Shape Computation Lab

n-Dipyramids

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01. Configurations 1-48 from the catalogue of the 78 non-equivalent two-coloring configurations of the n-dipyramids, for n = 5.

02. A table of non-equivalent configurations of k = 2 colors painted upon the faces of the n-dipyramids, for n<=12

03. Catalogue of the 13 non-equivalent two-coloring configurations of the n- dipyramid, for n = 3.

04. Configurations 1-32 from the catalogue of the 34 non-equivalent two-coloring configurations of the n-dipyramid, for n = 4

05. Configurations 1-48 from the catalogue of the 78 non-equivalent two-coloring configurations of the n-dipyramids, for n = 5.

06. Configurations 1-48 from the catalogue of the 237 non-equivalent two-coloring configurations of the n- dipyramids, for n = 6

07. Configurations 1-48 from the catalogue of the 687 non-equivalent two-coloring configurations of the n- dipyramids, for n = 7

08. Configurations 1-48 from the catalogue of the 2299 non-equivalent two-coloring configurations of the n- dipyramids, for n = 8

Athanassios Economou and Thomas Grasl

2017

Keywords: Configuration; Perfect coloring; Polya’s theorem of counting; Sieves, Xenakis; Counterpoint; Rhythm

The sieve is a powerful formal tool developed by the composer / architect Ioannis Xenakis to create integer-sequence generators that can be used for the generation of various numerical patterns to represent pitch scales, rhythm sequences, loudness progressions, density patterns, timber systems and so forth, The decomposition of a n-modulus of the sieve can be modeled as the arrangement of 2 colors x and y upon the faces of a regular n-polygon whereas the color x represents the sieve content and the color y represents the numerical distance between the sieve content.

Here, the work extends the model to capture spatial relations between sieves. The key idea to extend the model is to look at the possible spatial relations between sieves in Euclidean space and primarily in three-dimensional space. The candidate shapes to capture the notion of decomposition of a module are the extensions of the regular n-polygons in the Euclidean plane to the regular prisms in Euclidean space, and even more so, the regular n-dipyramids, the duals of the regular n-prisms. The computation here gives the complete number of all colorings of the faces of 12-dipyramids for up to 5 colors and the emphasis is given in the calculation of 2 colors as the three-dimensional extension of the model for the enumeration of the sieves wrapped around 2 n-polygons in space. More specifically it can be shown that there are 2 ways to color the 2 faces of an 1-dipyramid with 1 color, 6 ways to paint the 4 faces of a 2-dipyramid with 2 colors, 13 ways to paint the 6 faces of a 3-dipyramid with 2 colors, 34 ways to paint the 8 faces of the 4-dipyramid with 2 colors and so forth. The computation here foregrounds the calculation of the coloring of the 16 faces of the 8-dipyramid because this shape presents all possible symmetries found in the 7 infinite families of three-dimensional regular prisms. More specifically, the table shows that 16 faces of an 8-dipyramid can be painted with 2 colors in 2299 distinct ways. In a similar manner there are 2299 polyrhythms that consist of two rhythms played one against the other.