Hyperbolic Paradoxes

In this article, we explore the Banach-Tarski paradox within the context of the hyperbolic plane, utilizing both the half-plane model and the Poincaré disk model. The half-plane model represents the hyperbolic plane as the upper half of the Cartesian plane, where lines are either vertical rays or semicircles orthogonal to the x-axis. On the other hand, the Poincaré disk model depicts the hyperbolic plane within the unit disk, with lines represented by arcs of circles orthogonal to the boundary of the disk or diameters of the disk

If you are not familiar with these concepts, it is recommended that you read about them for a better understanding of this article: hyperbolic plane, half-plane model, and Poincaré disk model.

In this chapter we will show that Hyperbolic space is paradoxical. For that we will have base figure e, and two rotations: σ and τ as on picture:

this is σ in both the models with rotation points marked:

this is τ:

Easy to see that τ³ and σ² are identity movements. example of τ³

the interesting thing about these rotations is that any sequences of meaningful rotations gives us a unique nonitersecting with others result.

Now, as observed by Hausdorff, the abstract group \( \mathbb{Z}^2 \ast \mathbb{Z}^3 \). In fact, there is a partition of the group into \( A \cup B \cup C \) such that the Hausdorff relations:
\(\tau(A) = B,\)
\(\tau^2(A) = C,\)
\(\sigma(A) = B \cup C\)

and actually solution of this is:
A = {all \((\tau\sigma)^j\) and all of \(W_\sigma\), except \(\tau^2(\tau\sigma)^j\)};
B = {all \(\tau((\tau\sigma)^j)\) and all of \(W_\tau\), except \((\tau\sigma)^j\)};
C = {all \(\tau^2((\tau\sigma)^j)\) and all of \(W_{\tau^2}\), except \(\tau((\tau\sigma)^j)\)}.

and this is animation of this paradox

and other version of the paradox where \(\sigma(\text{red})= \text{green} \cup \text{blue}\) and \(\tau^2(\text{red})= green = \tau(\text{blue}\)):

and equality of entire space to \(\text{red})\):