Optimal Geometric Disks Covering using Tessellable Regular Polygons

Geometric Disks C overing (GDC) is one of the most typical and well studied problems in computational geometry. Geometric disks are well known 2-D objects which have surface area with circular boundaries but differ from polygons whose surfaces area are bounded by straight line segments. Unlike polygons covering with disks is a rigorous task because of the circular boundaries that do not tessellate. In this paper, we investigate an area approximate polygon to disks that facilitate tiling as a guide to disks covering with least overlap difference. Our study uses geometry of tessellable regular polygons to show that hexagonal tiling is the most efficient way to tessellate the plane in terms of the total perimeter per area coverage.


Introduction
Tiling differs from covering in that the former is a family of sets without overlap whereas the latter covers the entire plane with no gaps but with overlaps (Lessard, 2000, p.17) .Triangles, squares and hexagons are known to be the only Archimedean tiling's with lattice polygon (Ding, 2010, p.7).Any regular polygon that can tile has the property of covering.It is often useful to consider the single regular polygon whose area approximates that of a circle.This regular polygon could be a guide in our geometric disks covering problem.

Related Literature
Covering has been one of the most fundamental and yet challenging issues in wireless network and found many applications such as routing and broad casting (Xu and Whang 2011, pp. 108-118).A natural dual to covering is the corresponding tilling.Tiling is a countable family of closed sets * + which covers the Euclidean plane without any gaps or overlaps, (Keating and King, 1999, pp. 83-91).Here are known as the tiles of .When the set of polygons has the same shape and size then it is a monohedral tiling.The only edge-to-edge monohedral tiling's by regular polygons are tiling of squares, equilateral triangles and regular hexagons (Lessard, 2000, p. 17).square, equilateral triangle) to that of its circumcircle.

Proof
Consider the circle with radius  inscribed in a hexagon, the other by a square and the third by a triangle as shown in Figure 1 Case I: Hexagon    Comparing the areas obtained for the three geometrical shapes we conclude that hexagons approximate circles more closely than squares, regular triangles and generally than any other regular tessellable geometrical 2-dimensional polygon.
Corollary: Hexagons, because they approximate circles more closely are more compact than squares.This fact has direct application to any point of set sensors arranged on a plane or similar surface and can be reflected in nature (e.g.most animal vision organs have rods and cones arranged in nearly hexagonal tessellations in the eyes fovea), (Raposo , 2011, pp. 37).

Non -Tessellable Regular Polygon
Let  be the area of the inscribed polygon with  sides.As  increases, it appears that  becomes closer and closer to the area of the circle.We say that the area of the circle is the limit of the areas of the inscribed polygons, and we write Theorem 2: The area of an  sided non-tessellable regular polygon inscribed in a circle is closer to its circumcircle as  increases. Proof: Non-lattice Archimedean non-tillable regular polygon includes pentagon, heptagon, octagon, nonagon, decagon, etc.We shall give a proof that as the number of sides of a non-tessellable regular polygon increases its area approximates that of it circumcircle than any regular tessellable polygon.We shall give the percentage occupying area proof using geometry by considering the following cases.
Case I: Pentagon  Case IV:  sided polygon Generally, area of polygon inscribed in a circle with sides  is  =  × (area of a single triangle with origin as one vertex) The common area of circles based on some non-tessallable regular polygons and occupying ratio in a circle has been shown in Table 2. Table 2 shows the area on an  sided non-tessellable regular polygon approaching the area of a circle as n increases.
We shall state a theorem to that effect and give the first formal analytical proof.

Remark:
The above theorem states that the area  of an  −sided regular non-tessellable polygon inscribed in a circle with radius  approximates the area  of the circle as  becomes large. Proof.
We shall give the first analytical proof.
From equation ( 2) This establishes the fact that the area of a regular polygon with sides  inscribed in a circle of radius  approaches  as  becomes large.Table 4 illustrates the percentage occupying ratio of some non-tessellable regular polygon.

Geometry of Tessellable Polygons with Disks Covering
Geometrically, we illustrate a plane tiling with equilateral triangle, squares and hexagon.Hexagon is conveniently chosen because it is the only shape that is closest to being circular with the widest area.Figure 6 shows this concepts and the percentage overlap of a segment.

Overlap Difference for Optimal Disks Covering
We deduce formulas for calculating the width of any hexagonal disks covering in terms of the apothem ( ) or the radius of the disks ( ) or the height of the overlap area ().Consider two intersecting uniform disks shown in Figure 8. = (2√3 − 3) ( 7) Equation ( 5), ( 6) and ( 7) establishes the formula for calculating the width of a hexagonal disks covering.Table 5 shows the overlap difference and their percentage occupying ratio.
Consider Figure 9. Generally for an  sided regular polygon, Theorem 5: The total overlap difference created by  sided tessellable regular polygon inscribed in a disk for covering with radius  is 2 *1 −  (  )+.Each overlap difference is 2 *1 −  (  )+. Proof.
Let  denote the overlap difference of an  sided tessellable polygon.We shall prove by induction that when  ≥ 3 theorem 5 is true.
In telecommunication network design the overlap difference  help engineers to estimate before hand the overlap cost per choice of tessellable regular polygon.As the overlap difference increase with a decrease in the size of the regular polygon.
Theorem 6: The total overlap area created by  sided tessellable regular polygon inscribed in a disks for covering of radius  is * −  (  )+  . Proof.
From Figure 3.8, the area of each overlap difference is ∆ = area of disks − 2 ×(area of tessellable regular polygon)

Analysis of Results
Table 1 shows that it is only possible to compute the width of a hexagonal disks covering if either the radius of the disks is known or the apothem of the inscribed hexagon.It is also an establish fact that  >  and as  increases the overlap difference (width -) increases.This is due to the fact that the multiplier (2 − √3) is the least as compared to (2 − √2) and 1, hence in a regular tessellable polygon the number of sides increases with a decrease in the overlap difference.Thus, as  → ∞ for regular tessellable polygon, then  → 0. Our study also reveals that the area of a non-tessellable regular polygon inscribed in a circle can be calculated using the formula  =  1 2  (  ) where  is the number of sides and  as the radius of the circle.

Discussion
Geometry of tessellable regular polygon resulted in hexagonal area of √  which approximates closely the area of circle than any other tessellable regular polygon for disk covering.This is a 17.3% reduction over the disks area of  .
L' Hopital's rule was used to established the fact that the area of regular polygon inscribed in a disk limit to the area of a circle as n increases.Hexagon has a segment overlap of 8.655% compared to 18.175% for square and 29.33% for equi-triangular tilling in disk covering.Hence hexagon has the least overlap area therefore with least material cost for disk covering.Hexagonal tiling as a guide to disk covering is proved to have the least overlap difference of (2 − √3) which is 13.4% over the diameter of the disk.This implies that regular hexagon has the minimum width and therefore is the best geometric object for optimal disk covering in a plane A formulae for apothem  =   (  ) and total overlap difference  = 2 *1 −  (  )+ for tessellable regular polygon inscribed in disks for covering were put forward.
That of the area was found to be * −  (  )+  .

Conclusions and Recommendations
The findings in this study suggest that disks covering using hexagonal tessellation offers an optimal covering area of 82.7% per disks area.We use both geometry and analytical approach to establish the fact that the area of a regular polygon approximates the area of a circle as the number of sides increases.The study also shows a formulae for computing the overlap difference and the apothem of tessellable regular polygon inscribed in disks for covering.We establish formulae for computing the total overlap area for regular tessellable polygon and it is the first study to propound these formulae as well as use both geometry and analysis to establish approximation of regular polygon to that of a circle.Geometric disk covering which is an important study in computational geometry, geometric topology (rubber sheet geometry) as well as optimization of telecommunication network design can best be achieved in least time complexity using hexagonal tessellation.. Therefore, Pure and applied Mathematicians, Computer Scientists as well as Telecommunication engineers should not lose sight of this important finding when covering with disks.

Figure 2 .
Figure 2. Pentagon inscribed in a circle

Figure 7
Figure 7(a) illustrates the hexagonal cell layout.The inradius and the circumradius of the hexagonal cell are  and  , respectively.In Figure 7(b), cells are partially overlapped because  equals to the hexagon's circumradius.The model considers nodes not belonging to the cell of interest.Algebraically the best positioning of the GSM network is where the hexagonal and circular cells overlap to give us a difference of 2( −  ) as shown in Figure 7(b).

Figure 8 .
Figure 8. Overlap width for uniform disks Consider triangle  in Figure 8.

Figure 9 .
Figure 9. Apothem for regular polygon inscribed in disks Let the apothem of an  sided regular polygon be  .Case I: Equilateral triangle  .Consider ∆ in Figure 9. Then

Table 2 .
Occupying Ratio Comparison of non-tilling Archimedean shapes.

Table 4 .
Occupying Ratio Comparison of non-tessellable regular polygon.For any positive constant , the function  ↦ () given by

Table 5 .
Occupying overlap difference and ratio for uniform disks.