Detectors in Positron Emission Tomography (PET) scanners are usually arranged in rings with the specimen lying along the axial direction. This distribution, however, does not maximize the geometrical sensitivity unless the axial extent becomes much longer than the axial Field of View (FOV), a very costly solution. The optimal geometry for a PET scanner is a sphere with the specimen located at the center. The state of the art of scintillator crystals and detectors for -rays does not facilitate smooth spherical geometry design. In this work we propose a PET scanner of 4 steradian coverage for preclinical imaging as a proof of concept. As a first approximation to the ideal spherical topology, the scanner is shaped as an icosahedron. Furthermore we also present the new custom-tailored hexagonal Silicon Photomultiplier (SiPM) and a monolithic laser-engraved LYSO scintillator crystal that tiles the twenty scanner faces. Preliminary results of the scanner performance based on simulations point to enhanced sensitivity. Initial results on the characterization of the hexagonal photodetector by collection of pulses from a 22Na point source show similar performance as conventional square pixels with the benefit of the increased sensitivity and resolution.