# Fixed Point Theory for Digital k-Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces

## Abstract

**:**

^{n}− 1)-neighborhood; digital topology

## 1. Introduction

- (Q1)
- How to establish a geometric realization of an ${S}_{k}$?
- (Q2)
- Does the geometric realization transform an ${S}_{k}$ into a certain spherical (or a sphere-like) polyhedron in ${\mathbb{R}}^{3}$?
- (Q3)
- How to define the Euler characteristic of an ${S}_{k}$?
- (Q4)
- Are there certain relationships between the Euler characteristic of an ${S}_{k}$ and that of a geometric realization of an ${S}_{k}$?
- (Q5)
- What about the FPP or the AFPP for an ${S}_{k}$?

## 2. Basic Notions Related to Digital $\mathit{k}$-Surfaces and a Connected Sum for $\mathit{k}$-Surfaces

- We say that two subsets $(A,k)$ and $(B,k)$ of $(X,k)$ are k-adjacent if $A\cap B=\varnothing $ and that there are points $a\in A$ and $b\in B$ such that a and b are k-adjacent [17]. In particular, in case B is a singleton, say $B=\{x\}$, we say that A is k-adjacent to x.
- For a k-adjacency relation of ${\mathbb{Z}}^{n}$, a k-path with $l+1$ elements in ${\mathbb{Z}}^{n}$ is assumed to be a finite sequence ${({x}_{i})}_{i\in {[0,l]}_{\mathbb{Z}}}\subset {\mathbb{Z}}^{n}$ such that ${x}_{i}$ and ${x}_{j}$ are k-adjacent if $|\phantom{\rule{0.166667em}{0ex}}i-j\phantom{\rule{0.166667em}{0ex}}|=1$ [17].
- A digital image $(X,k)$ is said to be k-connected if, for any distinct points such as $x,y$ in $(X,k)$, there is a k-path ${({x}_{i})}_{i\in {[0,l]}_{\mathbb{Z}}}\subset X$ such that $x={x}_{0}$ and $y={x}_{l}$.
- For a digital image $(X,k)$, the k-component of $x\in X$ is defined to be the largest k-connected subset of $(X,k)$ containing the point x.
- We say that a simple k-path is a finite set ${({x}_{i})}_{i\in {[0,m]}_{\mathbb{Z}}}\subset {\mathbb{Z}}^{n}$ such that ${x}_{i}$ and ${x}_{j}$ are k-adjacent if and only if $|\phantom{\rule{0.166667em}{0ex}}i-j\phantom{\rule{0.166667em}{0ex}}|=1$ [17]. In the cases ${x}_{0}=x$ and ${x}_{m}=y$, we denote the length of the simple k-path with ${l}_{k}(x,y):=m$.
- A simple closed k-curve (or simple k-cycle) with l elements in ${\mathbb{Z}}^{n}$ [10], denoted by $S{C}_{k}^{n,l},l\ge 4,l\in {\mathbb{N}}_{0}\backslash \{2\}$, ${\mathbb{N}}_{0}$ is the set of even natural numbers [10,17] and is the finite set ${({x}_{i})}_{i\in {[0,l-1]}_{\mathbb{Z}}}$ such that ${x}_{i}$ and ${x}_{j}$ are k-adjacent if and only if $|\phantom{\rule{0.166667em}{0ex}}i-j\phantom{\rule{0.166667em}{0ex}}|=\pm 1(mod\phantom{\rule{0.166667em}{0ex}}l)$ [10].
- For a digital image $(X,k)$, a digital k-neighborhood of ${x}_{0}\in X$ with radius $\epsilon $ is defined in X as the following subset [10] of X:$${N}_{k}({x}_{0},\epsilon ):=\{x\in X\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{l}_{k}({x}_{0},x)\le \epsilon \}\cup \{{x}_{0}\},\phantom{\rule{2.em}{0ex}}$$$${N}_{k}(x,1)={N}_{k}(x)\cap X.\phantom{\rule{2.em}{0ex}}$$
- Rosenfeld [20] defined the notion of digital continuity of a map $f:(X,{k}_{0})\to (Y,{k}_{1})$ by saying that f maps every ${k}_{0}$-connected subset of $(X,{k}_{0})$ into a ${k}_{1}$-connected subset of $(Y,{k}_{1})$.

**Proposition**

**1**

**.**Let $(X,{k}_{0})$ and $(Y,{k}_{1})$ be digital images in ${\mathbb{Z}}^{{n}_{0}}$ and ${\mathbb{Z}}^{{n}_{1}}$, respectively. A function $f:(X,{k}_{0})\to (Y,{k}_{1})$ is (digitally) $({k}_{0},{k}_{1})$-continuous if and only if, for every $x\in X$, $f({N}_{{k}_{0}}(x,1))\subset {N}_{{k}_{1}}(f(x),1)$.

**Definition**

**1**

**.**Consider two digital images $(X,{k}_{0})$ and $(Y,{k}_{1})$ in ${\mathbb{Z}}^{{n}_{0}}$ and ${\mathbb{Z}}^{{n}_{1}}$, respectively. Then, a map $h:X\to Y$ is called a $({k}_{0},{k}_{1})$-isomorphism if h is a $({k}_{0},{k}_{1})$-continuous bijection, and further, ${h}^{-1}:Y\to X$ is $({k}_{1},{k}_{0})$-continuous. Then, we use the notation $X{\approx}_{({k}_{0},{k}_{1})}Y$. Moreover, in the case ${k}_{0}={k}_{1}:=k$, we use the notation $X{\approx}_{k}Y$.

**Definition**

**2**

**.**Let ${c}^{*}:=({x}_{0},{x}_{1},\cdots ,{x}_{n})$ be a closed k-curve in $({\mathbb{Z}}^{2},k,\overline{k},{c}^{*})$. A point x of $\overline{{c}^{*}}$, the complement of ${c}^{*}$ in ${\mathbb{Z}}^{2}$, is said to be interior to ${c}^{*}$ if it belongs to the bounded $\overline{k}$-connected component of $\overline{{c}^{*}}$.

- $(\star )$
- $MS{C}_{8}^{*}:=MS{C}_{8}\cup Int(MS{C}_{8})$ [4], where $MS{C}_{8}$ is a digital image 8-isomorphic to the digital image, i.e., $MS{C}_{8}:=S{C}_{8}^{2,6}:=\{{c}_{0}=(0,0),{c}_{1}=(1,1),{c}_{2}=(1,2),{c}_{3}=(0,3),{c}_{4}=(-1,2),{c}_{5}=(-1,1)\}$;
- $(\star )$
- $MS{C}_{4}^{*}:=MS{C}_{4}\cup Int(MS{C}_{4})$ [4], where $MS{C}_{4}$ is a digital image 4-isomorphic to the digital image, i.e., $MS{C}_{4}:=S{C}_{4}^{2,8}:=\{{v}_{0}=(0,0),{v}_{1}=(1,0),{v}_{2}=(2,0),{v}_{3}=(2,1),{v}_{4}=(2,2),{v}_{5}=(1,2),{v}_{6}=(0,2),{v}_{7}=(0,1)\}$; and
- $(\star )$
- $MS{C}_{8}^{\prime *}:=MS{C}_{8}^{\prime}\cup Int(MS{C}_{8}^{\prime})$ [4], where $MS{C}_{8}^{\prime}$ is a digital image 8-isomorphic to the digital image, i.e., $MS{C}_{8}^{\prime}:=S{C}_{8}^{2,4}:=\{{w}_{0}=(0,0),{w}_{1}=(1,1),{w}_{2}=(0,2),{w}_{3}=(-1,1)\}$.

**Definition**

**3**

**.**Let $((X,A),{k}_{0})$ and $(Y,{k}_{1})$ be a digital image pair and a digital image, respectively. Let $f,g:X\to Y$ be $({k}_{0},{k}_{1})$-continuous functions. Suppose there exist $m\in \mathbb{N}$ and a function $H:X\times {[0,m]}_{\mathbb{Z}}\to Y$ such that

- for all $x\in X,H(x,0)=f(x)$ and $H(x,m)=g(x)$;
- for all $x\in X$, the induced function ${H}_{x}:{[0,m]}_{\mathbb{Z}}\to Y$ given by ${H}_{x}(t)=H(x,t)$ for all $t\in {[0,m]}_{\mathbb{Z}}$ is $(2,{k}_{1})$-continuous;
- for all $t\in {[0,m]}_{\mathbb{Z}}$, the induced function ${H}_{t}:X\to Y$ given by ${H}_{t}(x)=H(x,t)$ for all $x\in X$ is $({k}_{0},{k}_{1})$-continuous. Then, we say that H is a $({k}_{0},{k}_{1})$-homotopy between f and g [24].
- Furthermore, for all $t\in {[0,m]}_{\mathbb{Z}}$, assume that the induced map ${H}_{t}$ on A is a constant which follows the prescribed function from A to Y. To be precise, ${H}_{t}(x)=f(x)=g(x)$ for all $x\in A$ and for all $t\in {[0,m]}_{\mathbb{Z}}$.Then, we call H a $({k}_{0},{k}_{1})$-homotopy relative to A between f and g and we say that f and g are $({k}_{0},{k}_{1})$-homotopic relative to A in Y, $f{\simeq}_{({k}_{0},{k}_{1})relA}g$ in symbols.

**Definition**

**4**

**.**For two digital images $(X,k)$ and $(Y,k)$ in ${\mathbb{Z}}^{n}$, if there are k-continuous maps $h:X\to Y$ and $l:Y\to X$ such that the composite $l\circ h$ is k-homotopic to ${1}_{X}$ and the composite $h\circ l$ is k-homotopic to ${1}_{Y}$, then the map $h:X\to Y$ is called a k-homotopy equivalence and is denoted by $X{\simeq}_{k\xb7h\xb7e}Y$. Moreover, we say that $(X,k)$ is k-homotopy equivalent to $(Y,k)$.

**Definition**

**5**

- In case X is pointed k-contractible, the k-fundamental group ${\pi}^{k}(X,{x}_{0})$ is trivial [24].

**Definition**

**6**

**.**Let $(X,k)$ be a digital image in ${\mathbb{Z}}^{3}$, and $\overline{X}:={\mathbb{Z}}^{3}\backslash X$. Then, X is called a closed k-surface if it satisfies the following:

- (1)
- In case $(k,\overline{k})\in \{(26,6),(6,26)\}$,
- (a)
- for each point $x\in X$, ${\left|X\right|}^{x}$ has exactly one k-component k-adjacent to x;
- (b)
- $|\overline{X}{|}^{x}$ has exactly two $\overline{k}$-components $\overline{k}$-adjacent to x; we denote by ${C}^{x\phantom{\rule{0.166667em}{0ex}}x}$ and ${D}^{x\phantom{\rule{0.166667em}{0ex}}x}$ these two components; and
- (c)
- for any point $y\in {N}_{k}(x)\cap X$ (or ${N}_{k}(x,1)$ in $(X,k)$), ${N}_{\overline{k}}(y)\cap {C}^{x\phantom{\rule{0.166667em}{0ex}}x}\ne \varphi $ and ${N}_{\overline{k}}(y)\cap {D}^{x\phantom{\rule{0.166667em}{0ex}}x}\ne \varphi $. Furthermore, if a closed k-surface X does not have a simple k-point, then X is called simple.

- (2)
- In case $(k,\overline{k})=(18,6)$,
- (a)
- X is k-connected,
- (b)
- for each point $x\in X$, ${\left|X\right|}^{x}$ is a generalized simple closed k-curve. Furthermore, if the image ${\left|X\right|}^{x}$ is a simple closed k-curve, then the closed k-surface X is called simple.

**Definition**

**7**

**.**In ${\mathbb{Z}}^{3}$, let ${S}_{{k}_{0}}$ (resp. ${S}_{{k}_{1}}$) be a closed ${k}_{0}$-(resp. a closed ${k}_{1}$-)surface, where ${k}_{0}={k}_{1}\in \{6,18,26\}$.

- Consider ${A}_{{k}_{0}}^{\prime}\subset {A}_{{k}_{0}}\subset {S}_{{k}_{0}}$ and take ${A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime}\subset {S}_{{k}_{0}}$, where ${A}_{{k}_{0}}{\approx}_{({k}_{0},4)}MS{C}_{4}^{*}$, ${A}_{{k}_{0}}{\approx}_{({k}_{0},8)}MS{C}_{8}^{*}$, or ${A}_{{k}_{0}}{\approx}_{({k}_{0},8)}MS{C}_{8}^{\prime *}$ and, further, ${A}_{{k}_{0}}^{\prime}{\approx}_{({k}_{0},4)}Int(MS{C}_{4})$, ${A}_{{k}_{0}}^{\prime}{\approx}_{({k}_{0},8)}Int(MS{C}_{8})$, or ${A}_{{k}_{0}}^{\prime}{\approx}_{({k}_{0},8)}Int(MS{C}_{8}^{\prime})$, respectively.
- Let $f:{A}_{{k}_{0}}\to f({A}_{{k}_{0}})\subset {S}_{{k}_{1}}^{\prime}$ be a $({k}_{0},{k}_{1})$-isomorphism. Remove ${A}_{{k}_{0}}^{\prime}$ and $f({A}_{{k}_{0}}^{\prime})$ from ${S}_{{k}_{0}}$ and ${S}_{{k}_{1}}$, respectively.
- Identify ${A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime}$ and $f({A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime})$ by using the $({k}_{0},{k}_{1})$-isomorphism f. Then, the quotient space ${S}_{{k}_{0}}^{\prime}\cup {S}_{{k}_{1}}^{\prime}/\sim $ is obtained by $i(x)\sim f(x)\in {S}_{{k}_{1}}^{\prime}$ for $x\in {A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime}$ and is denoted by ${S}_{{k}_{0}}\u266f{S}_{{k}_{1}}$, where ${S}_{{k}_{0}}^{\prime}={S}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime}$, ${S}_{{k}_{1}}^{\prime}={S}_{{k}_{1}}\backslash f({A}_{{k}_{0}}^{\prime})$, and the map $i:{A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime}\to {S}_{{k}_{0}}^{\prime}$ is the inclusion map.

**Remark**

**1**

**.**In the quotient space ${S}_{{k}_{0}}\u266f{S}_{{k}_{1}}:={S}_{{k}_{0}}^{\prime}\cup {S}_{{k}_{1}}^{\prime}/\sim $, the subsets ${S}_{{k}_{0}}^{\prime}\backslash ({A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime})$ and ${S}_{{k}_{1}}^{\prime}\backslash f({A}_{{k}_{0}}\backslash {A}_{{k}_{0}}^{\prime})$ in ${S}_{{k}_{0}}\u266f{S}_{{k}_{1}}$ are assumed to be disjoint and are not k-adjacent, where ${k}_{0}={k}_{1}:=k$. Then, the digital image $({S}_{{k}_{0}}\u266f{S}_{{k}_{1}},k)$ is called a (digital) connected sum of ${S}_{{k}_{0}}$ and ${S}_{{k}_{1}}$.

**Lemma**

**1.**

**Proof.**

- (1)
- for all $x\in MS{S}_{18}^{\prime},H(x,0)={1}_{MS{S}_{18}^{\prime}}$ as an identity map on the set $MS{S}_{18}^{\prime}$, say ${1}_{MS{S}_{18}^{\prime}}$, and $H(x,2)={C}_{\{{e}_{5}\}}$ as the constant map at the set $\{{e}_{5}\}$
- (2)
- for all $x\in MS{S}_{18}^{\prime}$, the induced function ${H}_{x}:{[0,2]}_{\mathbb{Z}}\to MS{S}_{18}^{\prime}$ given by ${H}_{x}(t)=H(x,t)$ for all $t\in {[0,2]}_{\mathbb{Z}}$ is $(2,18)$-continuous;
- (3)
- for all $t\in {[0,2]}_{\mathbb{Z}}$, the induced function ${H}_{t}:MS{S}_{18}^{\prime}\to MS{S}_{18}^{\prime}$ given by ${H}_{t}(x)=H(x,t)$ for all $x\in MS{S}_{18}^{\prime}$ is 18-continuous.Thus, we obtain H which is an 18-homotopy between ${1}_{MS{S}_{18}^{\prime}}$ and ${C}_{\{{e}_{5}\}}$.
- (4)
- Furthermore, for all $t\in {[0,2]}_{\mathbb{Z}}$, assume that the induced map ${H}_{t}$ on $\{{e}_{5}\}$ is a constant.

- Then, $MS{S}_{18}$ is indeed pointed 18-contractible (correction of the “non-18-contractibility” of $MS{S}_{18}$ in Theorem 4.3(3) of Reference [3] and Theorem 4.2(3) of Reference [4]). Moreover, it is proved to be a simple closed 18-surface (see Figure 1a) [3,4]. Using a method similar to the 18-homotopy of Equation (8), we observe that there is indeed an 18-homotopy relative to the set $\{{c}_{9}\}$ between ${1}_{MS{S}_{18}}$ and ${C}_{\{{c}_{9}\}}$, which is the constant map at $\{{c}_{9}\}$ (see Figure 2b),$$H:MS{S}_{18}\times {[0,3]}_{\mathbb{Z}}\to MS{S}_{18},\phantom{\rule{2.em}{0ex}}$$
- $MS{S}_{26}^{\prime}:=MS{S}_{18}^{\prime}$, which is 26-contractible [3,4] and is the minimal simple closed 26-surface (see Figure 1b). Finally, we obtain $(MS{S}_{26}^{\prime},26,6,{\mathbb{Z}}^{3})$ according to Equation (1). Moreover, the proof of the 26-contractibility of $MS{S}_{26}^{\prime}$ is trivially proceeded with the homotopy in Equation (8).

**Remark**

**2.**

- (1)
- the digital image $(T,6)$ is not a closed 6-surface.
- (2)
- $(T,6)$ is pointed 6-contractible.

**Proof.**

## 3. A Geometric Realization of a Simple Closed $\mathit{k}$-Surface

**Definition**

**8.**

**Remark**

**3.**

**Example**

**1.**

**Definition**

**9.**

**Example**

**2.**

- (1)
- Based on $MS{S}_{18}$, we observe that ${D}_{18}({c}_{1})$ is the set as the union of polygons formulated by the 18-cycles in ${M}_{18}({c}_{1})$, i.e., the union of the two triangles with boundary generated by the two 18-cycles $({c}_{0},{c}_{1},{c}_{9})$ and $({c}_{0},{c}_{1},{c}_{6})$ and the two rectangles with boundary formulated by the 18-cycles $({c}_{1},{c}_{2},{c}_{8},{c}_{9})$ and $({c}_{1},{c}_{2},{c}_{7},{c}_{6})$.
- (2)
- Based on $MS{S}_{18}^{\prime}$, we observe that ${D}_{18}({e}_{0})$ is the set which is the union of four triangles with boundary formulated by four 18-cycles in ${M}_{18}({e}_{0})$.
- (3)
- In terms of the methods used in Equations (1) and (2), based on $MS{S}_{6}$, we observe that ${D}_{6}({d}_{0})$ is the set as the union of twelve polygons (or regular rectangles) formulated by the twelve 6-cycles in ${M}_{6}({d}_{0})$.

**Definition**

**10.**

**Proposition**

**2.**

**Proof.**

**Remark**

**4.**

**Remark**

**5.**

**Remark**

**6**

**.**Unlike a typical surface (or a 2-dimensional topological manifold) in the Euclidean topological space $({\mathbb{R}}^{3},U)$, we observe that, given an ${S}_{k}$ and $x\in {S}_{k}$ motivated by the set, $|\phantom{\rule{0.166667em}{0ex}}{S}_{k}{\phantom{\rule{0.166667em}{0ex}}|}^{x}$, the sets ${M}_{k}(x)$ and ${D}_{k}(x),x\in {S}_{k}$ play important roles in establishing a geometric realization of the given ${S}_{k}$.

**Example**

**3.**

## 4. Euler Characteristics for Digital $\mathit{k}$-Surfaces and Connected Sums of Closed $\mathit{k}$-Surfaces

**Remark**

**7.**

**Proof.**

**Example**

**4.**

**Remark**

**8**

**.**Given an ${S}_{k}$ referred to in Example 4, Reference [14] considered only the simplicial complexes formulated by only 2-dimensional digital k-simplexes on ${S}_{k}$. Then, given an ${S}_{k}$ in ${\mathbb{Z}}^{3}$, it is obvious that it need not produce a polyhedron in ${\mathbb{R}}^{3}$. To be precise, according to the approach of Reference [14], since each of the sets

**Definition**

**11**

**.**For an ${S}_{k}$, the Euler characteristic of ${S}_{k}$ is defined by

**Remark**

**9**

**.**The approach using Definition 11 is consistent with the research of the Euler characteristic of a typical closed surface from algebraic topology and polyhedron geometry.

**Proposition**

**3.**

**Proof.**

**Example**

**5.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Example**

**6.**

## 5. The (Almost) Fixed Point Property for Digital $\mathit{k}$-Surfaces and Connected Sums of Closed $\mathit{k}$-Surfaces

- We say that a digital image $(X,k)$ in ${\mathbb{Z}}^{n}$ has the fixed point property (FPP) [30] if, for every k-continuous map $f:(X,k)\to (X,k)$, there is a point $x\in X$ such that $f(x)=x$.
- We say that a digital image $(X,k)$ in ${\mathbb{Z}}^{n}$ has the almost fixed point property (AFPP) [30,31] if, for every k-continuous self-map f of $(X,k)$, there is a point $x\in X$ such that $f(x)=x$ or $f(x)$ is k-adjacent to x. In general, we observe that the AFPP is a more generalized concept than the FPP.

**Theorem**

**2.**

**Proof.**

**Corollary**

**2.**

**Remark**

**11.**

**Remark**

**12.**

## 6. Conclusions and a Further Work

- a development of a new type digital surface associated with a Khalimsky manifold.
- fixed point theory for many kinds of digital topological structures on ${\mathbb{Z}}^{n}$ in [33].
- given a typical surface X in pure topology and geometry, after developing a new type of $LF$-topological structure on X, $T(X)$, we can explore some connections related to Euler characteristics between X and $T(X)$.
- after improving the earlier digital homology groups [14] for digital images, we can propose some relationships between the current Euler characteristic and a certain invariant involving new homology groups for digital closed k-surfaces.

## Funding

## Conflicts of Interest

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**Figure 2.**Configuration of the pointed 18-contractibility of $MS{S}_{18}^{\prime}$ (

**a**) and $MS{S}_{18}$ (

**b**).

**Figure 3.**(

**a**) Configuration of the elements of ${M}_{18}({c}_{1})$ in $MS{S}_{18}$ for the point ${c}_{1}\in MS{S}_{18}$; (

**b**) explanation of the elements of ${M}_{18}({e}_{0})$ in $MS{S}_{18}^{\prime}$ for the point ${e}_{0}\in MS{S}_{18}^{\prime}$ (or $MS{S}_{26}^{\prime}$); and (

**c**) configuration of the elements of ${M}_{6}({d}_{0})$ for the point ${d}_{0}\in MS{S}_{6}$.

**Figure 4.**Explanations of the non-almost fixed point property (AFPP) of $MS{S}_{18}$ (

**a**) $MS{S}_{18}^{\prime}$ (or $MS{S}_{26}^{\prime}$) (

**b**).

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**MDPI and ACS Style**

Han, S.-E.
Fixed Point Theory for Digital *k*-Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces. *Mathematics* **2019**, *7*, 1244.
https://doi.org/10.3390/math7121244

**AMA Style**

Han S-E.
Fixed Point Theory for Digital *k*-Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces. *Mathematics*. 2019; 7(12):1244.
https://doi.org/10.3390/math7121244

**Chicago/Turabian Style**

Han, Sang-Eon.
2019. "Fixed Point Theory for Digital *k*-Surfaces and Some Remarks on the Euler Characteristics of Digital Closed Surfaces" *Mathematics* 7, no. 12: 1244.
https://doi.org/10.3390/math7121244