By Symposium in Pure Mathematics Stanford University 1976, Visit Amazon's R. James Milgram Page, search results, Learn about Author Central, R. James Milgram, , American Mathematical Society
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Extra info for Algebraic and geometric topology
Moreover, each piece is invariant under multiples of 120 degree rotations and reflections, as we remarked above. Thus, in other words, symmetries which mayor may not be suggested by the first iteration and the arbitrary choice of an initial image - like the square in design 2 - of an MRCM are irrelevant. In fact, as we have seen they can be quite misleading. The only symmetries which provide us with choices for the transformations of an MRCM are those which are valid for the limit image. 13). 10.
J$) $ '-------' ! 33. The transformations WI, ... , W9 are chosen according to the blueprint. They all reduce by the factor 113. 38. Let us conclude this discussion with a network of MRCMs which can grow a twig. 39 shows the blueprint of two networked MRCMs which can be used as an example. The attractor of the first machine (L) is a straight line. Here we use two transformations which reduce by a factor of 112. Since in this setup we take just this line as the initial image the machine simply provides this image as a constant output which we use for the stem of the twig.
7 shows the blueprint of the MRCM. We use the word 'Cantor' as initial image. 8 shows the iteration (left) leading to the Cantor set as final image and compares this with the classical construction (right). Observe, that the classical construction seems to somehow capture the Cantor set C more accurately in every stage: the Cantor set is a subset of all the intermediate stages Co, C 1 , C 2 , ... This is not true for the MRCM representation when started with an arbitrary pattern IOThe relation between L-systems and IFSs was first studied in T.
Algebraic and geometric topology by Symposium in Pure Mathematics Stanford University 1976, Visit Amazon's R. James Milgram Page, search results, Learn about Author Central, R. James Milgram, , American Mathematical Society