Covers for Leo Moser's worm problem

Abstract

Leo Moser’s worm problem was posted in 1966 stating that “What is the region of the smallest area that can cover every unit are?”. In 2018, N. Ploymaklam and W. Wichiramala illustrated a new cover, which is currently smallest. Their cover was adapted from the cover of R. Norwood and G. Poole. In this work, we modify the region from the work of R. Norwood and G. Poole and the work of N. Ploymaklam and W. Wichiramala. Our regions are modified by changing their upper boundary. However, the regions remain satisfying properties in the work of R. Norwood and G. Poole. In addition, we construct the region using cosine function as the upper boundary. This region can cover every unit are which has area approximately 0.26009. Moreover, we try to construct the region starting from the lower boundary. However, we confront with the complex numerical computation. Thus, we cannot construct the region using this idea.