Journal of Applied Mathematics and Computing

, Volume 59, Issue 1–2, pp 661–676

# A note on the number of vertices of the Archimedean tiling

• Xianglin Wei
• Weiqi Wang
Original Research

## 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

52C20 52A10

## References

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