Golden rectangle

A series of golden rectangles. After cutting in each of them a square with a side equal to the short side of the rectangle remains the smaller golden rectangle. In the series are drawn two spirals, green circular circle built, red is a logarithmic spiral. Both are tangent to the sides of the rectangles in the places they divide.

Golden rectangle - a rectangle whose sides remain in golden ratio. It is characterized by the fact that after rectifying the square with the side equal to the long side of the rectangle, a new, larger golden rectangle is obtained.

From the definition of the golden rectangle and the property of the golden number φ it follows that:

If at the beginning the ratio of the sides is: a b = φ {\displaystyle {\frac {a}{b}}=\varphi } ,

When a square is added to a longer side, a rectangle with sides a + b and a is obtained, satisfying the following condition: a + b a = φ . {\displaystyle {\frac {a+b}{a}}=\varphi .}

In the opposite direction, by cutting off from the golden rectangle a square with a side equal to the short side of the rectangle is obtained a rectangle whose sides still remain in golden ratio.

Repeat these steps to get more or less golden rectangles.

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