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

By applying the Pythagoras Theorem, we know: square of blue line + square of red line  =  square of green lines and square of green line + square of purple line  =  square of magenta line therefore, square of blue line + square of red line +  square of purple line  =  square of magenta line Recalling that we have applied the Golden ratio property onto these three line segments, square of purple line  +  square of red line  =  3 x square of blue line and square of purple line  +  square of red line  +  square of blue line = 4 x square of blue line Therefore, square of the magenta line = 4 x the square of the blue line. Thus, taking the square root on both sides: the magenta line = 2 x blue line.

Back To Main Page