Part 9 (1/2)

where a, b, c a, b, c are arbitrary numbers. For example, in the equation are arbitrary numbers. For example, in the equation 2x 2x2+ 3x+1 = 0, a = 2, b = a = 2, b = 3, 3, c c= 1.

The general formula for the two solutions of the equation is

In the above example

In the equation we obtained for the Golden Ratio,

we have a = 1, b = -1, c = - a = 1, b = -1, c = - 1. The two solutions therefore are: 1. The two solutions therefore are:

APPENDIX 6.

The problem of the inheritance can be solved as follows. Let us denote the entire estate by E E and the share (in bezants) of each son by and the share (in bezants) of each son by x. x. (They all shared the inheritance equally.) (They all shared the inheritance equally.) The first son received:

The second son received:

Equating the two shares:

and arranging:

Therefore, each son received 6 bezants.

Subst.i.tuting in the first equation we have:

The total estate was 36 bezants. The number of sons was therefore 36/6 = 6. Fibonacci's solution reads as follows: The total inheritance has to be a number such that when 1 times 6 is added to it, it will be divisible by 1 plus 6, or 7; when 2 times 6 is added to it, it is divisible by 2 plus 6, or 8; when 3 times 6 is added, it is divisible by 3 plus 6, or 9, and so forth. The number is of 36 minus of 36 minus is is plus 1 is plus 1 is or 6; and this is the amount each son received; the total inheritance divided by the share of each son equals the number of sons, or or 6; and this is the amount each son received; the total inheritance divided by the share of each son equals the number of sons, or equals 6. equals 6.

APPENDIX 7.

The relation between the number of subobjects, n n, the length reduction factor, f f, and the dimension, D D, is

If a positive number A A is written as is written as A = A = 10 10L, then we call L the logarithm logarithm (base 10) of (base 10) of A A, and we write it as log A. A. In other words, the two equations In other words, the two equations A = A = 10 10L and L = log and L = log A A are entirely equivalent to each other. The rules of logarithms are: are entirely equivalent to each other. The rules of logarithms are: (i)The logarithm of a product of a product is the is the sum sum of the logarithms of the logarithms

(ii)The logarithm of a ratio ratio is the is the difference difference of the logarithms of the logarithms

(iii)The logarithm of a power of a number a power of a number is the power is the power times times the logarithm of the number the logarithm of the number

Since 100 = 1, we have from the definition of the logarithm that log 1 = 0. Since 10 = 1, we have from the definition of the logarithm that log 1 = 0. Since 101 = 10, 10 = 10, 102 = 100, and so on, we have that log 10 = 1, log 100 = 2, and so on. Consequently, the logarithm of any number between 1 and 10 is a number between 0 and 1; the logarithm of any number between 10 and 100 is a number between 1 and 2; and so on. = 100, and so on, we have that log 10 = 1, log 100 = 2, and so on. Consequently, the logarithm of any number between 1 and 10 is a number between 0 and 1; the logarithm of any number between 10 and 100 is a number between 1 and 2; and so on.

If we take the logarithm (base 10) of both sides in the above equation (describing the relation between n, f n, f and and D) D), we obtain

Therefore, dividing both sides by log f f

In the case of the Koch snowflake, for example, each curve contains four ”subcurves” that are one-third in size; therefore n n = 4, = 4,f = and we obtain = and we obtain

APPENDIX 8.

If we examine Figure 116 Figure 116(a), we see that the condition for the two branches to touch amounts to the simple requirement that the sum of all the horizontal horizontal lengths of the ever-decreasing branches with lengths starting with lengths of the ever-decreasing branches with lengths starting with f f3 would be equal to the horizontal component of the large branch of length would be equal to the horizontal component of the large branch of length f. f. All the horizontal components are given by the total length multiplied by the cosine of 30 degrees. We therefore obtain: All the horizontal components are given by the total length multiplied by the cosine of 30 degrees. We therefore obtain:

Dividing by cos 30 we obtain

The sum on the right-hand side is the sum of an infinite geometric geometric series (each term is equal to the previous term multiplied by a constant factor) in which the first term is series (each term is equal to the previous term multiplied by a constant factor) in which the first term is f f, and the ratio of two consecutive terms is f. f. In general, the sum In general, the sum S S of an infinite geometric sequence in which the first term is of an infinite geometric sequence in which the first term is a a, and the ratio of consecutive terms q q, is equal to

For example, the sum of the sequence

in which a = a = 1 and 1 and q q = is equal to = is equal to

In our case we find from the equation above:

Dividing both sides by f f, we get

Multiplying by (1-f) and arranging, we obtain the quadratic equation: