T H E
G O D S A N D B O O K O F D E A D A N C I E N T E G Y P T
A N D B O O K O F L I V I N G M O D E R N T O R U S
G O L D E N S P I R A L
Approximate logarithmic spirals can occur in nature, for example the arms of spiral galaxies - golden spirals are one special case of these logarithmic spirals, although there is no evidence that there is any general tendency towards this case appearing.
Phyllotaxis is connected with the golden ratio because it involves successive leaves or petals being separated by the golden angle; it also results in the emergence of spirals, although again none of them are (necessarily) golden spirals.
It is sometimes stated that spiral galaxies and nautilus shells get wider in the pattern of a golden spiral, and hence are related to both φ and the Fibonacci series.
In truth, spiral galaxies and nautilus shells (and many mollusk shells) exhibit logarithmic spiral growth, but at a variety of angles usually distinctly different from that of the golden spiral.
This pattern allows the organism to grow without changing shape.
In three-dimensional space, a Platonic solid is a regular, convex polyhedron.
It is constructed by congruent (identical in shape and size),
regular (all angles equal and all sides equal),
polygonal faces with the same number of faces meeting at each vertex.
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself.
For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum.
For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 * 28.
Euclid proved that 2p-1(2p-1) is an even perfect number whenever 2p-1 is prime.
For example, the first four perfect numbers are generated by the formula 2p-1(2p-1), with p a prime number, as follows:
for p = 2: 21(22 - 1) = 2 * 3 = 6
for p = 3: 22(23 - 1) = 4 * 7 = 28
for p = 5: 24(25 - 1) = 16 * 31 = 496
for p = 7: 26(27 - 1) = 64 * 127 = 8128
Owing to their form, 2p-1(2p - 1), every even perfect number is represented in binary form as p ones followed by p - 1 zeros.