A person is standing somewhere on Earth.

They walk 10km S, 10km E, 10km N.

They are back where they started.

Where are they?

\(\text{circumference} \propto e^\text{radius}\)

\(\text{area} \propto e^\text{radius}\)

\(\Downarrow\)

\(\text{circumference} \propto \text{area}\)

Vectors rotate when you translate them.

\(\text{NE} \neq \text{EN}\)

\(\Downarrow\)

Coordinates are very expensive relative to radius.

Velocities are points in

\[\mathbf{H}^3\]

Curvature is \(c\).

No need for “boosts”, Lorentz transforms, etc.

Relative velocity (**without hyperbolic geometry:**):

\[ \mathbf{v}_{1-2} = \frac{\sqrt{(\mathbf{v_1}-\mathbf{v_2})^2 - (\mathbf{v_1} \times \mathbf{v_2})^2}}{1 - \mathbf{v_1}\cdot\mathbf{v_2}} \]

Relative rapidity (**with hyperbolic geometry:**):

\[ \boldsymbol{\psi}_{1-2} = \boldsymbol{\psi}_1 - \boldsymbol{\psi}_2 \]

**Hierarchical data**

Concept spaces.

**Routing**

Every connected, finite graph has a greedy embedding in the hyperbolic plane. (Kleinberg 2007)

Kleinberg, R. 2007. “Geographic Routing Using Hyperbolic Space.” *Ieee Infocom 2007 - 26th Ieee International Conference on Computer Communications*. https://doi.org/10.1109/infcom.2007.221.