 Further collapse of Figure 143. 716 Another way of collapsing Figure 143. 716 B ...
Words of Thanks This book was brewing in my mind over a period of probably nearly twenty years-ever since I was thirteen and thi ...
forget to thank Don for the man-elous flexib\lity-in-inflexibility of his music-printing program, SMUT. He spent many long days ...
standing attitudes. In addition, Paul Csonka read an entire early version and made helpful comments. Thanks to E. O. Wilson for ...
PART I ...
FIGURE 1. Johann Sebastian Bach, tTl 1748. From a painting by Elias Gottlieb Haussmann. ...
Introduction: A Musico-Logical Offering Author: FREDERICK THE GREAT, King of Prussia, came to power in 1740. Although he is reme ...
pleased he would be to have the elder Bach come and pay him a visit; but this wish had never been realized. Frederick was partic ...
...
r r r It' FIGURE 3. The Royal Theme. When Bach sent a copy of his Musical Offering to the King, he included a dedicatory letter, ...
is of this opinion, and to prove it to me he sang aloud a chromatic fugue subject which he had given this old Bach, who on the s ...
able. Nevertheless, it too is based largely on the King's theme, chromatic and austere as it is. It is rather miraculous that Ba ...
inversion, try the tune "Good King Wenceslas". When the original and its inversion are sung together, starting an octave apart a ...
which many ideas and forms have been woven together, and in which playful double meanings and subtle allusions are commonplace. ...
FIGURE 5. Waterfall, by M. C. Escher (lithograph, 1961). most intellectually stimulating drawings of all time. Many of them have ...
FIGURE 6. Ascending and Descending, by M. C. Escher (lithograph, 1960). ...
remarkable. Bach and Escher are playing one single theme in two different "keys": music and art. Escher realized Strange Loops i ...
";;; '" ...c:: o 0.. .... o a r3 v ~ ...
haziness in level-counting, not only in Escher pictures, but in hierarchical, many-level systems. We will sharpen our understand ...
FIGURE 9. Kurt Gjjdel.. ...
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