The Great Icosahedron

I never knew what was so great about the ‘great icosahedron’. Now I do. Take a regular icosahedron whose vertices have coordinates in the field ℚ[√5], which consists of numbers a + b√5 with a and b rational. Apply the nontrivial element of the Galois group of this field: that is, simply replace √5 by -√5 in all your formulas. You get the great icosahedron: What do I mean by this, exactly? For example, take the regular icosahedron whose 12 vertices are (±1,±Φ,0), (0,±1,±Φ), (±Φ,0,±1) where Φ = (1 + √5)/2 is the golden ratio. Form a bunch of • points (all the vertices of the icosahedron), • lines (containing all the edges of the icosahedron), and • planes (containing all the faces of the icosahedron) Now replace √5 by -√5 in all your formulas. You get the equations for a new bunch of points, lines and planes. And: • the points are the vertices a great icosahedron; • the lines contain all the edges of this great icosahedron; • the planes contain all the faces of this great icosahedron.…