Advertisement

Journal of Applied Mathematics and Computing

, Volume 59, Issue 1–2, pp 661–676 | Cite as

A note on the number of vertices of the Archimedean tiling

  • Xianglin WeiEmail author
  • Weiqi Wang
Original Research
  • 45 Downloads

Abstract

There are 11 Archimedean tilings in \(\mathbb {R}^{2}\). Let E(n) denote the ellipse of short half axis length n\((n\in \mathbb {Z}^{+})\) centered at an arbitrary vertex of the Archimedean tiling by regular polygons of edge length 1, and let \(\mathcal {N}(E(n))\) denote the number of vertices of the Archimedean tiling that lie inside or on the boundary of E(n). In this paper, we present an algorithm to calculate the number \(\mathcal {N}(E(n))\), and get a unified formula \(\displaystyle \lim _{n\rightarrow \infty }\frac{\mathcal {N}(E(n))}{n^{2}}=m\cdot \frac{\pi }{S}\), where S is the area of the central polygon, and m is the ratio of long half axis length and short half axis length of the ellipse. Let \(\mathcal {C}\) be a cube-tiling by cubes of edge length 1 in \(\mathbb {R}^{3}\), and the vertex of cube-tiling is called a C-point. Let S(n) denote the sphere of radius \(n(n\in \mathbb {Z}^{+})\) centered at an arbitrary C-point, and let \(\mathcal {N}_{C}(S(n))\) denote the number of C-points that lie inside or on the surface of S(n). In this paper, we present an algorithm to calculate the number \(\mathcal {N}_{C}(S(n))\) and get a formula \(\displaystyle \lim _{n\rightarrow \infty }\frac{\mathcal {N}_{C}(S(n))}{n^{3}}=\frac{4\pi }{3V}\), where V is the volume of the cube.

Keywords

Discrete geometry Cube-tiling Archimedean tiling Central polygon 

Mathematics Subject Classification

52C20 52A10 

References

  1. 1.
    Olds, C., Lax, A., Davidoff, G.: The Geometry of Numbers. Mathematical Association of America, Washington (2000)zbMATHGoogle Scholar
  2. 2.
    Wei, X., Ding, R.: \(H\)-triangles with \(k\) interior \(H\)-points. Discrete Math. 308, 6015–6021 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ding, R., Reay, J., Zhang, J.: Areas of generalized \(H\)-polygons. J. Comb. Theory Ser. A 77, 304–317 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Rabinowitz, S.: On the number of lattice points inside a convex lattice \(n\)-gon. Congressus Numerantium 73, 99–124 (1990)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Kolodziejczyk, K.: Realizable quadruples for Hex-polygons. Graph Comb. 23, 61–72 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cao, P., Yuan, L.: The number of \(H\)-points in a circle. ARS Comb. 97A, 311–318 (2010)MathSciNetzbMATHGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.College of ScienceHebei University of Science and TechnologyShijiazhuangChina

Personalised recommendations