Golden Triangles and Infinite Series

Topic:Isosceles Triangles, Sequences and Series

A golden triangle is an isosceles triangle in which the ratio of one of the equal sides to the base is the golden ratio, that is,

. The angle at the vertex of the golden triangle is

. In this applet we start with an isosceles triangle whose equal sides have length

and the base is equal to

. The angles at the base are

Click the checkbox to show the first golden triangle.

Next, we generate a sequence of smaller embedded golden triangles. This sequence is constructed by rotating the previous triangle on

and rescaling it with factor

Click the Construction ON/OFF button or drag the slider.

Using the construction below, we can evaluate the series

, the series

and also the series

This applet is based on the note Proof Without Words: An Infinite Series Using Golden Triangles by Steven Edwards, The College Mathematics Journal Vol. 45, No. 2 (March 2014), p. 120.

Golden Triangles and Infinite Series

Golden Triangles and Infinite Series

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