On particular examples of planar integral point sets and their classification
OOn particular examples of planar integral point sets andtheir classification ∗Avdeev N.N. † , Momot E.A., Zvolinsky A.E. Voronezh State University
February 25, 2021
Abstract.
A planar integral point set is a set of non-collinear points in plane such that forany pair of the points the Euclidean distance between the points is integral. We discuss theclassification of planar integral point sets and provide examples of sets that are not covered bythe existent classification.
Definition 1.
A planar integral point set (PIPS) is a set P of non-collinear points in plane R such that for any pair of points P , P ∈ P the Euclidean distance | P P | between points P and P is integral. How do we describe an integral point set? For example, with the number of points in it,which is always finite [1, 2] and is said to be the cardinality of the IPS. Furthermore, we cannaturally define the diameter of a finite point set.
Definition 2.
The diameter of the integral point set P is defined as diam( P ) = max P ,P ∈P | P P | . (1)Every integral point set also has a characteristic [3, 4]. Characteristic of an IPS does notchange if the set is moved, dilated ot flipped; moreover, even addition or deletion of a pointdoes not change the characteristic of IPS.While the minimal possible diameter for planar integral point sets of given cardinality wasbeing computed, it was noticed [5] that such diameter is attained at sets with many points ona straight line; for some estimations on this tendency, we also refer the reader to [6]. Thus, thefollowing classification was introduced: Definition 3.
A planar integral point set M is said to be in semi-general position if no threepoints of M are located in a straight line. ∗ This work was carried out at Voronezh State University and supported by the Russian Science Foundationgrant 19-11-00197. † [email protected], [email protected] a r X i v : . [ m a t h . C O ] F e b he most dominating examples of PIPS in semi-general position are circular sets. Definition 4.
A planar integral point sets that is situated on a circle is said to be a circularpoint set.
So, the following constraint appeared.
Definition 5.
A planar integral point set M is said to be in general position if no three pointsof 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 knownexamples were presented in [7]. As for now, there is no known example of PIPS of cardinality8 in general position.The main purpose of this work is to provide examples of planar integral point sets that maygive the clue for development of further classification.For convenience, we use the notation [8–10]: √ p/q ∗ { ( x , y ) , ..., ( x n , y n ) } , which meansthat each abscissa is multiplied by /q and each ordinate is multiplied by √ p/q , i.e. √ p/q ∗ { ( x , y ) , ..., ( x n , y n ) } = (cid:26)(cid:18) x q , y √ pq (cid:19) , ..., (cid:18) x n q , y n √ pq (cid:19)(cid:27) . (2)Here q is the characteristic of the PIPS; every PIPS can be represented in such way [11][Theorem4]. It’s notable that any of the examples that are discussed below is located on a union of atmost three straight lines. For classification of planar IPS that are located on a union of twostraight lines, we refer the reader to [12].There are some examples of planar IPS that are not contained in the union of any threestraight lines: for examples, these are heptagons presented in [7] and 7-clusters from [13]. How-ever, we have to keep in mind that the circular inversion under certain conditions translatesan integral point set into an integral point set (although sometimes additional dilation is nec-essary). On the other hand, the circular inversion may translate a straight line into a circleand 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 ofthree generalized circles, because each seven points are located on a union of a circle and twostraight lines. Definition 6.
A planar integral point sets of n points with n − points on a straight line iscalled a facher set. Facher sets are very dominating examples of planar integral pont sets. In [14], facher setsof characteristic 1 are called semi-crabs . Definition 7 ( [12]) . A non-facher planar integral point sets situated in two parallel straightlines is called a rails set.
Among the rails sets, sets with 2 points on one line and all the other on another linedominate.Two IPSs below have been obtained by dilating [12, Fig. 34] by and resp.; third onehas been constructed by dilation and merge. 2igure 1: IPS of cardinality 42 and diameter 2473117504Figure 2: IPS of cardinality 46 and diameter 3118278592Figure 3: IPS of cardinality 48 and diameter 71720407616• Figure 1: P = √ / ∗ { ( ± − ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± } (3)• Figure 2: P = √ / ∗ { ( ± − ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± } (4)• Figure 3: P = √ / ∗ { ( ± − ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± } (5)3aking the examples into consideration, we can conjecture that there is an infinite pointset with rational distances that contains P . (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.) Figure 4 shows an example of three PIPS of cardinality 8, each pair of that shares 6 or 7 pointsbut cannot be united into another PIPS.a) b) c)Figure 4: IPSs with cardinality 8 and diameter 2520 with many common points• P = √ / ∗ { ( ± ± ± ± } • P = √ / ∗ { ( ± ± ± ± } • P = √ / ∗ { ( ± ± ± − } The distance between the non-adoptable points is (cid:115)(cid:18) − (cid:19) + (cid:18) (cid:19) ·
143 = 2 √ . (6)It’s notable that 320401 is a prime number. The set P shown on Figure 6 was obtained from the set shown on Figure 5 by dilation andlooking for points on x axis. P = √ / ∗ { (0; 0); ( ± ± − ± ± } (7)4igure 5: IPS of cardinality 9 and diameter 2890Figure 6: IPS of cardinality 19 and diameter 41327616148 P = √ / ∗ { ( ± ± − ± − − − − − − − } (8) Figure 7: P = √ / ∗ { ( − ± ± ± − − − − } (9) Figure 8 displays an IPS with no axis of symmetry; although the set is of characteristic 1, itcannot be extended by reflecting relatively to the x axis. Moreover, we failed to extend it by5igure 7: IPS of cardinality 17 and diameter 1024296dilation and looking for extra points on the x axis. P = √ / ∗ { (0; 0); (8450; 0); (12844; 0); (21294; 0); (29575; 0);( − − − − } (10)Figure 8: IPS of cardinality 8 and diameter 2535The set shown on Figure 9 has an axis of symmetry, but it is y axis, not x axis and dueto the fact that its characteristic is not 1, the set cannot be rotated by ◦ but still stay onlattice (2): P y = √ / ∗ { ( ± ± ± ± } (11) All the given planar integral point sets were obtained through a combination of computer searchand intuition of the authors.The source code can be obtained at https://gitlab.com/Nickkolok/ips-algo6igure 9: IPS of cardinality 8 and diameter 2400
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