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Recently, the existence of an Amplituhedron for tree level amplitudes in thebi-adjoint scalar field theory has been proved by Arkhani-Hamed et al. We arguethat hyperbolic geometry constitutes a natural framework to address the studyof positive geometries in moduli spaces of Riemann surfaces, and thus to try toextend this achievement beyond tree level. In this paper we begin anexploration of these ideas starting from the simplest example of hyperbolicgeometry, the hyperbolic plane. The hyperboloid model naturally guides us tore-discover the moduli space Associahedron, and a new version of itskinematical avatar. As a by-product we obtain a solution to the scatteringequations which can be interpreted as a special case of the two well knownsolutions in terms of spinor-helicity formalism. The construction is done in$1+2$ dimensions and this makes harder to understand how to extract theamplitude from the dlog of the space time Associahedron. Nevertheless, wecontinue the investigation accommodating a loop momentum in the picture. Bydoing this we are led to another polytope called Halohedron, which was alreadyknown to mathematicians. We argue that the Halohedron fulfils many criteriathat make it plausible to be understood as a 1-loop Amplituhedron for the cubictheory. Furthermore, the hyperboloid model again allows to understand that akinematical version of the Halohedron exists and is related to the one livingin moduli space by a simple generalisation of the tree level map.

Hyperbolic Geometry and Amplituhedra in 1+2 dimensions