Discrete Mathematics

Study of discontinuous numbers

Logic, Set Theory, Combinatorics, Algorithms,

Automata Theory, Graph Theory,

Number Theory, Game Theory, Information

Theory

RecreationalNumberTheory

Power of 9s

9 * 9 = 81

8 + 1 = 9

Multiply any number by 9Add the resultant digits

togetheruntil you get one digit

Always 9e.g.,

4 * 9 = 363 + 6 = 9

Square Root of Palendromic Numbers

Square Root of123454321

=11111

Square Root of1234567654321

=1111111

Leonardo of Pisa, known as Fibonacci. Series first stated in

1202 book Liber Abaci

0,1,1,2,3,5,8,13,21,34,55,89. . Each pair of previous numbers equaling the next number of

the Sequence.

Dividing a number in the sequence into the following

number produces theGolden Ratio

1.62

Debussy, Stravinsky, Bartókcomposed using

Golden mean (ratio, section).

Bartók’s Music for Strings, Percussion and Celeste

89

2134

21 13

13 21

55 34

Importance of number sequences to music.

After all, music is a sequence of numbers.

Pascal’s Triangle

• The sum of each row results in increasing powers of 2 (i.e., 1, 2, 4, 8, 16, 32, and so on).

• The 45° diagonals represent various number systems. For example, the first diagonal represents units (1, 1 . . .), the second diagonal, the natural numbers (1, 2, 3, 4 . . .), the third diagonal, the triangular numbers (1, 3, 6, 10 . . .), the fourth diagonal, the tetrahedral numbers (1, 4, 10, 20 . . .), and so on.

• All row numbers—row numbers begin at 0—whose contents are divisible by that row number are successive prime numbers.

• The count of odd numbers in any row always equates to a power of 2.

• The numbers in the shallow diagonals (from 22.5° upper right to lower left) add to produce the Fibonacci series (1, 1, 2, 3, 5, 8, 13 . . .), discussed in chapter 4.

• The powers of 11 beginning with zero produce a compacted Pascal's triangle (e.g., 110 = 1, 111 = 11, 112 = 121, 113 = 1331, 114 = 14641, and so on).

• Compressing Pascal's triangle using modulo 2 (remainders after successive divisions of 2, leading to binary 0s and 1s) reveals the famous Sierpinski gasket, a fractal-like various-sized triangles, as shown in figure 7.2, with the zeros (a) and without the zeros (b), the latter presented to make the graph clearer.

$1 million prize to createformula for creatingnext primes without

trial and error

• The sum of each row results in increasing powers of 2 (i.e., 1, 2, 4, 8, 16, 32, and so on).

• The 45° diagonals represent various number systems. For example, the first diagonal represents units (1, 1 . . .), the second diagonal, the natural numbers (1, 2, 3, 4 . . .), the third diagonal, the triangular numbers (1, 3, 6, 10 . . .), the fourth diagonal, the tetrahedral numbers (1, 4, 10, 20 . . .), and so on.

• All row numbers—row numbers begin at 0—whose contents are divisible by that row number are successive prime numbers.

• The count of odd numbers in any row always equates to a power of 2.

• The numbers in the shallow diagonals (from 22.5° upper right to lower left) add to produce the Fibonacci series (1, 1, 2, 3, 5, 8, 13 . . .), discussed in chapter 4.

• The powers of 11 beginning with zero produce a compacted Pascal's triangle (e.g., 110 = 1, 111 = 11, 112

= 121, 113 = 1331, 114 = 14641, and so on).• Compressing Pascal's triangle using modulo 2

(remainders after successive divisions of 2, leading to binary 0s and 1s) reveals the famous Sierpinski gasket, a fractal-like various-sized triangles, as shown in figure 7.2, with the zeros (a) and without the zeros (b), the latter presented to make the graph clearer.

1111 111111 111 111 1 1 1111111111 111 111 1 1 11111 11111 1 1 111 11 11 1 11 1 1 1 1 1 1 111111111111111111 111 111 1 1 11111 11111 1 1 111 11 11 1 11 1 1 1 1 1 1 111111111 1 11111111 1 1 111 11 11 111 1 1 1 1 1 1 11111 1111 1111 11111 1 1 1 1 1 1 111 11 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1111111111111111111111111111111111 111 111 1 1 11111 11111 1 1 111 11 11 1 11 1 1 1 1 1 1 111111111 111111111 1 1 111 11 11 111 1 1 1 1 1 1 11111 1111 1 111 11111 1 1 1 1 1 1 111 11 11 1 1 11 1 1 11 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11111111111111111 11111111111111111 1 1 111 11 1 1 1 11 1 1 1 1 1 1 11111 1111 1111 11111 1 1 1 1 1 1 111 11 11 1 1 11 1 1 11 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 111111111 1 1111111 1 1111111 1 11111111 1 1 1 1 1 1 111 11 11 11 11 11 11 111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11111 1111 1111 1111 1111 1111 1111 1111

Magic Squares

Square Matrixin which

all horizontal ranksall vertical columns

both diagonalsequal same number when

addedtogether

0-2 7 9 -9

-711 -5 2 4

6-1 13 -10 -3

-6-8 1 8 10

12 5 -11 -4 3

1

6-1 13 -10 -3

-6-8 1 8 10

12 5 -11 -4 3

0-2 7 9 -9

-711 -5 2 4

12 5 -11 -4 3

0-2 7 9 -9

-711 -5 2 4

6-1 13 -10 -3

-6-8 1 8 10

-711 -5 2 4

6-1 13 -10 -3

-6-8 1 8 10

12 5 -11 -4 3

0-2 7 9 -9

-6-8 1 8 10

12 5 -11 -4 3

0-2 7 9 -9

-711 -5 2 4

6 -1 13 -10 -3

0-2 7 9 -9

-711 -5 2 4

6-1 13 -10 -3

-6-8 1 8 10

12 5 -11 -4 3

1 2

3

4 5

Musikalisches Würfelspiele

Number of Possibilitiesof 2 matrixes

is1116

or45,949,729,863,572,161

45 quadrillion

Let’s hear a couple

Xn+1 = 1/cosXn2

(defun cope (n seed) (if (zerop n)() (let ((test (/ 1 (cos (* seed seed))))) (cons (round test) (cope (1- n) test)))))

? (cope 40 2)(-2 -1 -2 -4 -1 -11 -3 2 -1 10 1 -2 -1

2 -9 -2 1 2 29 1 -7 3 -9 -4 1 2 -2 -1 2 -1 3 1 -2 -1 2 4 1 2 -2 -1)

Tom Johnson’s

Formulas forString Quartet

No. 7

Iannis Xenakis

Metastasis