But considerably more common is for leaves to come out less than 180° apart, and in most plants the angle turns out to be essentially the same, and equal to almost exactly 137.5°. … For it turns out that an angle between successive elements of about 137.5° is equivalent to a rotation by a number of turns equal to the so-called golden ratio (1+Sqrt[5])/2 ≃ 1.618 which arises in a wide variety of mathematical contexts—notably as the limiting ratio of Fibonacci numbers.

Rule (f) has a period equal to the maximum of 16.

Nand and Nor are the only 2-input functions universal in this sense. ( {Equal} can for example reproduce only functions {9, 10, 12, 15} , {Implies} only functions {10, 11, 12, 13, 14, 15} , and {Equal, Implies} only functions {8, 9, 10, 11, 12, 13, 14, 15} .)

The first curve yields a perfect fifth, while the third curve yields a diminished fifth (or tritone) in an equal temperament scale.

The maximum possible repetition period for any system is always equal to the total number of possible states of the system. For the systems involving a single dot that we discussed above, the possible states correspond just to possible positions for the dot, and the number of states is therefore equal to the size of the system.

This means that every block with length up to n (except all 0's) must occur with equal frequency. … This quantity is -1 for all nonzero m for PN sequences (so that all but the first component in Abs[Fourier[(-1) list ]] 2 are equal), but has mean 0 for truly random sequences.

In all cases, the behavior is repetitive, and the maximum repetition period is equal to the number of possible positions.

With this initial condition, rule 184 exhibits an equal number of black and white stripes, which annihilate in pairs so as to yield a regular nested pattern.

And as Claude Shannon argued in the 1940s, the length of message needed to be reasonably certain that only one key will satisfy this criterion is equal to the length of the key divided by the redundancy of the language in which the message is written—equal to about 0.5 for English (see below ).

Up to limited m nested sequences can contain all k m possible blocks, and can do so with asymptotically equal frequencies. … In both cases equal frequencies of blocks are obtained only for sequences of length exactly 2 j .