Read e-book online Looking at Numbers PDF

By Tom Johnson

ISBN-10: 3034805535

ISBN-13: 9783034805537

ISBN-10: 3034805543

ISBN-13: 9783034805544

Galileo Galilei acknowledged he used to be “reading the e-book of nature” as he saw pendulums swinging, yet he may additionally easily have attempted to attract the numbers themselves as they fall into networks of variations or shape loops that synchronize at diversified speeds, or connect themselves to balls passing out and in of the palms of fine jugglers. Numbers are, in the end, part of nature. As such, taking a look at and wondering them is a manner of figuring out our courting to nature. but if we accomplish that in a technical, expert approach, we have a tendency to forget their simple attributes, the issues we will be able to comprehend through easily “looking at numbers.”

Tom Johnson is a composer who makes use of common sense and mathematical versions, resembling combinatorics of numbers, in his track. The styles he unearths whereas “looking at numbers” is additionally explored in drawings. This booklet makes a speciality of such drawings, their good looks and their mathematical that means. The accompanying reviews have been written in collaboration with the mathematician Franck Jedrzejewski.

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Washington, DC: Mathematical Association of America. 2 Sums Another interesting way of looking at numbers is simply to put them together when they have the same sum. I first became interested in this when I wanted to construct groups of chords having the same average height, that is, when the sums of the notes would all be the same. That would permit me to write harmonies that would move a lot without ever really going up or down. To make the music even more immobile, I wanted to link these chords by minimal differences, so that with each move one voice would move up a notch and one would move down a notch, and the rest would not change.

O’Rourke, P. Taslakian and G. Toussaint established the pumping lemma [10]. The vertices a 2 A and b 2 B are isospectral if they have the same histogram of distances to all other vertices in their respective sets. Let A; B be homometric sets with isospectral vertices a 2 A and b 2 B: Then the sets A0 obtained from mA by replacing ma with fma; ma Æ 1; . ; ma Æ rg and B0 obtained from mB by replacing mb with fmb; mb Æ 1; . ; mb Æ rg have the same interval content in Zmn with r þ 1 m. For example, for m ¼ 2, r ¼ 0; Æ1, we have seen that ð0; 1; 2; 5Þ is homometric with ð0; 1; 3; 4Þ in Z8 .

6 Sums of 6 to 21 2 Sums Integer Partitions Fig. 7 Sums of 10 to 26 Fig. 8 Sums of 15 to 30 27 28 Fig. 9 Sums of 21 to 33 Fig. 10 Sums of 6 to 18 2 Sums Integer Partitions References Aigner, M. 2007. A Course in Enumeration. Berlin: Springer. , and K. Eriksson. 2004. Integer Partitions. Cambridge: Cambridge University Press. Bóna, M. 2002. A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory. Singapore: World Scientific Publishing. 29 Bryant, V. 1993. Aspects of Combinatorics.

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Looking at Numbers by Tom Johnson

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