Non-equivalent coloring schemes for the regular n-gons, for n=4 and n=5

Spatial Sieves

Economou A and Grasl T

Xenakis’ sieves are a pow­er­ful for­mal tool to cre­ate integer-sequence gen­er­a­tors that can be used for the gen­er­a­tion of var­i­ous numer­i­cal pat­terns to rep­re­sent pitch scales, rhythm sequences, as well as pat­terns of loud­ness, den­sity, tim­ber and so forth. The key idea in the design of the sieves is the notion of a vocab­u­lary of ele­men­tary mod­ules that can be com­bined one with another under the oper­a­tions of union and inter­sec­tion in var­i­ous ways to pro­duce a series of pat­terns that rise from the sim­plest pos­si­ble ones to highly expres­sive struc­tures sim­u­lat­ing almost random-like lin­ear dis­tri­b­u­tions of points on a line. Sig­nif­i­cantly Xenakis asserted that his alge­bra of sieves revealed insight­ful views to fun­da­men­tal aspects of com­po­si­tion by “giv­ing when it exists, a more hid­den sym­me­try derived from the decom­po­si­tion of a mod­u­lus”. It is this ana­lyt­i­cal power of the for­mal­ism to reveal hid­den sym­me­tries, emer­gent struc­tures, and order­ing sys­tems that is revis­ited and reworked here.

The key idea for the work out­lined here is that the con­struc­tion of the sieves when it is pro­duced by divi­sion rather that by aggre­ga­tion and con­cate­na­tion pro­vides read­ily avail­able nested pat­terns that clearly fore­ground var­i­ous emer­gent numer­i­cal and spa­tial struc­tures. The for­mal tool for this inquiry is taken by Polya’s the­o­rem of count­ing non-equivalent con­fig­u­ra­tions with respect to a given per­mu­ta­tion group. Here the the­o­rem is applied within an auto­mated com­pu­ta­tional frame­work to pro­duce all pos­si­ble non-equivalent con­fig­u­ra­tions that can be extracted from a reg­u­lar divi­sion of a lin­ear mod­ule into an n-number of aliquot parts. The soft­ware gen­er­ates auto­mat­i­cally the cyclic index of any sym­me­try group of any finite lin­ear shape, allows for the com­pu­ta­tion of a fig­ure inven­tory of vari­ables upon the cycle index and visu­al­izes the result of all non-equivalent con­fig­u­ra­tions with cor­re­spond­ing iso­mor­phic two-dimensional sub­shapes of reg­u­lar n-gons. Sig­nif­i­cantly the soft­ware illus­trates all such pos­si­ble con­fig­u­ra­tions in par­tial order lat­tices too to show the nested orders and sym­me­tries of the spa­tial sieves.

Xenakis Past, Present, and Future
28-30 January 2010
New York (USA)