Construction of a Pentagon on the cube Proof of Part I: Pentagon is Equilateral Proof of Part II: Pentagon is in One Plane Proof of Part III: Pentagon is Equiangular Proof of Part IV: Side of Dodec Form Apotome Construction of the Dodecahedron

Now, let us examine the Golden Ratio again. In this picture, on the square on the left hand side: square of light green line  +  square of red line  =  3 x square of green line Since the red lines are of equal length, as are the green lines, we can transform our arguments into: square of vertical light green line  +  square of blue line  =  3 x square of green line moreover, by the triangle property: square of the vertical light green line  +  square of the blue line  =  square of the thin yellow line thus, the golden ratio argument has finally been transformed into: vertical light green line  =  3 x the horizontal blue line.

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