Masteries · Jason GlowneyArithmoí Book 1

Masteries

1.3.1 Curvature and Light

Jason Glowney

The stan­dard ap­proach to matrix groups in­volves con­sid­er­a­tions about the type of space and how one might mea­sure dis­tance in this space. A gen­er­al prac­tice is to con­sid­er spaces as Eu­clid­ean or non-Eu­clid­ean, along with the number of rep­re­sent­ed di­men­sions. Eu­clid­ean spaces are flat, and an ex­cel­lent way to test for this flat­ness is to add up the three angles of a tri­an­gle and ensure it sums to π\pi. As the most in­tu­itive non-Eu­clid­ean geom­e­try, the sphere's sur­face has pos­i­tive cur­va­ture, where the sum of the angles of the tri­an­gle's angles is a value greater than π\pi. Hy­per­bol­ic spaces may be less in­tu­itive for some, having neg­a­tive cur­va­ture and tri­an­gu­lar angle sums less than π\pi.

Figure
Figure 1.3.1 – Tri­an­gle in three-di­men­sion­al Eu­clid­ean space where α+β+γ=π.\alpha + \beta + \gamma = \pi.
Figure
Figure 1.3.2 – Tri­an­gle on the sur­face of a sphere where α+β+γ>π.\alpha + \beta + \gamma > \pi.
Figure
Figure 1.3.3 – A hy­per­bol­ic tri­an­gle on the sur­face of a pseu­dos­phere where α+β+γ<π.\alpha + \beta + \gamma < \pi.

Fig­ures 1.3.1-3 show tri­an­gles on sur­faces with zero, pos­i­tive, and neg­a­tive cur­va­tures, rep­re­sent­ing Eu­clid­ean space, the sur­face of the sphere, and the sur­face of the pseu­dos­phere. An es­sen­tial math­e­mat­i­cal con­sid­er­a­tion is to ask how dif­fer­ent geome­tries and cur­va­tures are re­lat­ed? A metric is a term used by geome­ters, rep­re­sent­ing the math­e­mat­i­cal means to mea­sure things like dis­tances in these var­i­ous spaces. These met­rics can take on trigono­met­ric forms to define the in­vari­ance of a mea­sured dis­tance, where the fa­mil­iar ex­am­ple of the circle is

cos2⁡(α)+sin2⁡(α)=ρ,\begin{array}{r} \cos^{2}{(\alpha) + \sin^{2}{(\alpha) = \rho}}, \end{array}
(1.3.1)

and per­haps, the less fa­mil­iar hy­per­bol­ic metric,

cosh2⁡(β)−sinh2⁡(β)=ρ.\begin{array}{r} \cosh^{2}\begin{array}{r} (\beta) - \sinh^{2}(\beta) = \rho. \end{array} \end{array}
(1.3.2)

In Figure 1.3.4, the unit circle rep­re­sents the dis­tance metric solved for ρ=1\rho = 1, re­sult­ing in the straight­for­ward metric cos2⁡(α)+sin2⁡(α)=1\cos^{2}{(\alpha) + \sin^{2}{(\alpha) = 1}}.

Figure
Figure 1.3.4 – The unit circle's dis­tance in­vari­ance for ρ=1\rho = 1.

The in­vari­ant dis­tance ρ=1\rho = 1 also de­fines the unit hy­per­bo­la, which, like the unit circle, has a trigono­met­ric metric rep­re­sen­ta­tion cosh2⁡(β)−sinh2⁡(β)=1.\cosh^{2}\begin{array}{r} (\beta) - \sinh^{2}(\beta) = 1. \end{array} The hor­i­zon­tal axis of this hy­per­bol­ic tri­an­gle is the cosh⁡(β)\cosh(\beta) func­tion, and the ver­ti­cal axis is the sinh⁡(β)\sinh(\beta) func­tion. The hy­per­bol­ic hy­potenuse rep­re­sents the metric's in­vari­ance, found by eval­u­at­ing ρ=cosh2(β)−sinh2⁡(β).{\rho = cosh}^{2}\begin{array}{r} (\beta) - \sinh^{2}(\beta) \end{array}.These dis­tance met­rics are well-es­tab­lished for both Eu­clid­ean and non-Eu­clid­ean geome­tries, but what is less es­tab­lished is the math­e­mat­i­cal ground­work for what angles are in non-Eu­clid­ean spaces and how the varied geome­tries relate to one an­oth­er by the angle ar­gu­ment space. Taking the ap­proach that angle mea­sure­ments should not be con­fined to real values ∠∈ℝ\mathbb{\angle \in R}, and in­stead take on com­plex values ∠∈ℂ\mathbb{\angle \in C}, re­veals in­cred­i­ble topo­log­i­cal in­sights about how geo­met­ric spaces relate. The in­ter­con­nect­ed­ness of these geo­met­ric spaces with cur­va­tures ∈ℝ\in \mathbb{R}, rep­re­sents a re­cur­rent theme for the sub­ject matter found in this book.

Figure
Figure 1.3.5 – The unit hy­per­bo­la's dis­tance in­vari­ance for ρ=1\rho = 1.