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I tried to find out how to plot a pentagram in the complex plane. Let the radius of the pentagram be 1, then all the corners of the pentagram satisfy the equation

x^5 = 1 (where x is a complex number).

The answer is, of course, x = exp (2n*pi*i/5).

But I wanted to express that in roots, not in exponentials.

I could not find anything on the Web, so I tried it myself.

It can be done, but the answer is not beautiful.

The prime root is:

x = a + i*a / sqrt(1-2/sqrt(5)),

where a = [ (9-4*sqrt(5)) / (16*sqrt(5)-16) ]^(1/5).

Next, I tried to express this in terms of the "Golden Ratio" Phi. Because I knew that Phi appears in the pentagram a lot.

Since

Phi^2 = 1*Phi + 1

Phi^3 = 2*Phi + 1

Phi^4 = 3*Phi + 2

Phi^5 = 5*Phi + 3

... and so on (that's the Fibonacci series twice),

we get

x = (Phi - 1)/2 + sqrt(Phi + 2)*i/2

and also

x^2 = -Phi/2 + sqrt(-Phi + 3)*i/2.

(x^3 and x^4 are obviously symmetrical wrt. the real axis)

That looks better.

But not perfect. I tried to get rid of the square root in the imaginary part, because I wanted everything to be linear in Phi. But no success.

Any help?

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# Pentagram coordinates

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