an approximate logarithmic spiral generated by the distribution
of primesThe prime number theorem, which describes the approximate rate at which the prime numbers 'thin out' tells us that where is the number of primes less than or equal tox.
The fact that the primes are in this way 'logarithmically distributed' can be exploited in order to construct an approximate logarithmic spiral from the primes themselves. Recall that a logarithmic (or equiangular) spiral can be described most easily in polar coordinates as: Herea and k are arbitrary constants. For example,
if we take a = 1 and k = 1/4, we get the spiral
which looks like this:
Now if we rearrange the formula like this:
we see that we can graph the spiral by plotting points (r, 4logr)
in polar coordinates over some range of r > 0.
We can also rearrange the PNT formula as follows: and so we see that In other words, we can produce an approximation of the above spiral, which is described in polar coordinates as (r, 4logr), by plotting points
To clarify - at radial distance Below we see this plot over four different ranges, and how it quite clearly produces an approximate spiral which appears to 'smooth itself out' as the range increases.
It is important to note that these 'spirals' are generated without
any direct reference to a logarithm. The only 'input' is the data contained in
the prime counting function , We should also note that the choice Often, when explaining the basic facts of the distribution of primes to interested non-mathematicians, I come up against a significant obstacle: The primes thin out, asymptotically, at a particular rate
(this much people can generally understand), but the rate is governed by
a Although mathematically unremarkable, nothing like the above images have
ever been published anywhere that I am aware of. I feel that they provide
a very pleasing illustration of my favoured explanation, and may perhaps
provide those people unable (or unwilling) to grasp the concept of a logarithm
an opportunity to witness directly the beauty of the prime number
theorem.
It might help to imagine actually physically marking these figures
out on a plane. Suppose we were to mark out We then imagine ourselves walking out along the positive We then return to the point This might actually make quite a nice piece of 'land art', involving
the placing of stones or the planting of trees, if anyone feels motivated
to actually carry this out.
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