Gödel, Escher, Bach An Eternal Golden Braid by Douglas R. Hofstadter
G(n) = n - G(G(n - 1» G(O) = 0 for n > 0 How does this function G(n) code for the tree-structure? Quite simply, if you constr ...
It is reminiscent of the Fibonacci definition in that each new value is a sum of two previous values-but not of the immediately ...
1.0 .9 .8 .7 .6 , ~\ .. ,: .... -/ \' / FIGURE 32. Graph of the function INT(x). There is a jump discontinuity at every rational ...
What corresponds to the bottom in the definition of INT is a picture (Fig. 33a) composed of many boxes, showing where the copies ...
(a) FIGURE 33(a) The skeleton from which [NT can be constructed by recursive substitutions. (b) The skeleton from which Gplot ca ...
Gplot shows that distribution. The horizontal axis represents energy, and the vertical axis represents the above-mentioned ratio ...
1.0 .9 .8 . 7 .6 .5 .4 .3 .2 1 .0 ~ -:::::~:~~~~::~::=.:;:~:::~~~~:::j?{:~??}!tf!:?0::~~~~ ~~!'l~~:?T?~ \~:???::~?E~.~~:~::::: ...
Let us be a little more concrete, now. Let's limit ourselves to only two kinds of particles: electrons and photons. We'll also h ...
As you might have anticipated, these virtual processes can be nested inside each other to arbitrary depth. This can give rise to ...
the physicist has to be able to take a sort of average of all the infinitely many different possible drawings which involve virt ...
FIGURE 36. Fish and Scales, by M. C. Escher (woodcut, 1959). luted "copy" of the entire fish-and so there is more than a grain o ...
FIGURE 37. Butterflies, by M. C. Escher (wood-engraving, 1950). Escher drawing or a Bach piece gives it away. Just as a fish's D ...
ent levels at once. But the events on different levels aren't exactly the same-rather, we find some invariant feature in them, d ...
repeated several times in larger loops, which in turn are carried out re- peatedly ... While the result of a low-level loop migh ...
notice that we have now defined "best move" recursively, simply using the maxim that what is best for one side is worst for the ...
program) was world champion. But after ten years had passed, it seemed that the day a computer would become world champion was s ...
Canon by I ntervallic Augmentation Achilles and the Tortoise have just finished a delicious Chinese banquet for two, at the best ...
less as cookies than as message bearers. Unfortunately you seem to have swallowed some of your fortune. What does the rest say? ...
Achilles: What? A jukebox with only one record? That's a contradiction in terms. Why is the jukebox so big, then? Is its single ...
Achilles: I don't understand that at all. You CAN'T pull different songs out of the same record! Tortoise: That's what I thought ...
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