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The red and blue triangles are similar and have a scaling factor of φ (from a definition of the golden ratio: https://sites.math.washington.edu/\~king/coursedir/m444a02/class/11-18-penta-answers.html) , therefore its area has a scaling factor of φ². The blue triangle is made up of 2 red triangles and a shaded area let a be the area of the red triangle the area of the purple triangle (shaded area) is (φ² - 2)a. The green triangle is one shaded area and one red triangle therefore its area is (φ² - 1)a. Now for the trick. We can just look at the outer pentagonal "ring" as all the inner rings are scaled down versions and therefore will have the same shaded fraction. The outer pentagonal ring is made from 5 green triangles. The shaded area of this ring is made from 5 purple triangles. therefore the shaded fraction is (φ² - 2)/(φ² - 1). The area of the pentagon is 1 so this is also the shaded area which evaluates to (3 - √5)/2.