Abstract
In this study we study overlap dimensions in cyclic tessellable regular polygons. Overlap difference and area created by tessellable regular polygons inscribed in disks for covering play a significant role in computational geometry and signal interference in telecommunication network design. Regular triangles and squares are no exceptions except for optimality. We propose general formulae for computing the dimensions of a regular polygon inscribed in a disk. The study also leads to formulae for computing the overlap difference of tessellable regular polygons in disk covering. We realize that the cyclic regular hexagon has both optimal covering area and minimal overlap difference of 17.3 and 86.6% reduction over the original 100% disks size, respectively.
Highlights
Any tessellable regular polygon inscribed in a disk is a cyclic tessellable regular polygon
The relationship between the overlaps created by equilateral triangles, squares and hexagons inscribed in disks has not yet been studied
Ratio of overlap difference and area for tesselable regular polygons inscribed in disks: Table 1 shows the relationship between the overlap difference and area for three tessellable regular polygons inscribed in a disk with radius RR1, hexagonal apothem rr1 and their occupying ratio or covering fraction to that of the disks
Summary
Any tessellable regular polygon inscribed in a disk is a cyclic tessellable regular polygon. Any regular polygon that can tile has the property of covering These tessellable regular polygons have a lot of geometric and algebraic properties when inscribed in a disk with fixed radius. Optimally packed in a hexagon to be closest to that of a circle with the same number of disks packed. This geometrically confirms the proof that hexagon approximates circle closely than any tessellable regular polygon. The relationship between the overlaps created by equilateral triangles, squares and hexagons inscribed in disks has not yet been studied. This research paper aim at this relationship as well as the dimensions of a regular tessellable polygon that can be inscribed in disks with a fixed radius RR1
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