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calculation φ

rectangle Golden Ratio

For a Golden Ratio rectangle this applies: B/A = (A+B)/B.
Presume A = 1. We are looking for length B with the property above.
Then the following applies:

B = (1 + B)/B

B2 = B + 1

B2 - B -1 = 0

This is a simple quadratic equation with solution B = $\frac{1+\sqrt{5}}{2}$ (The other solution is negative and therefore without sense.)

 

Now the pentagram.

pentagon03 pentagon04

Take the side of the regular pentagon equal to 1. With simple trigonometry (symmetries, parallelograms) it is clear that also B = 1, C + E = 1 and C + D = 1. The length of diagonal A = D + C + E. So D = A -1

Here applies the following: A/B = (C + D)/D. I don't give the proof written out in full here, but the core is the uniformity of the triangles STR and PQR (right figure).

So: A/1 = 1/(A - 1)

A(A - 1) = 1

A2 - A - 1 = 0, which is the same quadratic equation as above.

 

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