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Multibeam analog arrays have been proposed for millimeter-wave joint communication and sensing (JCAS). We study multibeam planar arrays for JCAS, providing time division duplex communication and full-duplex sensing with steerable beams. In order to have a large aperture with a narrow beamwidth in the radiation pattern, we propose to design a sparse tiled planar array (STPA) aperture with affordable number of phase shifters. The modular tiling and sparse design of the array are non-convex optimization problems, however, we exploit the fact that the more irregularity of the antenna array geometry, the less the side lobe level. We propose to first solve the optimization by the maximum entropy in the phase centers of tiles in the array; then we perform sparse subarray selection leveraging the geometry of the sunflower array. While maintaining the same spectral efficiency in the communication link as conventional uniform planar array (CUPA), the STPA improves angle of arrival estimation when the line-of-sight path is dominant, e.g., the STPA with 125 elements distinguishes two adjacent targets with 20$^\circ$ difference in the proximity of boresight whereas CUPA cannot. Moreover, the STPA has a 40$\%$ shorter blockage time compared to the CUPA when a blocker moves in the elevation angles.