# On a conjecture of L. Fejes Tóth and J. Molnár about circle coverings of the plane

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## Abstract

Tóth and Molnár (Math Nachr 18:235–243, 1958) formulated the conjecture that for a given homogeneity *q* the thinnest covering of the Euclidean plane by arbitrary circles is greater or equal a function *S*(*q*). Florian (Rend Semin Mat Univ Padova 31:77–86, 1961) proved that if the covering consists of only two kinds of circles then the conjecture is true supposed that \(S(q)\le S(1/q)\) what can be easily verified by a computer. In this paper we consider the general case of circles with arbitrary radii from an interval of the reals. We set up two further functions \(M_0(q)\) and \(M_1(q)\) and prove that the conjecture is true if *S*(*q*) is less than or equal to *S*(1 / *q*), \(M_0(q)\) and \(M_1(q)\). As in the case of two kinds of circles this can be readily confirmed by computer calculations. (For \(q\ge 0.6\) we even do not need the function \(M_1(q)\) for computer aided comparisons.) Moreover, we obtain Florian’s result in a shorter different way.

## Keywords

Covering of the plane Incongruent discs Minimum density Conjecture of Tóth and Molnár Minimal density of three circles covering a triangle## 1 Introduction

A circle covering \(\mathbf{K(q)}\) of homogeneity *q* of the Euclidean plane is a countable set of closed circular discs \(C_i\) with radii \(r_i\) such that every point of the plane belongs to at least one circle of \(\mathbf{K(q)}\) and \(q:=\inf (r_i/r_j),i,j=1,2,\ldots \). Given *q* with \(0\le q\le 1\) it is of interest to determine the density *D*(*q*) of a thinnest covering of the plane. (For a formal definition of a thinnest covering see, for example, the classical book [8].)

*D*(

*q*) or accurate bounds of it. For upper bounds of

*D*(

*q*) cf. [2, 5, 6, 7, 8], for lower bounds [1, 3, 4, 9, 10]. The most outstanding open conjecture concerning a lower bound (considered by G. Blind in 1974 as “hopeless to prove analytically” (cf. [1])) had been proposed by Tóth and Molnár (first published 1958, [10]). It states that for \(0<q\le 1\)

*S*(

*q*) is based on the following setting. The centres \(O_1,O_2,O_3\) of three closed circular discs \(C_1,C_2,C_3\) with radii \(r_1,r_2,r_3\) which have exactly one common point (but no common inner point) form a triangle

*T*with angles \(\alpha ,\beta ,\gamma \) (see Fig. 1; to choose a scale b is set to be 1).

*T*the quotient of the area covered by \(C_1,C_2,C_3\) within

*T*and the area \(\varDelta T\) of

*T*is given by

*T*can be neglected if it turns out in the course of determining the minimum of the function \(\delta (q)\) that this assumption is satisfied by itself.

L. Fejes Tóth and J. Molnár had shown in [10] that \(\min \delta (q)\le D(q)\) and assumed that \(\min \delta (q)=S(q)\).

In this paper we consider a representation of \(\delta (q)\) together with its adhering boundary conditions derived by the author in [2]. In this representation \(\delta (q)\) is expressed as a function of \(q,\beta ,r_1\) and \(\gamma \) (see Fig. 1). We prove that for given *q* and \(\beta \) \(\delta (q,\beta ,r_1,\gamma )\) has a unique stationary point at which \(\delta (q,\beta ,r_1,\gamma )\) assumes its minimum \(\delta _0(q,\beta )\). However, this minimum is not always assumed within the limits of the domain *B* defined by all boundary conditions. If the minimum is attained within *B* computer calculations show that \(M_0(q):= \min _{\beta }\delta _0(q,\beta ) > S(q)\). As for the boundary values of *B* we prove that in order to find min\(\delta (q)\) only two further functions remain which have to be compared to *S*(*q*) by computer aid. First, if \(q<0.6\) and \(|O_1,O_3|=r_1+r_3\), the minimum of the covering density will yield a function \(M_1(q)\) which has to be taken into account. Second, if \(0<q\le 1\) and two radii of the circles \(C_1, C_2, C_3\) coincide we will prove that min\(\delta (q)=S(q)\) or min\(\delta (q)= S(1/q)\), this way recovering a result by Florian [3], but in a totally different (and shorter) way. As already pointed out in [3] computer calculations establish that \(S(q)\le S(1/q)\). Showing by computer aid that also \(M_1(q)>S(q)\) for \(0<q<0.6\) we therefore obtain that *S*(*q*) is the smallest of the four functions \(M_0(q)\), \(M_1(q)\), *S*(1 / *q*) and *S*(*q*) for any homogeneity *q*, this way confirming the conjecture of Tóth and Molnár.

## 2 \(\delta (q)\) for constant *q* and \(\beta \)

*T*. In view of \(r_1\ge r_2\ge r_3\) we can further conclude that

*W*is well defined.

We now agree to keep the parameters *q* and \(\beta \) fixed with \(0<q<1\) and \(0<\beta <\pi \).

*R*, so that we consider \(\delta \) as a function \(\delta (\gamma ,R)\) subject to the boundary conditions (2.1) and (2.5). Condition (2.1) then assumes the form

*B*. If we only consider the rectangle defined by (2.6) we will refer to it as

*A*. — A typical shape of

*B*is illustrated by Fig. 2 (for which \(q=3/5\) and \(\beta =0.628\)).

### Lemma 2.1

*G*is a positive, strongly convex function of \(\gamma \), \(\gamma \in (0,\pi -\beta )\). It assumes its minimium for \(\gamma = \gamma _0\) which is the unique solution of the equation

### Proof

From \(\delta (q)=RG+\beta \cot \beta -\beta W\ge 1\) and \(\beta \cot \beta < 1\) we infer \(G> \beta W/R\ge 0\).

*g*(

*x*):

From \(f'(x)=-2((\pi -x)\sin (2x)+1-\cos (2x))=-2\sin (2x)((\pi -x)-\tan (\pi -x))\le 0\) we obtain that f is decreasing and therefore, because of \(f(\pi /2)=\pi /2\) and \(f(\pi )=0\) we can conclude that \(f\ge 0\).

Due to \(g'''\le 0\) the second derivative \(g''\) is decreasing, hence \(G''(\gamma )=g''(\gamma )-g''(\beta +\gamma ) > 0\), i.e, *G* is strongly convex.

Equation (2.8) follows from \(G'(\gamma )=g'(\gamma )-g'(\beta +\gamma )=0\). Because \(g''(x)\ge 0\) for \(x\le \pi /2\) the first derivative of g is increasing so that \(\gamma _0\le \pi /2\) and \(\gamma _0+\beta \le \pi /2\) cannot hold together, from which we get that in any case \(\gamma _0\ge \pi /2-\beta \). \(\square \)

### Theorem 2.2

The function \(\delta (\gamma ,R)\) has exactly one stationary point \((\gamma _0,R_0)\in A\) at which it assumes its minimum within the domain *A*.

### Proof

\(\frac{\partial \delta }{\partial \gamma } = 0\) implies Eq. (2.8) from which we implicitly obtain the unique point \(\gamma _0\) and \(G_0:=G(\gamma _0)\).

*R*we get

*R*. By taking into account \(\gamma _0\) and setting \(R_0:=R_2\) we thus obtain

*A*. \(\square \)

*W*one obtains

If \((\gamma _0,R_0)\) satisfies condition (2.6) then the minimum \(\delta (\gamma _0,R_0)\) will be assumed within *B* in which case we introduce the following function:

### Definition 2.3

Let \(M_0(q):=\min _{\beta }\delta _0(q,\beta )\) for \(\gamma _0 \le \mathrm {arccot}(\frac{W_0-\cot (\beta )}{R_0p})\le \beta +\gamma _0\), besides the assumption that \(\gamma _0\) and \(W_0\) can both be expressed by \(G_0\) according to (2.11) and (2.12), respectively.

*B*, what we will do aside the assumption that \((\gamma _0,R_0)\notin B\):

- (a)
\(\gamma =\pi /2-\beta \), if \(\beta \le \pi /2\) and \(\gamma =0\) for \(\beta >\pi /2\),

- (b)
\(\gamma =\pi -\beta \),

- (c)
\(R=1/(1+q)^2\),

- (d)
\(R=1/(1-q)^2\),

- (e)
\(Rp\cot (\beta +\gamma )=W-\cot \beta \), which stands for \(r_1=r_2\) and

- (f)
\(Rp\cot \gamma =W-\cot \beta \), which represents the case \(r_3=r_1q=r_2\).

*T*with \(\alpha \le \gamma \) or \(\alpha =\beta \) or \(r_2=r_3\) (the latter leading to case (f)). For discussing the cases (a) and (b) we will assume these conditions.

Using \(\rightarrow \) to indicate that a variable approaches to a point and also for a function to converge to a limit, we thus obtain

(a) If \(\gamma =\pi /2-\beta \) for \(\beta \le \pi /2\) then \(\alpha \le \gamma \) or \(\alpha =\beta \) implies \(\alpha =\gamma =\pi /2\) or \(\alpha =\beta = \pi /2\), respectively. If both, \(\alpha \) and \(\gamma \), approach \(\pi /2\), then \(R \rightarrow \infty \) which because of \(R\le 1/(1-q)^2\) means that \(q\rightarrow 1\). Therefore we can conclude that \(\delta (q)\ge \) min \(\delta (1)\). If \(\alpha \) and \(\beta \) approach \(\pi /2\) then \(\gamma \rightarrow 0\) and by Eq. (2.2) we obtain \(\delta (q) \rightarrow \infty \).

If \(\pi /2<\beta <\pi \) then \(\gamma =0\) implies \(\alpha =0\) for \(\alpha \le \gamma \) from which we can infer \(\delta (q)\rightarrow \infty \), and with respect to \(\alpha =\beta \) the outcome will be the same.

*B*in respect to all angles \(\beta \), we will make use of the assumption within conjecture (1.1) that no circle must cut into the opposite side of the triangle

*T*. However, in case (d) this will alter the upper bound of R.

## 3 Two radii in a line

If \(r_1=1/(1+q)\) and no circle cuts into an opposite side of *T*, \(r_1\) and \(r_3\; (=r_1q^2)\) form the segment of a straight line, and the third radius is orthogonal to this line (see Fig. 4). We will establish a representation for \(\delta (q)\) anew on the basis of Fig. 4. From this drawing we can read off readily:

The area \(\varDelta T\) of the triangle *T* is given by \(2\varDelta T=r_2 \; \tan (\alpha )=r_2(1+q)=q\tan (\gamma )\), \(\arctan (q)\le \alpha \le \pi /4\) and, looking for the smallest possible and the largest \(\beta \), we get \(\pi /4+\arctan (q)\le \beta \le \pi /4+\arctan (1/q)\) which is equivalent to \(\arctan q\le \alpha \le \pi /4\).

### Theorem 3.1

If \(q\ge 0.6\) then \(\delta (q,\beta )\ge 1.219...>S(1)\) for every \(\beta \).

### Proof

*q*(easily checked for \(q\ge 0.5\)), and \(\tilde{\delta }_1(0.6)>1.219>S(1)\).\(\square \)

### Definition 3.2

For \(0<q<0.6\) and \(\bar{\delta }_1(q,\alpha )\) according to (3.2) we set \(M_1(q):=\min _{\alpha }\bar{\delta }_1(q,\alpha )\) for \(\arctan (q)\le \alpha \le \pi /4\).

### Remark 3.3

*T*. Then due to \(r_1\) being as long as possible the situation illustrated in Fig. 5 occurs.

From Fig. 5 one immediately realizes substituting \(r_1\) with *r* and \(a=rq+r_2\) that \(r_2=r\cot \beta \), \(a=r(q+\cot \beta )\), \(2\varDelta T=ar\), \(\sin \gamma =r=1/\sqrt{1+q^2}\), \(\gamma =\mathrm {arccot}(q)\) and the range of \(\beta \) is \(\pi /4\le \beta \le \mathrm {arccot}(q)\).

*q*and \(\beta \), referred to as \(\delta _2(q,\beta )\), we therefore obtain:

*S*(

*q*) (see Eq. (1.1)), one gets

### Theorem 3.4

\(\min _{\beta }\delta _2(q,\beta )\ge S(q)\).

Next we will discuss the cases (e) and (f) of the boundary conditions stated above, yet without the assumption that no circle must cut into an opposite side of the triangle *T*, since looking for minima this fact will turn out by itself.

## 4 Coverings with only two kind of circles

### Theorem 4.1

If \(r_1=r_2\) then \(\min \delta (q)=S(q)\), and if \(r_2=r_3\) then \(\min \delta (q)=S(1/q)\).

### Proof

*R*which entails the solution ([2],(10))

*R*within \(\delta (q)=RG+\beta \cot (\beta )-\beta W\) and writing \(\pi -\alpha -\gamma \) instead of \(\beta \) entails

As also conveyed in [2], interpreting geometrically *x* and *q* within the function \(s(q,x):= \frac{\pi -2(1-q^2)\arctan (\sqrt{1-x^2}/(x+q))}{2\sqrt{1-x^2}(x+q)}\) constituting *S*(*q*), straightforward calculations link \(\beta \) to *x* (see Fig. 6 for \(\alpha =\beta \)): One obtains \(\min _{0<\beta <\pi /2} \delta _{12}(q,\beta )=S(q)\) and \(\min _{0<\beta <\pi /2}\delta _{2,3}(q,\beta )=S(1/q)\). \(\square \)

## 5 Conclusions

We have come so far to know that for all \(q\ge 0.6\) \(\min \delta (q)\) is to be found among the minima of the densities \(\delta _0(q,\beta )\) (Eq. 2.13), \(\delta _{12}(q,\beta )\) (Eq. 4.1) and \(\delta _{23}(q,\beta )\) (Eq. 4.2) described by the functions \(M_0(q)\) (Definition 2.3), *S*(*q*) and *S*(1 / *q*) (see Eq.(1.1)).

For \(0<q<0.6\) we have not only to take into account the minima of \(\delta _0(q,\beta )\), \(\delta _{12}(q,\beta )\) and \(\delta _{23}(q,\beta )\) but also the minimum of \(\bar{\delta }_1(q,\alpha )\) which gives rise to \(M_1(q)\) (Definition 3.2). With this in mind we can formulate the following

### Theorem 5.1

If \(M_0(q)\) and *S*(1 / *q*) are less than or equal to *S*(*q*) for \(0.6\le q\le 1\) — this can be verified by computer calculations — the conjecture of L. Fejes Tóth and J. Molnár can be considered as confirmed.

If \(M_0(q)\), \(M_1(q)\) and *S*(1 / *q*) are less than or equal to *S*(*q*) for \(0<q<0.6\) — this can also be shown by computer calculations — the correctness of the conjecture ensues.

The functions \(M_0(q)\), \(M_1(q)\), *S*(*q*) and *S*(1 / *q*) are visualized in Fig. 7.

## Notes

### Acknowledgements

Open access funding provided by TU Wien (TUW).

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