Lie Sphere Geometry Part 15: Exercise with the pole shift

 How will the formulas of Part 14 change if North and South Pole are exchanged due to a pole shift? Immanuel Velikovsky’s hypothesis in Worlds in Collision imagines a cataclysmic pole shift triggered by planetary near-misses, such as Venus and Mars, dramatically altering Earth’s axis and climate, showcasing his bold independence of thought and vivid imagination in reinterpreting ancient myths as cosmic events. His work reflects a daring willingness to challenge conventional science, prioritizing creative synthesis over established norms.

Shifting the poles cataclysm

 In this post we follow Velikowsky's idea. Choosing the origin of the projection is our arbitrary choice. A really true geometry should not depend on such arbitrary choices. We need to look into this issue closer, and this involves straightforward calculation following the onc we have already done, but with a different formula for the stereographic projection. So, there will be nothing new in this part - just a simple exercise in applying again the techniques that we already know.

We start with repeating, almost verbatim, the derivation of the formulas for spheres

S_r(m) = {x ∈ S³: x · m = cos(r)},                 (15.1)

with normal vector field

n(x) = (m − cos(r)x)/sin(r).                 (15.2)

as in Part 9. The South Pole projection lead us, in Part 13, to the formula (13.7):

τ_s (c, ρ) = ([1 − q(c, ρ))/2 e₀ + c +(1 + q(c, ρ))/2 e₄ + ρ e₅ ],

or, using e_± = (e₄ ± e₀ )/2, to

τ_s (c, ρ) = [e₊ + q(c, ρ)e₋ + c + ρ e₅ ].                 (a)

As we will see below, the North Pole projection gives us, instead, the formula

τ'_s(c, ρ) = [(q(c, ρ) − 1)/2 e₀ + c +(q(c, ρ) + 1)/2 e₄ + ρ e₅ ],

or

τ'_s(c, ρ) = [q(c, ρ)e₊ + c + e₋ + ρ e₅ ]                 (b)

There are two distinguished points: the origin of space, and the infinity. In the first case, that of τ_s , the origin, corresponding to c = 0, ρ = 0, is represented by p_o = [e₊], while the infinity is represented by p_∞ = [e₋]. In the second case, that of τ'_s, the roles dramatically change: now p_o = [e₋], while p_∞ = [e₊ ].

The roles dramatically change: now p_o = [e₋], while p_∞ = [e₊ ].

This observation tells us that a different approach is needed. Q and Q + are homogeneous spaces. Thus any point can be chosen to represent the infinity. We know that (cf. (13.22)

(e₊, e₋ ) = − 1/2.

This suggests that if we decide to choose p_∞ = [u], then p_o = [u'] should be chosen arbitrarily, so that (u, u') ≠ 0.

This is the idea that we will follow in the future. For now, however, we need to do our boring homework. You can see it done in the complete Notes or, only Cj. 15, in the pdf displayed  below.