The critical points of fractal formulae are usually just a set of numbers (mostly including complex numbers). The following formula is different:

z_{n+1} = cos(cz_{n}) + c

As usual c is a constant representing the location in the complex plane.

This formula has critical points that are an expression.

To find the critical points for this fractal the following equation needs to be solved:

f'(z) = 0

Now

f(z) = cos(cz) + c

so

f'(z) = c(-sin(cz)) = 0

The c outside the brackets can be ignored and the minus sign disappears because we’re dealing with zero (hint: multiply both sides by -1).

When does the sin of a value equal zero?

sin(0) = 0

sin(π) = 0

sin(2π) = 0

In fact the sin of any integer multiple of π is zero.

So the critical point must be when

z = n&pi/c

where n is an integer.

When n is zero the critical point is zero which is a nice simple number, if n is not zero there is an expression, effectively the critical point is dependent on the location in the complex plan.

The picture for a critical point of zero is:

Now what does the picture for π/c look like? My software Saturn only allows the initial value of z to be set to a number so the expression can’t be used. I’ll look at the allowing the initial value of z to be set to an expression in a future version of Saturn.

In order to produce a picture using the critical points other than 0 for this fractal I’m resorting to a program called Gnofract4d.

Critical Point π/c

The next critical value at 2π/c results in:

Critical Point 2π/c

The picture looks the same as the picture for the critical point at zero. The critical points at 2nπ/c produce the same pictures as do the critical points at (2n + 1)π/c. So only two critical points are needed to give an idea of what the multiple critical point version of the fractal looks like.

Multiple critical point version

Note: a relatively high bailout of 300 is needed to produce these pictures.

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