# The Local Theory for Regular Systems in the Context of t-Bonded Sets

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## Abstract

**:**

## 1. Introduction

## 2. Delone Sets, Crystals and $\mathit{t}$-Bonded Sets

**Definition**

**1.**

- (i)
- (r-condition) any open ball of radius r has at most one point from X;
- (ii)
- (R-condition) any closed ball of radius R has at least one point from X.

**Remark**

**1.**

**Definition**

**2.**

**Remark**

**2.**

**Definition**

**3.**

**Definition**

**4.**

- (i)
- for any point $x\in {\mathbb{R}}^{d}$, the intersection of $F\left(G\right)$ and the orbit $G\xb7x$ is not empty;
- (ii)
- for any point $x\in {\mathbb{R}}^{d}$, the interior of $F\left(G\right)$ contains at most one point from $G\xb7x$.

**Definition**

**5.**

**Statement**

**1.**

**Definition**

**6.**

**Theorem**

**1.**

**Theorem**

**2.**

- (i)
- G is a crystallographic group by Definition 5 above.
- (ii)
- G is a space group. That is, the subgroup of all pure translations T in G is a d-dimensional lattice, which is necessary for the maximal abelian subgroup of G.
- (iii)
- There exists $x\in {\mathbb{R}}^{d}$ such that $G\xb7x$ is a Delone set of ${\mathbb{R}}^{d}$.
- (iv)
- For any $x\in {\mathbb{R}}^{d}$, the G-orbit $G\xb7x$ is a Delone set in ${\mathbb{R}}^{d}$.

**Statement**

**2.**

**Definition**

**7.**

**Definition**

**8.**

**Theorem**

**3.**

## 3. Clusters in $\mathit{t}$-Bonded and Delone Sets

**Definition**

**9.**

**Statement**

**3.**

**Example**

**1.**

**Theorem**

**4.**

**Theorem**

**5.**

**Definition**

**10.**

**Definition**

**11.**

**Definition**

**12.**

**Definition**

**13.**

**Lemma**

**1.**

- (1)
- If aff${C}_{x}\left(\rho \right)={\mathrm{\Pi}}_{x}^{n}$ and $n<d$, then ${S}_{x}\left(\rho \right)={\overline{S}}_{x}\left(\rho \right)\oplus O(x,d-n)$, where ${\overline{S}}_{x}\left(\rho \right)\subset O(x,n)$, and $O(x,d-n)$ is the group of isometries of the orthogonal complement ${Q}_{x}^{d-n}$ that leave point x fixed.
- (2)
- The group ${\overline{S}}_{x}\left(\rho \right)$ is a finite subgroup of $O(x,n)$. Particularly, if aff${C}_{x}\left(\rho \right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{\mathbb{R}}^{d}$, then group ${S}_{x}\left(\rho \right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{\overline{S}}_{x}\left(\rho \right)$ is a finite subgroup of $O(x,d)$.
- (3)
- The group ${S}_{x}\left(\rho \right)$ is finite if and only if the rank of ${C}_{x}\left(\rho \right)$ is equal to d or $d-1$.

## 4. Local Criterion for Regular $\mathit{t}$-Bonded Systems

**Theorem**

**6.**

**Proof.**

**Definition**

**14.**

- (1)
- if for a t-bonded set X in ${\mathbb{R}}^{d}$$N\left({\widehat{\rho}}_{d}\right)=1$, then X is a regular system;
- (2)
- for any $0<{\rho}^{\prime}<{\widehat{\rho}}_{d}$, there is a t-bonded set Y in ${\mathbb{R}}^{d}$ with $N\left({\rho}^{\prime}\right)=1$, which is not a regular system.

**Theorem**

**7.**

- (1)
- $N({\rho}_{0}+t)=1$;
- (2)
- for some point ${x}_{0}\in X$ ${S}_{{x}_{0}}\left({\rho}_{0}\right)={S}_{{x}_{0}}({\rho}_{0}+t)$.

**Theorem**

**8.**

- (1)
- $N({\rho}_{0}+2R)=1$;
- (2)
- ${S}_{x}\left({\rho}_{0}\right)={S}_{x}({\rho}_{0}+2R)$.

**Corollary**

**1.**

**Corollary**

**2.**

**Definition**

**15.**

**Theorem**

**9.**

**Theorem**

**10.**

**Theorem**

**11.**

**Theorem**

**12.**

## 5. $\mathit{t}$-Bonded and Delone Sets in ${\mathbb{R}}^{\mathbf{2}}$ and ${\mathbb{R}}^{\mathbf{3}}$

**Statement**

**4.**

**Proof.**

**Lemma**

**2.**

**Lemma**

**3.**

- Then, X is either a non-collinear set for which $q\le 6$ or a collinear set.
- If X is collinear, it is a regular system of three types:
- X is a two-point set;
- X is a lattice on a line;
- X is a bi-lattice, i.e., the union of two congruent and parallel lattices on a line.

**Definition**

**16.**

**Remark**

**3.**

**Remark**

**4.**

**Theorem**

**13.**

**Theorem**

**14.**

## 6. Local Criteria for Multiregular $\mathit{t}$-Bonded Systems and Crystals

**Definition**

**17.**

**Theorem**

**15.**

- (1)
- $N\left({\rho}_{0}\right)=N({\rho}_{0}+t)=m;$
- (2)
- ${S}_{x}\left({\rho}_{0}\right)={S}_{x}({\rho}_{0}+t),\forall x\in X.$

- (i)
- g is a symmetry of the entire set X;
- (ii)
- For any $i\in [1,m]$, the group ${G}_{i}$ generated by all symmetries g of X with the conditions above acts transitively on every set ${X}_{j}$, $j\in [1,m];$
- (iii)
- ${G}_{i}$ does not depend on i, i.e., ${G}_{i}=\mathrm{Sym}\left(\mathrm{X}\right)$.

**Proof.**

**Theorem**

**16.**

- (1)
- $N\left({\rho}_{0}\right)=N({\rho}_{0}+t)=m;$
- (2)
- ${S}_{x}\left({\rho}_{0}\right)={S}_{x}({\rho}_{0}+t),\forall x\in X.$

**Theorem**

**17.**

- (1)
- $N\left({\rho}_{0}\right)=N({\rho}_{0}+2R)=m;$
- (2)
- ${S}_{x}\left({\rho}_{0}\right)={S}_{x}({\rho}_{0}+2R),\forall x\in X.$

## 7. On Delone Sets of Finite Type and $\mathit{t}$-Bonded Sets of Infinite Type

**Statement**

**5.**

**Theorem**

**18.**

**Proof.**

- ${x}_{i}=(ia/2,0)$, for all $i\in 2\mathbb{Z}$ (i.e., i is even) and
- ${x}_{i}=(ia/2,-{\theta}_{1})$, for all $i\in 2\mathbb{Z}+1$ (i.e., i is odd) and ${\theta}_{1}>0$.

- ${y}_{i}=(ib/2,kt)$, for all $i\in 2\mathbb{Z}$ and
- ${y}_{i}=(ib/2,kt+{\theta}_{2})$ for all $i\in 2\mathbb{Z}+1$, ${\theta}_{2}>0$, $i\in Z$.

- ${z}_{i}=(0,ci/2)$ if i is even, where, we note, ${z}_{0}={x}_{0}$ and ${z}_{2n}={y}_{0}$;
- ${z}_{i}=({\theta}_{3},ci/2)$ if i is odd.

## 8. Summary

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Feynmann, R.; Leighton, R.; Sands, M. The Feynmann Lectures on Physics; Addison-Wesley: London, UK, 1963; Volume 2. [Google Scholar]
- Delone, B.N.; Dolbilin, N.P.; Stogrin, M.I.; Galiuilin, R.V. A local criterion for regularity of a system of points. Soviet Math. Dokl.
**1976**, 17, 319–322. [Google Scholar] - Dolbilin, N.P. On Local Properties for Discrete Regular Systems. Sov. Math. Dokl.
**1976**, 230, 516–519. [Google Scholar] - Dolbilin, N.P.; Stogrin, M.I. A local criterion for a crystal structure. In Proceedings of the IXth All-Union Geometrical Conference, Chisinau, Moldova, 20–22 September 1988; p. 99. (In Russian). [Google Scholar]
- Stogrin, M.I. On the upper bound for the order of axis of a star in a locally regular Delone set, Geometry, Topology, Algebra and Number Theory, Applications. In Proceedings of the International Conference Dedicated to the 120th Anniversary of Boris Nikolaevich Delone (1890–1980), Moscow, Russia, 16–20 August 2010; Abstracts. Steklov Mathematical Institute: Moscow, Russia, 2010; pp. 168–169. (In Russian). [Google Scholar]
- Dolbilin, N.P.; Lagarias, J.C.; Senechal, M. Multiregular point systems. Discret Comput. Geometry
**1998**, 20, 477–498. [Google Scholar] [CrossRef] - Lagarias, J. Geometric Models for Quasicrystals I. Delone Sets of Finite Type. Discret Comput. Geometry
**1999**, 21, 161–191. [Google Scholar] [CrossRef] - Fedorov, E.S. Elements of the Study of Figures. Zap. Mineralog. Obsc.
**1985**, 21, 1–279. [Google Scholar] - Delaunay, B. Sur la sphère vide. A la mémoire de Georges Voronoï. Bull. Acad. Sci. l’URSS
**1934**, 6, 793–800. [Google Scholar] - Delone, B.N. Geometry of positive quadratic forms. Uspekhi Matematicheskikh Nauk
**1937**, 3, 16–62. (In Russian) [Google Scholar] - Hilbert, D.; Cohn-Vossen, S. Geometry and Imagination; AMS Chelsea Pub.: Providence, RI, USA, 1999; p. 367. [Google Scholar]
- Charlap, L. Bieberbach Groups and Flat Manifolds; Springer: New York, NY, USA, 1986. [Google Scholar]
- Schoenflies, A. Kristallsysteme und Kristallstruktur; Druck und verlag von BG Teubner: Leipzig, Germany, 1891. [Google Scholar]
- Milnor, J. Hilbert’s problem 18: On crystallographic groups, fundamental domains, and on sphere packing, in: Mathematical Developments Arising From Hilbert Problems. In Proceedings of the Symposia in Pure Mathematics; American Math. Soc.: Providence, Rhode Island, 1976; Volume 28, pp. 491–506. [Google Scholar]
- Bieberbach, L. Ueber die Bewegungsgruppen des n-dimensionalen Euklidischen Raumes I. Math. Ann.
**1911**, 70, 207–336. [Google Scholar] [CrossRef] - Dolbilin, N.P. A Criterion for crystal and locally antipodal Delone sets. Vestnik Chelyabinskogo Gos Universiteta
**2015**, 3, 6–17. (In Russian) [Google Scholar] - Dolbilin, N.P.; Magazinov, A.N. Locally antipodal Delauney Sets. Russ. Math. Surv.
**2015**, 70, 958–960. [Google Scholar] [CrossRef] - Dolbilin, N.P.; Magazinov, A.N. The Uniqueness Theorem for Locally Antipodal Delone Sets. Proc. Steklov Inst. Math.
**2016**, 294, 215–221. [Google Scholar] [CrossRef] - Bouniaev, M.; Dolbilin, N. Regular and Multiregular t-bonded Systems. J. Inf. Process. Jpn.
**2017**, 25, 735–740. [Google Scholar] - Dolbilin, N. Delone Sets in ${\mathbb{R}}^{3}$: The Regularity Conditions. Fundam. Prikl. Mat.
**2016**, 21. An English translation will appear in the English version of ‘Journal of Mathematical Sciences’ in 2018. (In Russian) [Google Scholar] - Baburin, I.; Bouniaev, M.; Dolbilin, N.; Erochovets, N.; Garber, A.; Krivovichev, S.; Schulte, E. A Lower Bound for the Regularity Radius of Delone Sets. Acta Cryst.
**2017**. submitted. [Google Scholar] - Bouniaev, M.M.; Dolbilin, N.P. Regular t-bonded Systems in ${\mathbb{R}}^{3}$. Eur. J. Comb. (Mem. Michele Deza)
**2018**, in press. [Google Scholar] - Mikhail, M.; Bouniaev, N.P. PDolbilin, Local Theory of Crystals: Development and Current Status. In Proceedings of the 4-th Annual International Conference on Computational Mathematics, Computational Geometry and Statistics, Singapore, 26–27 January 2015; pp. 39–45. [Google Scholar]
- Dolbilin, N. Delone Sets: Local Identity and Global Symmetry. In Discrete Geometry and Symmetry; Memory Karoly Bezdek’s and Egon Schulte’s 60th Birthdays; Proceedings in Mathematics and Statistics, (to Appear); Conder, M.D.E., Deza, A., Ivich, A., Eds.; Springer: Weiss, UK, 2018. [Google Scholar]

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**MDPI and ACS Style**

Bouniaev, M.; Dolbilin, N.
The Local Theory for Regular Systems in the Context of *t*-Bonded Sets. *Symmetry* **2018**, *10*, 159.
https://doi.org/10.3390/sym10050159

**AMA Style**

Bouniaev M, Dolbilin N.
The Local Theory for Regular Systems in the Context of *t*-Bonded Sets. *Symmetry*. 2018; 10(5):159.
https://doi.org/10.3390/sym10050159

**Chicago/Turabian Style**

Bouniaev, Mikhail, and Nikolay Dolbilin.
2018. "The Local Theory for Regular Systems in the Context of *t*-Bonded Sets" *Symmetry* 10, no. 5: 159.
https://doi.org/10.3390/sym10050159