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Ultrasound transducers designed for therapeutic purposes such as tissue ablation, histotripsy, or drug delivery require large apertures for adequate spatial localization while providing sufficient power and steerability without the presence of secondary grating lobes. In addition, it is highly preferred to minimize the total number of channels and to maintain simplicity in electrical matching network design. To this end, we propose array designs that are both space-filling and non-periodic in the placement of the elements. Such array designs can be generated using the mathematical concept of non-periodic or aperiodic tiling (tessellation) and can lead to reduced grating lobes while maintaining full surface area coverage to deliver maximum power. For illustration, we designed two 2-D space-filling therapeutic arrays with 128 elements arranged on a spherical shell. One was based on the two-shape Penrose rhombus tiling, and the other was based on a single rectangular shape arranged non-periodically. The steerability performance of these arrays was studied using acoustic field simulations. For comparison, we also studied two other arrays, one with circular elements distributed randomly, and the other a periodic array with square elements. Results showed that the two space-filling non-periodic arrays were able to steer to treat a volume of 16 × 16 × 20 mm while ensuring that the grating lobes were under -10 dB compared with the main lobe. The rectangular non-periodic array was able to generate two and half times higher power than the random circles array. The rectangular array was then fabricated by patterning the array using laser scribing methods and its steerability performance was validated using hydrophone measurements. This work demonstrates that the concept of space-filling aperiodic/non-periodic tiling can be used to generate therapy arrays that are able to provide higher power for the same total transducer area compared with random arrays while - aintaining acceptable grating lobe levels.