![]() ![]() For example, circle packing into a larger circle or a square is well addressed in the literature. The shape of the container and the packed items may vary from a fixed-size circle, square, rectangle, triangle, etc. There are applications with different optimization goals, like minimizing the size of the container, or maximizing the number of arranged inner components. ![]() Also, arranged components can be nonoverlapping or there could be overlap allowed. For example, inner shapes can be the same in area or mixed shapes can be considered. Basic requirements must be considered depending on the specific application to find the optimal packing efficiency (PE). It consists of embedding smaller shapes into a larger one called container. Packing is a common problem in many different fields, such as computer science, design, manufacturing, and engineering. Efficiency remains satisfactory, as we show that, by producing the proposed irregular hexagon sensors from the same wafer as a regular hexagon, we can obtain almost the same SE. It enables the sensor to remain symmetric and hexagonal in shape, even though irregular, and produced with minimal number of cuts with respect to dodecagons. Hence, we construct an irregular convex hexagon that is semiregularly tessellating the targeted area. Archimedean semiregular tessellation and its more flexible variants with irregular dodecagons can provide these triangular spacings but with larger number of sensor cuts. The reason why we have replaced the “perfect” regular tessellation with semiregular one is the need to provide spacings at the sensor vertices for placing mechanical apertures in the design of the new CMS detector. It is well-known that semiregular tessellation will cause larger silicon waste, but it is important to formulate the ratio between the two, as it affects the sensor production cost. We provide mathematical expressions to formulate the difference in efficiency between regular and semiregular tessellations. We revisit this problem by using some well-known formulations concerning regular hexagons. Even though packing problems are common in many fields of research, not many authors concentrate on packing polygons of known dimensions into a circular shape to optimize a certain objective. Also, a specific application is considered when produced sensors need to cover the circular area of interest with the largest packing efficiency (PE). We concentrate on the sensor manufacturing application, where sensors need to be produced from a circular wafer with maximal silicon efficiency (SE) and minimal number of sensor cuts. In this paper, a problem of packing hexagonal and dodecagonal sensors in a circular container is considered. ![]()
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