# On Three Constructions of Nanotori

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## Abstract

**:**

## 1. Introduction

## 2. Three Constructions of Nanotori

#### 2.1. The Four Parameters Construction, $M(a,b,c,d)$

**Proposition**

**1.**

**Proof.**

#### 2.2. Altshuler’s Construction, ${M}^{*}(r,n,{m}^{*})$

**Proposition**

**2.**

**Observation**

**1.**

#### 2.3. Generalized Honeycomb Torus Construction, $\phantom{\rule{3.33333pt}{0ex}}\mathrm{GHT}(a,b,c)$

- (i)
- $i=k$ and $j={(\ell \pm 1)}_{b}$;
- (ii)
- $j=\ell $, $k=i-1$, $i+j$ is even, and $i\ge 1$;
- (iii)
- $i=0,\phantom{\rule{0.166667em}{0ex}}k=a-1$, and $\ell ={(j+c)}_{b}$ and j is even.

**Proposition**

**3.**

- –
- for $a=1$, b even, and $c\in \{1,b-1\}$; or
- –
- for $a\in N$, $b=2$ and $c\in \{0,1\}$.

## 3. Relation between the Constructions

#### 3.1. Relation between $\phantom{\rule{3.33333pt}{0ex}}\mathrm{GHT}$ and ${M}^{*}$

**Proposition**

**4.**

**Proof.**

- (i*)
- $i=k$ and $j=\ell \pm 1$; and
- (ii*)
- $j=\ell $, and $k=i-1$ and $i+j$ is even.

**Proposition**

**5.**

#### 3.2. Relation between Constructions M and ${M}^{*}$

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

#### 3.3. Relation between $\phantom{\rule{3.33333pt}{0ex}}\mathrm{GHT}$ and M

**Proposition**

**8.**

**Proposition**

**9.**

**Proposition**

**10.**

**Proposition**

**11.**

## 4. Isomorphisms of Nanotori through Construction M

**Proposition**

**12.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**2.**

**Theorem**

**3.**

- 1:
- $({a}^{\prime},{b}^{\prime})={x}_{1}(a,b)+{y}_{1}(c,d)$ and $({c}^{\prime},{d}^{\prime})={x}_{2}(a,b)+{y}_{2}(c,d)$;
- 2:
- $({a}^{\prime},{b}^{\prime})={x}_{1}(-b,a+b)+{y}_{1}(-d,c+d)$ and $({c}^{\prime},{d}^{\prime})={x}_{2}(-b,a+b)+{y}_{2}(-d,c+d)$;
- 3:
- $({a}^{\prime},{b}^{\prime})={x}_{1}(-a-b,a)+{y}_{1}(-c-d,c)$ and $({c}^{\prime},{d}^{\prime})={x}_{2}(-a-b,a)+{y}_{2}(-c-d,c)$;
- 4:
- $({a}^{\prime},{b}^{\prime})={x}_{1}(a+b,-b)+{y}_{1}(c+d,-d)$ and $({c}^{\prime},{d}^{\prime})={x}_{2}(a+b,-b)+{y}_{2}(c+d,-d)$;
- 5:
- $({a}^{\prime},{b}^{\prime})={x}_{1}(b,a)+{y}_{1}(d,c)$ and $({c}^{\prime},{d}^{\prime})={x}_{2}(b,a)+{y}_{2}(d,c)$;
- 6:
- $({a}^{\prime},{b}^{\prime})={x}_{1}(-a,a+b)+{y}_{1}(-c,c+d)$ and $({c}^{\prime},{d}^{\prime})={x}_{2}(-a,a+b)+{y}_{2}(-c,c+d)$.

**Corollary**

**1.**

#### Redundancy of the Parameters of $M(a,b,c,d)$

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

- ${m}_{1}\equiv {x}_{1}a+{y}_{1}c\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}{n}_{1})$, where ${x}_{1},{y}_{1}$ are integer solutions of ${x}_{1}b+{y}_{1}d={r}_{1}$;
- ${m}_{2}\equiv -{x}_{2}b-{y}_{2}d\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}{n}_{2})$, where ${x}_{2},{y}_{2}$ are integer solutions of ${x}_{2}(a+b)+{y}_{2}(c+d)={r}_{2}$;
- ${m}_{3}\equiv {x}_{3}(-a-b)+{y}_{3}(-c-d)\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}{n}_{3})$, where ${x}_{3},{y}_{3}$ are integer solutions of ${x}_{3}a+{y}_{3}c={r}_{3}$.

**Corollary**

**4.**

**Corollary**

**5.**

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**Construction of a nanotorus ${M}^{*}(3,4,1)$. Points A, ${B}^{*}$, and ${E}^{*}$ coincide. Blue segments form a hexagon on ${M}^{*}(3,4,1)$.

**Figure 4.**(

**a**) Normal cycle in $M(3,1,-1,3)$, $r=1$. (

**b**) Normal cycles in $M(3,-2,-1,4)$, $r=2$. The characteristic parallelogram for $M(3,-2,-1,4)$ is obtained by rotation of the characteristic parallelogram $M(3,1,-1,3)$ for 60${}^{\circ}$.

**Figure 7.**Characteristic parallelogram (gray) for $M(3,1,-1,3)$ and characteristic parallelogram (green) for $M(10,0,3,1)$. The nanotori they define are isomorphic.

**Figure 8.**Characteristic parallelogram (gray) for $M(3,-2,-1,4)$ and characteristic parallelogram (green) for $M(5,0,2,2)$. These parallelograms define isomorphic nanotori.

**Figure 9.**The gray parallelogram, $ABCD$ is defining $M(3,1,-1,3)$ nanotori. The green parallelogram ${A}^{\prime}{B}^{\prime}{C}^{\prime}{D}^{\prime}$, in (

**a**) is obtained by reflection and in (

**b**) by rotation. (

**a**) $\rho \left(M\right(3,1,-1,3\left)\right)=M(-1,4,-3,2)$; (

**b**) $\sigma \left(M\right(3,1,-1,3\left)\right)=M(4,-1,2,-3)$.

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Andova, V.; Dimovski, P.; Knor, M.; Škrekovski, R.
On Three Constructions of Nanotori. *Mathematics* **2020**, *8*, 2036.
https://doi.org/10.3390/math8112036

**AMA Style**

Andova V, Dimovski P, Knor M, Škrekovski R.
On Three Constructions of Nanotori. *Mathematics*. 2020; 8(11):2036.
https://doi.org/10.3390/math8112036

**Chicago/Turabian Style**

Andova, Vesna, Pavel Dimovski, Martin Knor, and Riste Škrekovski.
2020. "On Three Constructions of Nanotori" *Mathematics* 8, no. 11: 2036.
https://doi.org/10.3390/math8112036