#  On particular examples of planar integral point sets and their
classification 111 This work was carried out at Voronezh State University and
supported by the Russian Science Foundation grant 19-11-00197.

Avdeev N.N<EMAIL_ADDRESS><EMAIL_ADDRESS>, Momot E.A., Zvolinskiy
A.E.

_Voronezh State University_

### Abstract.

A planar integral point set is a set of non-collinear points in plane such
that for any pair of the points the Euclidean distance between them is
integral. We discuss the classification of planar integral point sets and
provide examples of sets that are not covered by the existent classification.

### Keywords.

integral point sets, planar integral point sets, classification of planar
integral point sets, diameter of an integral point set

##  1 Introduction

######  Definition 1.

A planar integral point set (PIPS) is a set $\mathcal{P}$ of non-collinear
points in plane $\mathbb{R}^{2}$ such that for any pair of points
$P_{1},P_{2}\in\mathcal{P}$ the Euclidean distance $|P_{1}P_{2}|$ between
points $P_{1}$ and $P_{2}$ is integral.

How do we describe an integral point set? For example, by the number of points
in it, which is always finite [1, 2] and is said to be the _cardinality_ of
the IPS. Furthermore, we can naturally define the diameter of a finite point
set.

######  Definition 2.

The diameter of the integral point set $\mathcal{P}$ is defined as

$\operatorname{diam(\mathcal{P})}=\underset{P_{1},P_{2}\in\mathcal{P}}{\max}|P_{1}P_{2}|.$
(1)

Every planar IPS aso has a characteristic [3, 4].

######  Definition 3.

The characteristic of a planar integral point set $M$ is the least positive
integer $q$ such that the area of any triangle with vertices
$M_{1},M_{2},M_{3}\in M$ is commensurable with $\sqrt{q}$.

Characteristic of an IPS does not change if the set is moved, dilated ot
flipped; moreover, even addition or deletion of a point does not change the
characteristic of IPS.

While the minimal possible diameter for planar integral point sets of given
cardinality was being computed, it was noticed [5] that such diameter is
attained at sets with many points on a straight line; for some estimations on
this tendency, we also refer the reader to [6]. Thus, the following
classification was introduced:

######  Definition 4.

A planar integral point set $M$ is said to be in _semi-general position_ if no
three points of $M$ are located in a straight line.

The most dominating examples of PIPS in semi-general position are circular
sets.

######  Definition 5.

A planar integral point sets that is situated on a circle is said to be a
circular point set.

So, the following constraint appeared.

######  Definition 6.

A planar integral point set $M$ is said to be in _general position_ if no
three points of $M$ are located in a straight line and no four points of $M$
are located in a circle.

Planar integral point sets in general position are very difficult to find; the
first known examples were presented in [7]. As for now, there is no known
example of PIPS of cardinality 8 in general position.

The main purpose of this work is to provide examples of planar integral point
sets that may give the clue for development of further classification.

For convenience, we use the notation [8, 9, 10]:
$\sqrt{p}/q*\\{(x_{1},y_{1}),...,$ $(x_{n},y_{n})\\}$, which means that each
abscissa is multiplied by $1/q$ and each ordinate is multiplied by
$\sqrt{p}/q$, i.e.

$\sqrt{p}/q*\\{(x_{1},y_{1}),...,(x_{n},y_{n})\\}=\left\\{\left(\frac{x_{1}}{q},\frac{y_{1}\sqrt{p}}{q}\right),...,\left(\frac{x_{n}}{q},\frac{y_{n}\sqrt{p}}{q}\right)\right\\}.$
(2)

Here $q$ is the characteristic of the PIPS; every PIPS can be represented in
such way [11][Theorem 4].

Note that all examples that are discussed below are located on a union of at
most three straight lines. For classification of planar IPS that are located
on a union of two straight lines, we refer the reader to [12].

There are some examples of planar IPS that are not contained in the union of
any three straight lines: for examples, these are heptagons presented in [7]
and 7-clusters from [13]. However, we have to keep in mind that the circular
inversion under certain conditions translates an integral point set into an
integral point set (although sometimes additional dilation is necessary). On
the other hand, the circular inversion may translate a straight line into a
circle and vice versa. Thus, we can consider _generalized circles_ , that are
circles or straight lines; obviously, in that point of view all the examples
from [7] and [13] are located on a union of three generalized circles, because
each seven points are located on a union of a circle and two straight lines.

##  2 Rails sets

######  Definition 7.

A planar integral point sets of $n$ points with $n-1$ points on a straight
line is called a facher set.

Facher sets are predominant examples of planar integral point sets. In [14],
facher sets of characteristic 1 are called semi-crabs.

######  Definition 8 ([12]).

A non-facher planar integral point sets situated in two parallel straight
lines is called a rails set.

Among the rails sets, sets with 2 points on one line and all the other on
another line dominate.

Two IPSs below have been obtained by dilating [12, Fig. 34] by $23$ and $29$
resp.; third one has been constructed by dilation and merge.

Figure 1: IPS of cardinality 42 and diameter 2473117504
Figure 2: IPS of cardinality 46 and diameter 3118278592
Figure 3: IPS of cardinality 48 and diameter 71720407616

  * •

Figure 1:

$\mathcal{P}_{42}=\sqrt{154}/{1}*\\{(\pm 219513840;-15069600);(\pm
345596160;0);(\pm 260201760;0);\\\ (\pm 225792840;0);(\pm 213234840;0);(\pm
153961080;0);(\pm 144668160;0);(\pm 25116000;0);\\\ (\pm 694026840;0);(\pm
514710560;0);(\pm 359116940;0);(\pm 13423904;0);(\pm 75682880;0);\\\ (\pm
464143680;0);(\pm 827069880;0);(\pm 92144325;0);(\pm 1195180740;0);(\pm
1236558752;0);\\\ (\pm 44590560;0);(\pm 339925740;0);(\pm 117312468;0)\\}$ (3)

  * •

Figure 2:

$\mathcal{P}_{46}=\sqrt{154}/{1}*\\{(\pm 276778320;-19000800);(\pm
435751680;0);(\pm 328080480;0);\\\ (\pm 268861320;0);(\pm 194124840;0);(\pm
182407680;0);(\pm 1559139296;0);(\pm 284695320;0);\\\ (\pm 1506967020;0);(\pm
1042827240;0);(\pm 875077320;0);(\pm 648982880;0);(\pm 585224640;0);\\\ (\pm
452799620;0);(\pm 95426240;0);(\pm 16925792;0);(\pm 116181975;0);(\pm
428602020;0);\\\ (\pm 56222880;0);(\pm 769560480;0);(\pm 626458560;0);(\pm
31668000;0);(\pm 130761918;0)\\}$ (4)

  * •

Figure 3:

$\mathcal{P}_{48}=\sqrt{154}/{1}*\\{(\pm 6365901360;-437018400);(\pm
10022288640;0);(\pm 23985026520;0);\\\ (\pm 389293216;0);(\pm
6183810360;0);(\pm 4464871320;0);(\pm 4195376640;0);(\pm 728364000;0);\\\ (\pm
35860203808;0);(\pm 34660241460;0);(\pm 7545851040;0);(\pm 20126778360;0);\\\
(\pm 14926606240;0);(\pm 13460166720;0);(\pm 10414391260;0);(\pm
2194803520;0);\\\ (\pm 2672185425;0);(\pm 9857846460;0);(\pm
1293126240;0);(\pm 17699891040;0);\\\ (\pm 14408546880;0);(\pm
3007524114;0);(\pm 6547992360;0);(\pm 3402061572;0)\\}$ (5)

Taking the examples into consideration, we can conjecture that there is an
infinite point set with rational distances that contains $\mathcal{P}_{48}$.
(However, it is known [15] that if a point set $S$ with rational distances has
infinitely many points on a line or on a circle, then all but 4 resp. 3 points
of $S$ are on the line or on the circle.)

##  3 Example of sets with many common points that cannot be merged

Figure 4 shows an example of three PIPS of cardinality 8, each pair of that
shares 6 or 7 points but cannot be combined into another PIPS.

a)

b)

c)

Figure 4: IPSs with cardinality 8 and diameter 2520 with many common points

  * •

$\mathcal{P}=\sqrt{143}/2*\\{(\pm 1620;0);(\pm 1920;300);(\pm 735;75);(\pm
340;0);\\}$

  * •

$\mathcal{P}=\sqrt{143}/2*\\{(\pm 1620;0);(\pm 1920;300);(\pm 735;75);(\pm
1767;147);\\}$

  * •

$\mathcal{P}=\sqrt{143}/2*\\{(\pm 1620;0);(\pm 1920;300);(\pm
735;75);(-340;0);(1767;147);\\}$

The distance between the non-adoptable points is

$\sqrt{\left(\frac{1767}{2}-\frac{340}{2}\right)^{2}+\left(\frac{147}{2}\right)^{2}\cdot
143}=2\sqrt{320401}.$ (6)

It’s notable that 320401 is a prime number.

##  4 Integral point sets with two axes of symmetry

The set $\mathcal{P}_{19}$ shown on Figure 6 was obtained from the set
$\mathcal{P}_{9}$ shown on Figure 5 by dilation and looking for points on $x$
axis.

Figure 5: IPS of cardinality 9 and diameter 2890

$\mathcal{P}_{9}=\sqrt{1}/1*\\{(0;0);(\pm 1445;0);(\pm 1085;0);(-455;\pm
528);(455;\pm 528)\\}$ (7)

Figure 6: IPS of cardinality 19 and diameter 20663808074

$\mathcal{P}_{19}=\sqrt{1}/1*\\{(0;0);(-987843675;\pm
1146332880);(987843675;\pm 1146332880);\\\ (\pm 729918777;0);(\pm
972103809;0);(\pm 1113030324;0);(\pm 1400170149;0);\\\ (\pm 3137217825;0);(\pm
2355627225;0);(\pm 10331904037;0);\\}$

##  5 Arrow-like integral point sets with one axis of symmetry

On Figure 7, the following IPS is shown:

$\mathcal{P}_{17}=\sqrt{1}/1*\\{(-2847;\pm 72072);(47073;\pm
124488);(47073;0);(\pm 98943;0);\\\
(-694668;0);(-71487;0);(-50367;0);(-14943;0);(15457;0);\\\
(23582;0);(63073;0);(125307;0);(172207;0)(329628;0);\\}$ (8)

Note that the axis of symmetry for $\mathcal{P}_{17}$ is the $x$ axis; all
such sets are of characteristic 1. Moreover, note that the set contains three
points with the same first coordinates.

Figure 7: IPS of cardinality 17 and diameter 1024296

On Figures 8 and 9, other examples of arrow-like IPSs are shown. The one on
Figure 9 is obtained from the one on Figure 8 by dilation and looking for
points on $x$ axis.

Figure 8: IPS of cardinality 10 and diameter 2431
Figure 9: IPS of cardinality 15 and diameter 19203744

Figure 8:

$\mathcal{P}_{10}=\sqrt{1}/1*\\{(0;0);(0;\pm 252);(960;\pm
468);(-1120;0);(-405;0);(336;0);(561;0);(1311;0)\\}$

Figure 9:

$\mathcal{P}_{15}=\sqrt{1}/1*\\{(0;0);(0;\pm 1413720);(5385600;\pm
2625480);\\\
(-6283200;0);(-2272050;0);(-1971915;0);(-635040;0);(1884960;0);\\\
(3147210;0);(4558176;0);(5976333;0);(7354710;0);(12920544;0);\\}$

##  6 Other examples

Figure 10 displays an IPS with no axis of symmetry; although the set is of
characteristic 1, it cannot be extended by the reflection in the $x$ axis.
Moreover, we failed to extend it by dilation and looking for extra points on
the $x$ axis.

$\mathcal{P}_{8}=\sqrt{1}/13*\\{(0;0);(8450;0);(12844;0);(21294;0);(29575;0);\\\
(-2366;-8112);(10647;-14196);(15022;-3696)\\}$ (9)

Figure 10: IPS of cardinality 8 and diameter 2535

The set shown on Figure 11 has an axis of symmetry, but it is $y$ axis, not
$x$ axis. Due to the fact that its characteristic is not 1, the set cannot be
rotated by $90^{\circ}$ but still be on lattice (2):

$\mathcal{P}_{8y}=\sqrt{42}/1*\\{(\pm 1200;0);(\pm 529;182);(\pm 814;152);(\pm
440;80)\\}$ (10)

Figure 11: IPS of cardinality 8 and diameter 2400

##  7 Final remarks

All the given planar integral point sets were obtained through a combination
of computer search and intuition of the authors.

The source code can be obtained at https://gitlab.com/Nickkolok/ips-algo

##  8 Acknowledgements

The authors thank Dr. Prof. E.M. Semenov and Dr. A.S. Usachev for
proofreading.

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The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.