Next Article in Journal
Application of the Concept of Statistical Causality in Integrable Increasing Processes and Measures
Next Article in Special Issue
On the Potential Vector Fields of Soliton-Type Equations
Previous Article in Journal
Lax Pairs for the Modified KdV Equation
Previous Article in Special Issue
A Novel Robust Topological Denoising Method Based on Homotopy Theory for Virtual Colonoscopy
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

On the Noteworthy Properties of Tangentials in Cubic Structures

by
Vladimir Volenec
1 and
Ružica Kolar-Šuper
2,*
1
Department of Mathematics, University of Zagreb, Bijenička Cesta 30, 10 000 Zagreb, Croatia
2
Faculty of Education, University of Osijek, Cara Hadrijana 10, 31 000 Osijek, Croatia
*
Author to whom correspondence should be addressed.
Axioms 2024, 13(2), 122; https://doi.org/10.3390/axioms13020122
Submission received: 6 December 2023 / Revised: 9 February 2024 / Accepted: 11 February 2024 / Published: 16 February 2024
(This article belongs to the Special Issue Discrete Curvatures and Laplacians)

Abstract

:
The cubic structure, a captivating geometric structure, finds applications across various areas of geometry through different models. In this paper, we explore the significant characteristics of tangentials in cubic structures of ranks 0, 1, and 2. Specifically, in the cubic structure of rank 2, we derive the Hessian configuration ( 12 3 , 16 4 ) of points and lines. Finally, we introduce and investigate the de Vries configuration of points and lines in a cubic structure.

1. Introduction

The close connection between the cubic structure and algebraic structures on cubic curves was studied in [1,2,3], and this correlation was further examined in [4]. The cubic structure was defined in [4]. Let Q be a nonempty set whose elements are called points, and let [ ] Q 3 be a ternary relation on Q. Such a relation and the ordered pair ( Q , [ ] ) will be called a cubic relation and a cubic structure, respectively, if the following properties are satisfied:
C1.
For any two points a , b Q , there is a unique point c Q such that [ a , b , c ] (i.e., ( a , b , c ) [ ] ).
C2.
The relation [ ] is totally symmetric (i.e., [ a , b , c ] implies [ a , c , b ] , [ b , a , c ] , [ b , c , a ] , [ c , a , b ] , and [ c , b , a ] ).
C3.
[ a , b , c ] , [ d , e , f ] , [ g , h , i ] , [ a , d , g ] , and [ b , e , h ] imply [ c , f , i ] , which can be clearly written in the form of the following table:
abc.
def
ghi
In [4], numerous examples of cubic structures are presented, with one notable example in which Q is the set of all non-singular points of a cubic curve in the plane. In this context, the notation [ a , b , c ] signifies that points a, b, and cQ are collinear. Therefore, in a general cubic structure ( Q , [ ] ) , if [ a , b , c ] holds true, then we will also say that points a, b, and c form a line. If this statement is not valid, then we say that ( a , b , c ) is a triangle.
The concept of tangentials of points was introduced in [5]. The point a is said to be the tangential of the point a if [ a , a , a ] holds true. If a is the tangential of the point a, then we will also say that the point a is an antecedent of the point a . It is obvious that every point has one and only one tangential a . The tangential a of the tangential a of a point a is called its second tangential. We will always denote the tangential and the second tangential of any point x as x and x . The validity of [ a , b , c ] implies the validity of [ a , b , c ] . Two distinct points having the same tangential are called corresponding points. All points such that any two of them are corresponding points (i.e., have the same tangential) are said to be associated. The maximal number of associated points always equals 2 m for some m N { 0 } , and the number m is called the rank of the observed cubic structure. In the case of a cubic structure where collinear triples of non-singular points are observed on a cubic curve in the complex plane, ranks 0, 1, or 2 appear, depending on whether the cubic has a spike, an ordinary double point, or is without singular points. We will mention here some of the results from [5] in the form of several lemmas.
Lemma 1.
Let a 1 and a 2 be corresponding points with the common tangential a , let o be any point, and let b 1 and b 2 be points such that [ o , a 1 , b 1 ] and [ o , a 2 , b 2 ] . Then, b 1 and b 2 are corresponding points with the common tangential b such that [ o , a , b ] . In addition, there is a point c such that [ a 1 , b 2 , c ] and [ a 2 , b 1 , c ] , and o and c are corresponding points.
Lemma 2.
If [ a , b , c ] ; [ a , e , f ] ; [ b , f , d ] ; and [ c , d , e ] , then a , d ; b , e ; and c , f are pairs of corresponding points, and the tangentials a , b , and c satisfy [ a , b , c ] .
Lemma 3.
If a 1 , a 2 , a 3 , and a 4 are associated points with the common tangential a , then there exist points p, q, and r such that [ a 1 , a 2 , p ] , [ a 3 , a 4 , p ] , [ a 1 , a 3 , q ] , [ a 2 , a 4 , q ] , [ a 1 , a 4 , r ] , and [ a 2 , a 3 , r ] , and points a , p, q, and r are associated.
Lemma 4.
Suppose [ a , b , c ] holds, where a , b , and c are mutually different points. All different antecedents of points a , b , and c in a cubic structure of rank 2 can be denoted by a 1 , a 2 , a 3 , and a 4 ; b 1 , b 2 , b 3 , and b 4 ; and c 1 , c 2 , c 3 , and c 4 such that the following hold:
[ a 1 , b 1 , c 1 ] , [ a 1 , b 2 , c 2 ] , [ a 1 , b 3 , c 3 ] , [ a 1 , b 4 , c 4 ] , [ a 2 , b 1 , c 2 ] , [ a 2 , b 2 , c 1 ] , [ a 2 , b 3 , c 4 ] , [ a 2 , b 4 , c 3 ] , [ a 3 , b 1 , c 3 ] , [ a 3 , b 2 , c 4 ] , [ a 3 , b 3 , c 1 ] , [ a 3 , b 4 , c 2 ] , [ a 4 , b 1 , c 4 ] , [ a 4 , b 2 , c 3 ] , [ a 4 , b 3 , c 2 ] , [ a 4 , b 4 , c 1 ] .
Points a, b, c, d, e, and f are said to form a quadrilateral { a , d ; b , e ; c , f } if there exist lines [ a , b , c ] , [ a , e , f ] , [ d , b , f ] , and [ d , e , c ] , and we say that the points from each pair of points a , d ; b , e ; and c , f are opposite. Lemma 2 actually asserts that pairs of opposite vertices of a quadrilateral are corresponding.
We will say that a, d; b, e; and c, f are pairs of opposite vertices of a complete quadrilateral ( a , d ; b , e ; c , f ) if there exist lines [ a , b , c ] , [ a , e , f ] , [ d , b , f ] , and [ d , e , c ] . According to [5] (Theorem 3.5), pairs of opposite vertices have common tangentials which are collinear points.
The motivation for this paper is drawn from classical books [6,7,8,9] which extensively covered the properties of cubics. Additionally, a wealth of various research papers on this subject exist, although we will not provide a detailed list.

2. Some Properties of Tangentials in a General Cubic Structure

The following theorem allows the construction of the tangential of a point when the tangential of another given point is known:
Theorem 1.
Let a point a and its tangential a be given. For each point p, let q, r, s, and t be points such that [ a , p , q ] , [ a , p , r ] , [ q , r , s ] , and [ a , s , t ] hold. Then, point t is the tangential of point p.
Proof. 
This statement follows from the table
arp.
aqp
ast
Theorem 2.
Let q and r be fixed points, and let a 1 , a 2 , b 1 , and b 2 be points such that [ r , a 1 , a 2 ] , [ q , a 1 , b 1 ] , and [ q , a 2 , b 2 ] . Then, point s such that [ s , b 1 , b 2 ] is also a fixed point.
Proof. 
Let q be the tangential of q. From the table
qqq
a1a2r
b1b2s
we obtain [ q , r , s ] . Let c 1 , c 2 , d 1 , and d 2 be points such that [ r , c 1 , c 2 ] , [ q , c 1 , d 1 ] , and [ q , c 2 , d 2 ] . Then, the statement [ s , d 1 , d 2 ] , which should be proven, follows from the table
qc1d1.
qc2d2
qrs
Theorem 3.
If p, q, r, and s are points such that [ a , p , q ] , and [ a , r , s ] hold, and u and v are points such that [ p , r , u ] and [ q , s , v ] hold, then for the tangential a of point a, the statement [ u , v , a ] holds.
Proof. 
This statement follows from the table
pru.
qsv
aaa
Theorem 4.
Let a 1 , b 1 , c 1 , and d 1 be points such that there is a point e 1 which satisfies [ a 1 , b 1 , e 1 ] and [ c 1 , d 1 , e 1 ] , and let t be any point. If points a 2 , b 2 , c 2 , and d 2 satisfy [ t , a 1 , a 2 ] , [ t , b 1 , b 2 ] , [ t , c 1 , c 2 ] , and [ t , d 1 , d 2 ] , then there is a point e 2 satisfying [ a 2 , b 2 , e 2 ] and [ c 2 , d 2 , e 2 ] .
Proof. 
If t is the tangential of point t, and e 2 is the point such that [ t , e 1 , e 2 ] , then from the tables
ta1a2tc1c2
tb1b2td1d2
te1e2te1e2
we obtain [ a 2 , b 2 , e 2 ] and [ c 2 , d 2 , e 2 ] . □
Theorem 5.
Let points a 1 and a 2 have the common tangential a , and let b be any point. If c and d are points such that [ a 1 , b , c ] and [ a 2 , b , d ] hold, then there is a point e satisfying [ a 1 , d , e ] and [ a 2 , c , e ] .
Proof. 
Let point e be such that [ a 1 , d , e ] . Then, [ a 2 , c , e ] follows from the table
aa2a2.
a1bc
a1de
Theorem 6.
In a cubic structure, there are as many triangles with vertices a, b, and c whose sides pass through the three given points d, e, and f of this cubic structure as there are antecedents of each point (i.e., in a structure of rank m, there are 2 m such triangles).
Proof. 
Let g be the point such that [ e , f , g ] , and let h be the point satisfying [ d , g , h ] . If a is some antecedent point of the point h (i.e., if h is the tangential a of the point a), then a is one of the required points. Indeed, if b and c are points such that [ a , f , b ] and [ a , e , c ] , then we obtain [ b , c , d ] from the table
afb.
aec
hgd
The number of solutions is equal to the number of antecedents of point h. □
Theorem 7.
If [ a 1 , a 2 , a 3 ] , [ b 1 , b 2 , b 3 ] , and [ a i , b i , c i ] , where i = 1 , 2 , 3 , then the table
a1b1c1
a2b2c2
a3b3c3
implies [ c 1 , c 2 , c 3 ] . If points a 1 , b 2 , and c 3 have the common tangential, then points a 2 , b 3 , and c 1 have the common tangential, as do points a 3 , b 1 , and c 2 .
Proof. 
Here, [ b 1 , a 1 , c 1 ] and [ c 2 , b 2 , a 2 ] imply [ b 1 , a 1 , c 1 ] and [ c 2 , b 2 , a 2 ] , and since [ b 2 , b 2 , a 1 ] and [ c 3 , c 3 , b 2 ] also hold true, the tables
b 1 a 1 c 1 c 2 b 2 a 2
b 1 b 2 b 3 c 2 c 3 c 1
b 1 b 2 b 3 c 2 c 3 c 1
imply [ b 3 , b 3 , c 1 ] and [ c 1 , c 1 , a 2 ] (i.e., b 3 = c 1 and c 1 = a 2 , respectively). In the same way, the second claim follows from the first statement by cyclically replacing the indices 1 2 3 1 . □
Theorem 8.
Let points a , b , and c be tangentials of points a, b, and c. Then, [ a , b , c ] and [ a , b , c ] imply [ a , b , c ] .
Proof. 
This statement follows from the table
aaa.
bbb
ccc
The next theorem is the converse of Lemma 2.
Theorem 9.
If [ a , b , c ] and [ c , d , e ] hold, and if points a and b have the common tangential a , then there is a point f satisfying [ a , e , f ] and [ b , d , f ] (i.e., there exists the quadrilateral { a , d ; b , e ; c , f } ).
Proof. 
Let f be a point such that [ a , e , f ] . From the table
acb
add
aef
we obtain [ b , d , f ] .
Theorem 10.
Let a, b, c, d, e, and f be points such that [ b , c , d ] , [ c , a , e ] , and [ a , b , f ] . The following four statements are equivalent: a = d , b = e , c = f , and [ d , e , f ] .
Proof. 
Because of the symmetry of our ternary relation, it suffices to prove that a = d if and only if [ d , e , f ] . If a = d , then [ d , e , f ] follows from the table
add.
ace
abf
Conversely, if [ d , e , f ] , then a = d follows from the table
cbd.
efd
aaa
The following theorem is the converse of Lemma 3.
Theorem 11.
If there exist lines [ a , b , p ] , [ c , d , p ] , [ a , c , q ] , [ b , d , q ] , [ a , d , r ] , and [ b , c , r ] , then points a, b, c, and d have the common tangential t, and points p, q, r, and t have the common tangential.
Proof. 
If t is the tangential of point a, then the fact that point t is the common tangential of points b, c, and d follows from the tables
qdb,pdc,pcd.
crbbrcbqd
aataataat
According to the table
abp,
abp
ttt
the point t is the tangential of p. The proof for points q and r is similar. □
Theorem 12.
Let [ b , c , d ] , [ c , a , e ] , and [ a , b , f ] hold. Points a, b, and c have the common tangential if and only if there exists a point p such that [ a , d , p ] , [ b , e , p ] , and [ c , f , p ] .
Proof. 
Let a = b = c = q , and let p be a point where [ a , d , p ] . Statements [ b , e , p ] and [ c , f , p ] follow from the tables
qbbqcc.
aceabf
adpadp
Conversely, if there is a point p such that [ a , d , p ] , [ b , e , p ] , and [ c , f , p ] , and if q is the tangential of the point a, then from the tables
pebpfc
dcbdbc
aaqaaq
we find that q is also the tangential of b and c. □
Theorem 13.
If [ b , c , d ] , [ c , a , e ] , and [ a , b , f ] , then there exists a point p satisfying [ a , d , p ] and [ e , f , p ] .
Proof. 
The proof follows by applying the table
aaa.
cbd
efp
Theorem 14.
If [ b , c , d ] , [ c , a , e ] , and [ a , b , f ] , then [ a , b , c ] and [ d , e , f ] are equivalent statements.
Proof. 
Suppose [ a , b , c ] holds. Since [ b , c , d ] implies [ b , c , d ] , we find that d = a (i.e., [ d , d , a ] ). Due to Theorem 13, there is a point p such that [ a , d , p ] and [ e , f , p ] , which imply p = d (i.e., [ e , f , d ] holds true). Conversely, suppose [ d , e , f ] holds. From the table
efd,
cbd
aaa
we find that d = a . As [ b , c , d ] implies [ b , c , d ] , we obtain [ b , c , a ] . □
Following [10], a trio of points ( a , b , c ) is called a triad if for points d, e, and f such that [ b , c , d ] , [ c , a , e ] , and [ a , b , f ] , the statement [ d , e , f ] also holds true (i.e., there is a quadrilateral { d , a ; e , b ; f , c } in which the triad ( a , b , c ) is a triangle). We will call this quadrilateral the circumscribed quadrilateral and line [ d , e , f ] the complementary line of the triad ( a , b , c ) . Obviously, the quadrilateral { d , a ; e , b ; f , c } is also circumscribed to three other triads ( a , e , f ) , ( b , d , f ) , and ( c , d , e ) , to which the lines [ b , c , d ] , [ c , a , e ] , and [ a , b , f ] are complementary. Due to Lemma 1, the pairs of points a , d ; b , e ; and c , f have the common tangentials a , b , and c belonging to one line.
Theorem 15.
If { d , a ; e , b ; f , c } is the circumscribed quadrilateral of the triad ( a , b , c ) , and if g, h, and i are points such that [ a , d , g ] , [ b , e , h ] , and [ c , f , i ] hold, then these points also form the triad ( g , h , i ) whose complementary line is the line ( d , e , f ) , where d , e , and f are the tangentials of points d, e, and f.
Proof. 
We already have [ b , c , d ] , [ c , a , e ] , [ a , b , f ] , and [ d , e , f ] , and the last relation implies [ d , e , f ] . From the tables
beh,adg,andadg,
cficfibeh
dddeeefff
we obtain [ h , i , d ] , [ g , i , e ] , and [ g , h , f ] which, together with [ d , e , f ] , yields the statement of the theorem. □
Theorem 16.
Let a 1 , a 2 , and a 3 be given points, let ( i , j , k ) be any cyclic permutation of ( 1 , 2 , 3 ) , and let points b i and c i ( i = 1 , 2 , 3 ) be defined in such a way that [ a i , a j , b k ] and [ b i , b j , c k ] hold. Then, [ b i , c i , a i ] holds for i = 1 , 2 , 3 .
Proof. 
The statement follows using the table
akajbi.
bjbkci
aiai a i

3. Properties of the Tangentials in the Cubic Structures of Ranks 0 and 1

Theorem 17.
In a cubic structure of rank 0, the antecedents of three collinear points are also collinear.
Proof. 
Let a, b, and c be antecedents of points a , b , and c , respectively, and let [ a , b , c ] hold true. Suppose that there is a point d such that [ a , b , d ] . This implies [ a , b , d ] , and thus d = c . Since each point has only one antecedent, we conclude that d = c . □
Theorem 18.
In a cubic structure of rank 1, let a 1 and a 2 be two different points having the common tangential a, and let b be a point different from a which has the common tangential with a. Then, [ a 1 , a 2 , b ] holds true.
Proof. 
Let c be a point such that [ a 1 , a 2 , c ] , and let a be the tangential of a. From the table
a1a2c
a1a2c
aaa
it follows that point c has the tangential a . If c = a , then we would have [ a 1 , a 2 , a ] and [ a 1 , a 1 , a ] . This would lead to the contradiction a 1 = a 2 . Consequently, c = b . □
Theorem 19.
If points a and b have the tangentials a and b , respectively, and if point c is such that [ a , b , c ] , then there exists a point c to which point c is tangential and which satisfies [ a , b , c ] .
Proof. 
Let c be a point such that [ a , b , c ] . This implies [ a , b , c ] , and the point c is uniquely determined. □
Theorem 20.
In a cubic structure of rank 1, let [ a , b , c ] hold true, and let a 1 , a 2 ; b 1 , b 2 ; and c 1 , c 2 be pairs of different points with common tangentials a , b , and c , respectively. Then, the indices of these points can be chosen such that [ a 1 , b 1 , c 1 ] , [ a 1 , b 2 , c 2 ] , [ a 2 , b 1 , c 2 ] , and [ a 2 , b 2 , c 1 ] (i.e., such that { a 1 , a 2 ; b 1 , b 2 ; c 1 , c 2 } is a quadrilateral).
Proof. 
Let us choose arbitrary labeled points with tangentials a and b . Due to Theorem 19, one of the points with the tangential c lies on the same line with points a 1 and b 1 . Let us label this point with c 1 and the other point with c 2 . We therefore have [ a 1 , b 1 , c 1 ] . Then, because of axiom C1, [ a 1 , b 2 , c 1 ] and [ a 2 , b 1 , c 1 ] cannot be valid. According to Theorem 19, [ a 1 , b 2 , c 2 ] and [ a 2 , b 1 , c 2 ] must hold. From any of these two statements, due to C1, it follows that [ a 2 , b 2 , c 2 ] cannot be valid, and according to Theorem 19, [ a 2 , b 2 , c 1 ] must hold. □

4. The Properties of Tangentials in Cubic Structures of Rank 2

Theorem 21.
If non-collinear points a, b, and c in a cubic structure of rank 2 have collinear tangentials a , b , and c , respectively, then ( a , b , c ) is a triad.
Proof. 
Each of the points a , b , and c has four antecedents a i , b i , and c i ( i = 0 , 1 , 2 , 3 ) . According to Lemma 4, the indices of these points can be chosen such that, among others, [ a 0 , b 0 , c 0 ] , [ a 0 , b 1 , c 1 ] , [ a 1 , b 0 , c 1 ] , and [ a 1 , b 1 , c 0 ] hold, where points a 1 , b 1 , and c 1 are the original points a, b, and c. □
Theorem 22.
Let the triad ( a , b , c ) and the line [ d , e , f ] be given. If g, h, and i are points such that [ a , d , g ] , [ b , e , h ] , and [ c , f , i ] , then ( g , h , i ) is also a triad.
Proof. 
Here, [ a , b , c ] holds, and the existence of four lines in the theorem implies the existence of the lines [ d , e , f ] , [ a , d , g ] , [ b , e , h ] , and [ c , f , i ] . From the table
adg
beh
cfi
we obtain [ g , h , i ] . Points g, h, and i are not collinear, because otherwise [ a , b , c ] would follow from the table
dga,
ehb
fic
which is not true. Due to Theorem 20, ( g , h , i ) is a triad. □
Theorem 23.
Let a, b, and c be three non-collinear points in a cubic structure of rank 2. Then, there are four triples of the form ( d , e , f ) such that [ e , f , a ] , [ f , d , b ] , and [ d , e , c ] hold true.
Proof. 
Let g be a point such that [ b , c , g ] , and additionally, let h be a point such that [ a , g , h ] . If we assume that [ e , f , a ] , [ f , d , b ] , and [ d , e , c ] hold, then from the table
fbd
ecd
agh
it follows that point h is necessarily the tangential of d. Therefore, for point d, we have to take one of the four antecedents of h. Let us take one such point d, and then let e and f be points such that [ c , d , e ] and [ b , d , f ] hold. Then [ e , f , a ] follows from the table
dce.
bdf
hga
In the previous inference, if points a, b, and c are collinear, then g = a , and thus [ a , a , h ] . Therefore, point a is one of the antecedents of point h, and any of the other three can be taken as point d. In case d = a , we obtain e = b and f = c . Theorem 16 solves the problem of finding a triangle whose “sides” pass through the given points.
Theorem 24.
If a 1 , a 2 , a 3 , and a 4 are different points with the common tangential a , then [ a 1 , a 2 , a 3 , a 4 , b , c ] implies [ b , c , a ] , where a is the tangential of a .
Proof. 
According to Lemma 2, there is a point p such that [ a 1 , a 2 , p ] , [ a 3 , a 4 , p ] , and points p an a have the common tangential a . Let t be a point such that [ b , c , t ] . As [ a 1 , a 2 , a 3 , a 4 , b , c ] , [ a 1 , a 2 , p ] , [ a 3 , a 4 , p ] , and [ b , c , t ] imply [ p , p , t ] , we obtain that t is the tangential of p (i.e., t = a ). □
Let a, b, c, and d be associated points with the common tangential p. According to Lemma 3, there are points d, e, and f such that [ a , b , e ] , [ c , d , e ] , [ a , c , f ] , [ b , d , f ] , [ a , d , g ] , and [ b , c , g ] . Points p, e, f, and g are associated with the common tangential p . According to the same lemma, there are points e 1 , f 1 , and g 1 such that [ p , e , e 1 ] , [ f , g , e 1 ] , [ p , f , f 1 ] , [ e , g , f 1 ] , [ p , g , g 1 ] , and [ e , f , g 1 ] . Points p , e 1 , f 1 , and g 1 are associated with the common tangential p .
Theorem 25.
Using the same labeling and the results from above, there exist points q, r, s, and t such that [ p , a , q ] , [ b , e 1 , q ] , [ c , f 1 , q ] , [ d , g 1 , q ] , [ p , b , r ] , [ a , e 1 , r ] , [ d , f 1 , r ] , [ c , g 1 , r ] , [ p , c , s ] , [ d , e 1 , s ] , [ a , f 1 , s ] , [ b , g 1 , s ] , [ p , d , t ] , [ c , e 1 , t ] , [ b , f 1 , t ] , and [ a , g 1 , t ] hold true, and points q, r, s, and t are associated.
Proof. 
Let point q be such that [ p , a , q ] . Then, [ b , e 1 , q ] , [ c , f 1 , q ] , and [ d , g 1 , q ] follow from the tables
fdb,edc,andecd.
fge1egf1efg1
paqpaqpaq
Similarly, [ p , b , r ] implies [ a , e 1 , r ] , [ d , f 1 , r ] , and [ c , g 1 , r ] , and from [ p , c , s ] , we obtain [ d , e 1 , s ] , [ a , f 1 , s ] , and [ b , g 1 , s ] , and [ p , d , t ] implies [ c , e 1 , t ] , [ b , f 1 , t ] , and [ a , g 1 , t ] . Finally, switching to the tangentials, from [ p , a , q ] , [ p , b , r ] , [ p , c , s ] , and [ p , d , t ] , we obtain [ p , p , q ] , [ p , p , r ] , [ p , p , s ] , and [ p , p , t ] (i.e., q = r = s = t ). □

5. Hesse Configuration

The configuration ( 12 3 , 16 4 ) of points and lines obtained in Lemma 4 is called the Hesse configuration. In this configuration, we have the line [ a 1 , b 1 , c 1 ] , nine lines of the forms [ a 1 , b i , c i ] , [ a i , b 1 , c i ] , and [ a i , b i , c 1 ] ( i = 2 , 3 , 4 ) , and six lines of the form [ a i , b j , c k ] , where ( i , j , k ) is any permutation of ( 2 , 3 , 4 ) . The lines of this configuration can be divided into 4 quadruplets of lines, each of which contains all 12 points of the configuration:
[ a 1 , b 1 , c 1 ] , [ a 2 , b 3 , c 4 ] , [ a 3 , b 4 , c 2 ] , [ a 4 , b 2 , c 3 ] ,
[ a 1 , b 2 , c 2 ] , [ a 2 , b 4 , c 3 ] , [ a 3 , b 3 , c 1 ] , [ a 4 , b 1 , c 4 ] ,
[ a 1 , b 3 , c 3 ] , [ a 2 , b 1 , c 2 ] , [ a 3 , b 2 , c 4 ] , [ a 4 , b 4 , c 1 ] ,
[ a 1 , b 4 , c 4 ] , [ a 2 , b 2 , c 1 ] , [ a 3 , b 1 , c 3 ] , [ a 4 , b 3 , c 2 ] .
Moreover, because of the following two theorems, the points of the Hesse configuration lie with some other points on some more lines.
Theorem 26.
Using the notation from Lemma 4, there are points x i , y i , and z i ( i = 2 , 3 , 4 ) such that there are 18 lines [ a 1 , a i , x i ] , [ a j , a k , x i ] , [ b 1 , b i , y i ] , [ b j , b k , y i ] , [ c 1 , c i , z i ] , and [ c j , c k , z i ] , where i = 2 , 3 , 4 , j , k i , and j < k .
Proof. 
Let x i , y i , and z i ( i = 2 , 3 , 4 ) be points such that [ a 1 , a i , x i ] , [ b 1 , b i , y i ] , and [ c 1 , c i , z i ] hold. From the table
b1cjaj
c1bkak
a1aixi
we obtain [ a j , a k , x i ] . Cyclically permuting letters a, b, and c in the previous table proves [ b j , b k , y i ] , and repeating it proves [ c j , c k , z i ] . □
Theorem 27.
For each permutation ( i , j , k ) of ( 2 , 3 , 4 ) , for the points from Theorem 22 there is a line [ x i , y j , z k ] .
Proof. 
The proof follows by applying the table
a1aixi.
b1bjyj
c1ckzk
Theorem 28.
Let a be the common tangential of mutually different points a 1 , a 2 , a 3 , and a 4 in a cubic structure of rank 2. If o is any point, and if b 1 , b 2 , b 3 , and b 4 are points such that [ o , a i , b i ] ( i = 1 , 2 , 3 , 4 ) , then b 1 , b 2 , b 3 , and b 4 are mutually different points with the common tangential b such that [ o , a , b ] , where o is the tangential of point o. In addition, there are mutually different points c, d, and e which are different from point o such that points c, d, e, and o have the common tangential o and [ a 1 , b 2 , c ] , [ a 2 , b 1 , c ] , [ a 3 , b 4 , c ] , [ a 4 , b 3 , c ] , [ a 1 , b 3 , d ] , [ a 3 , b 1 , d ] , [ a 2 , b 4 , d ] , [ a 4 , b 2 , d ] , [ a 1 , b 4 , e ] , [ a 4 , b 1 , e ] , [ a 2 , b 3 , e ] , and [ a 3 , b 2 , e ] hold true.
Proof. 
Let a 1 and a 2 be different points with the common tangential a , let o be any point, and let b 1 and b 2 be points such that [ o , a 1 , b 1 ] and [ o , a 2 , b 2 ] . Then, according to Lemma 1, points b 1 and b 2 are different and have the common tangential b such that [ o , a , b ] , where o is the tangential of o. Aside from that, there is a point c such that [ a 1 , b 2 , c ] and [ a 2 , b 1 , c ] , and points o and c have the common tangential o . Points o and c are different because otherwise, we would have [ a 1 , b 1 , o ] and [ a 1 , b 2 , o ] , where b i b 2 . In a cubic structure of rank 2, each point has four different antecedent points. Let a 1 , a 2 , a 3 , and a 4 be different points with the common tangential a . If o is any point, and b 1 , b 2 , b 3 , and b 4 are points such that [ o , a i , b i ] ( i = 1 , 2 , 3 , 4 ) , then points b 1 , b 2 , b 3 , and b 4 are mutually different. Due to the previous facts, points b 1 , b 2 , b 3 , and b 4 have the common tangential b such that [ o , a , b ] , and there are points c, d, and e such that [ a 1 , b 2 , c ] , [ a 2 , b 1 , c ] , [ a 1 , b 3 , d ] , [ a 3 , b 1 , d ] , [ a 1 , b 4 , e ] , and [ a 4 , b 1 , e ] and which have the tangential o . Points c, d, and e are mutually different because, for example, c = d would imply [ a 1 , b 3 , c ] , which contradicts [ a 1 , b 2 , c ] , where b 2 b 3 , and they are also different from point o. There is also point f with tangential o and different from o such that [ a 2 , b 3 , f ] and [ a 3 , b 2 , f ] . It must coincide with one of points c, d, or e. Due to [ a 2 , b 1 , c ] and [ a 1 , b 3 , d ] , it can be neither c nor d, and thus f = e (i.e., [ a 2 , b 3 , e ] and [ a 3 , b 2 , e ] hold true). Similarly, one can show that [ a 2 , b 4 , d ] , [ a 4 , b 2 , d ] , [ a 3 , b 4 , c ] , and [ a 4 , b 3 , c ] . □
If we now rename points o, c, d, and e as c 0 , c 1 , c 2 , and c 3 , respectively, then we have 16 lines of the form [ a i , b j , c k ] , where the indices i , j , k { 0 , 1 , 2 , 3 } are such that either all three are different or all three are equal to zero. Otherwise, one index is equal to zero, and the other two are equal and different from zero. We find the Table 1 from which one can see which point c k lies on the same line with some point a i and some point b j . We once again obtain the Hesse configuration ( 12 4 , 16 3 ) from Lemma 4.
Theorem 29.
The following quadrilaterals exist in the Hesse configuration shown in the previous table:
{ a 0 , a 1 ; b 0 , b 1 ; c 0 , c 1 } , { a 0 , a 1 ; b 2 , b 3 ; c 2 , c 3 } , { a 2 , a 3 ; b 0 , b 1 ; c 2 , c 3 } , { a 2 , a 3 ; b 2 , b 3 ; c 0 , c 1 } ,
{ a 0 , a 2 ; b 0 , b 2 ; c 0 , c 2 } , { a 0 , a 2 ; b 1 , b 3 ; c 1 , c 3 } , { a 1 , a 3 ; b 0 , b 2 ; c 1 , c 3 } , { a 1 , a 3 ; b 1 , b 3 ; c 0 , c 2 } ,
{ a 0 , a 3 ; b 0 , b 3 ; c 0 , c 3 } , { a 0 , a 3 ; b 1 , b 2 ; c 1 , c 2 } , { a 1 , a 2 ; b 0 , b 3 ; c 1 , c 2 } , { a 1 , a 2 ; b 1 , b 2 ; c 0 , c 3 } .
Proof. 
The proof is obvious when referring to the above table. For example, the last statement is a consequence of the existence of lines [ a 1 , b 1 , c 0 ] , [ a 1 , b 2 , c 3 ] , [ a 2 , b 1 , c 3 ] , and [ a 2 , b 2 , c 0 ] . □

6. The de Vries Configuration

In [11,12], it is claimed (and in [13], it is proven) that for an elliptic cubic, there are only two non-isomorphic configurations ( 12 4 , 16 3 ) in which there are three disjoint quadruples of points such that no two points from a particular quadruple are on one of the 16 lines of the configuration. One of these configurations is the Hesse configuration, and the other can be called the de Vries configuration. Both configurations were also observed in [14,15,16]. All five authors used the properties of the ambient space (i.e., the properties of the projective plane in which the cube is embedded). In this paper, we will observe the de Vries configuration in any cubic structure by means of that structure (i.e., using only axioms C1– C3). The observed cubic structure should be of at least rank 1.
We start from three non-collinear points a 0 , a 1 , and b 0 . Let c 0 and a 01 be points such that [ a 0 , b 0 , c 0 ] and [ a 0 , a 1 , a 01 ] , and let a 2 and a 3 be two different points with the common tangential a 01 . Let b 2 , b 3 , c 2 , and c 3 be points such that [ c 0 , a 2 , b 3 ] , [ c 0 , a 3 , b 2 ] , [ b 0 , a 2 , c 2 ] , and [ b 0 , a 3 , c 3 ] hold, and let b 1 be the point such that [ a 2 , c 3 , b 1 ] . Then, by using the tables
a01a0a1,a01a0a1,a01a3a3.
a2c0b3a3c0b2a2b0c2
a2b0c2a3b0c3a2c3b1
we obtain [ a 1 , b 3 , c 2 ] , [ a 1 , b 2 , c 3 ] , and [ a 3 , c 2 , b 1 ] . If b 1 and c 1 are points such that [ a 1 , c 0 , b 1 ] and [ a 1 , b 0 , c 1 ] , then from the tables
b1c3a2,b1c2a3
c0a3b2c0a2b3
a1b0c1a1b0c1
we have [ a 2 , b 2 , c 1 ] and [ a 3 , b 3 , c 1 ] . Finally, from the tables
c0b0a0,c0b0a0,c0b0a0.
a3c2b1a2c1b2a1c2b3
b2a2c1b3a1c2b1a2c3
we obtain [ a 0 , b 1 , c 1 ] , [ a 0 , b 2 , c 2 ] , and [ a 0 , b 3 , c 3 ] . Therefore, we proved the following theorem:
Theorem 30.
There exists a configuration ( 12 4 , 16 3 ) of points a i , b i , and c i , where i = 0 , 1 , 2 , 3 , with the corresponding Table 2.
The configuration from Theorem 30 is the de Vries configuration. To create it, we begin such that points a 2 and a 3 have the common tangential a 01 satisfying [ a 0 , a 1 , a 01 ] . However, such a property appears several times more in the configuration (i.e., the following holds true):
Theorem 31.
In the configuration from Theorem 30, pairs of points a 0 , a 1 ; a 2 , a 3 ; b 0 , b 1 ; b 2 , b 3 ; c 0 , c 1 ; and c 2 , c 3 have common tangentials a 23 , a 01 , b 23 , b 01 , c 23 , and c 01 , respectively, such that [ a 2 , a 3 , a 23 ] , [ a 0 , a 1 , a 01 ] , [ b 2 , b 3 , b 23 ] , [ b 0 , b 1 , b 01 ] , [ c 2 , c 3 , c 23 ] , and [ c 0 , c 1 , c 01 ] hold true.
Proof. 
We will prove all statements using only Table 2 and statements [ a 2 , a 3 , a 23 ] , [ a 0 , a 1 , a 01 ] , [ b 2 , b 3 , b 23 ] , [ b 0 , b 1 , b 01 ] , [ c 2 , c 3 , c 23 ] , and [ c 0 , c 1 , c 01 ] (i.e., independent from the construction method of the observed configuration). Statements [ a 0 , a 0 , a 23 ] , [ a 1 , a 1 , a 23 ] , [ a 2 , a 2 , a 01 ] , and [ a 3 , a 3 , a 01 ] can be derived from the tables
b0c0a0,b0c1a1,b0c2a2,b0c3a3.
c2b2a0c2b3a1c0b3a2c0b2a3
a2a3a23a2a3a23a0a1a01a0a1a01
We obtain [ b 0 , b 0 , b 23 ] , [ b 1 , b 1 , b 23 ] , [ b 2 , b 2 , b 01 ] , and [ b 3 , b 3 , b 01 ] from the tables
c0a0b0,c0a1b1,c0a3b2,c0a2b3.
a3c3b0a3c2b1a0c2b2a0c3b3
b2b3b23b2b3b23b0b1b01b0b1b01
Then, [ c 0 , c 0 , c 23 ] , [ c 1 , c 1 , c 23 ] , [ c 2 , c 2 , c 01 ] , and [ c 3 , c 3 , c 01 ] follow from the tables
a0b0c0,a0b1c1,a0b2c2,a0b3c3.
b2a3c0b2a3c1b0a2c2b0a3c3
c2c3c23c2c3c23c0c1c01c0c1c01
Theorem 32.
Under the conditions of Theorem 30, there exist complete quadrilaterals ( a 0 , a 1 ; b 0 , b 1 ; c 0 , c 1 ) , ( a 2 , a 3 ; b 2 , b 3 ; c 1 , c 0 ) , ( a 2 , a 3 ; b 0 , b 1 ; c 2 , c 3 ) , and ( a 0 , a 1 ; b 2 , b 3 ; c 2 , c 3 ) .
Proof. 
Using Table 2, it is easy to check the existence of quadruples of the required lines. For instance, for the last quadrilateral, we have the lines [ a 0 , b 2 , c 2 ] , [ a 0 , b 3 , c 3 ] , [ a 1 , b 2 , c 3 ] , and [ a 1 , b 3 , c 2 ] . □
From Table 2, we see that the lines
[ a 0 , b 0 , c 0 ] , [ a 2 , c 0 , b 3 ] , [ a 0 , b 3 , c 3 ] , [ a 2 , c 3 , b 1 ] , [ a 0 , b 1 , c 1 ] , [ a 2 , c 1 , b 2 ] , [ a 0 , b 2 , c 2 ] , [ a 2 , c 2 , b 0 ]
pass alternately through points a 0 and a 2 , while the lines
[ a 1 , b 0 , c 1 ] , [ a 3 , c 1 , b 3 ] , [ a 1 , b 3 , c 2 ] , [ a 3 , c 2 , b 1 ] , [ a 1 , b 1 , c 0 ] , [ a 3 , c 0 , b 2 ] , [ a 1 , b 2 , c 3 ] , [ a 3 , c 3 , b 0 ]
pass alternately through points a 1 and a 3 . In fact, we find two Steiner octagons with fundamental points a 0 , a 2 and a 1 , a 3 , and all 16 lines of the configuration were used. However, it is also possible to form two Steiner octagons from all 16 lines of configuration with fundamental points a 0 , a 3 and a 1 , a 2 , 2 with fundamental points b 0 , b 2 and b 1 , b 3 , 2 with fundamental points b 0 , b 3 and b 1 , b 2 , 2 with fundamental points c 0 , c 2 and c 1 , c 3 , and 2 with fundamental points c 0 , c 3 and c 1 , c 2 . In the two observed octagons, opposing vertices have the common tangential, and the same holds for the other Steiner octagons.
Using the following theorems, we will show how to associate yet another de Vries configuration to the one from Theorem 30.
Theorem 33.
There exists a complete quadrilateral ( a 23 , a 01 ; b 23 , b 01 ; c 23 , c 01 ) .
Proof. 
From the tables
a3a2a23,a3a2a23,b3b2b23,c3c2c23
b3b2b23b1b0b01c1c0c01a1a0a01
c1c1c23c2c2c01a3a3a01b2b2b01
we obtain the lines [ a 23 , b 23 , c 23 ] , [ a 23 , b 01 , c 01 ] , [ a 01 , b 23 , c 01 ] , and [ a 01 , b 01 , c 23 ] , which proves the statement. □
Since the opposite vertices of the quadrilaterals have common tangentials, we have the following:
Corollary 1.
The pairs of points a 23 , a 01 ; b 23 , b 01 ; and c 23 , c 01 have common tangentials.
Theorem 34.
There exist points x 2 , x 3 , y 2 , y 3 , z 2 , and z 3 such that there exist lines [ a 0 , a 2 , x 2 ] , [ a 1 , a 3 , x 2 ] , [ a 0 , a 3 , x 3 ] , [ a 1 , a 2 , x 3 ] , [ b 0 , b 2 , y 2 ] , [ b 1 , b 3 , y 2 ] , [ b 0 , b 3 , y 3 ] , [ b 1 , b 2 , y 3 ] , [ c 0 , c 2 , z 2 ] , [ c 1 , c 3 , z 2 ] , [ c 0 , c 3 , z 3 ] , and [ c 1 , c 2 , z 3 ] . Furthermore, there exist quadrilaterals ( a 0 , a 1 ; a 2 , a 3 ; x 2 , x 3 ) , ( b 0 , b 1 ; b 2 , b 3 ; y 2 , y 3 ) , and ( c 0 , c 1 ; c 2 , c 3 ; z 2 , z 3 ) . Finally, the pairs of points x 2 , x 3 ; y 2 , y 3 ; and z 2 , z 3 have common tangentials.
Proof. 
Let x 2 , x 3 , y 2 , y 3 , z 2 , and z 3 be such that there are lines [ a 0 , a 2 , x 2 ] , [ a 0 , a 3 , x 3 ] , [ b 0 , b 2 , y 2 ] , [ b 0 , b 3 , y 3 ] , [ c 0 , c 2 , z 2 ] , and [ c 0 , c 3 , z 3 ] . The existence of the remaining six lines can be inferred from the following tables:
c0b1a1,b0c1a1,a0c1b1,
b0c3a3c0b3a2c0a2b3
a0a2x2a0a3x3b0b2y2
 
a0c1b1,a0b1c1,a0b1c1,
c0a3b2b0a3c3b0a2c2
b0b3y3c0c2z2c0c3z3
The existence of these 12 lines proves the existence of the three mentioned quadrilaterals, and the last statement is an immediate consequence of [5] (Theorem 3.4). □
Theorem 35.
There exists a de Vries configuration ( 12 4 , 16 3 ) of points a 01 , a 23 , x 2 , x 3 , b 01 , b 23 , y 2 , y 3 , c 01 , c 23 , z 2 , and z 3 with the corresponding Table 3.
Proof. 
Except for the lines from Theorem 33, one should prove the existence of yet another 12 lines. But this follows from the following tables:
a0a1a01,a0a1a01,a2a3a23,a2a3a23,
b0b2y2b0b3y3b0b2y2b0b3y3
c0c3z3c0c2z2c2c0z2c2c1z3
 
a0a2x2,a0a2x2,a0a3x3,a0a3x3,
b0b1b01b2b3b23b0b1b01b2b3b23
c0c3z3c2c0z2c0c2z2c2c1z3
 
a0a2x2,a0a2x2,a0a3x3,a0a3x3,
b0b2y2b2b1y3b2b0y2b0b3y3
c0c1c01c2c3c23c2c3c23c0c1c01
Comparing Table 3 in the text of this theorem with Table 2 in Theorem 30 reveals that they represent the same configuration. □
Using Table 2, we proved Theorems 31 and 32 concerning the existence of some common tangentials and quadrilaterals. Similarly, by employing Table 3, we can prove analogous theorems about the existence of common tangentials and quadrilaterals in this second de Vries configuration.

7. Conclusions

The concept of a tangential in a general cubic structure was introduced and studied in [5]. In this paper, we explored the noteworthy properties of tangentials in cubic structures of ranks 0, 1, and 2. We investigated the relationships between tangentials and various other concepts in cubic structures of specific ranks. Additionally, we constructed the Hesse configuration of points and lines in a cubic structure of rank 2. We obtained and explored the de Vries configuration of points and lines in a cubic structure. The authors’ future research aims to conduct a more detailed investigation into admissible and non-admissible configurations in cubic structures. However, in order to accomplish this, it is important to first dig up some additional significant properties of the tangentials in cubic structures beyond those already discovered in [5]. This paper used the cubic structure to demonstrate how the results can be reached with this quite simple structure. These findings were expressed in the language of models in the most well-known cubic structure: the geometry on cubic curves. However, certain additional cubic structure models were considered in [4], and therefore the findings achieved using a cubic structure were also readily obtained in these models.

Author Contributions

Conceptualization, V.V. and R.K.-Š.; validation and writing—original draft preparation, V.V. and R.K.-Š. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data are contained within the article.

Acknowledgments

The authors would like to thank the referees for their valuable comments which helped to improve the manuscript.

Conflicts of Interest

The authors declare no conflicts of interest.

References

  1. Etherington, L.M.H. Quasigroups and cubic curves. Proc. Edinb. Math. Soc. 1965, 14, 273–291. [Google Scholar] [CrossRef]
  2. Khalid, M. Group law on the cubic curve. Irish Math. Soc. Bull. 2007, 60, 67–89. [Google Scholar] [CrossRef]
  3. Zirilli, F. C-struttura associata ad una cubica piana e C-struttura astratta. Ricerche Mat. 1967, 16, 202–232. [Google Scholar]
  4. Volenec, V.; Kolar-Begović, Z.; Kolar-Šuper, R. Cubic structure. Glas. Mat. Ser. III 2017, 52, 247–256. [Google Scholar] [CrossRef]
  5. Volenec, V.; Kolar-Begović, Z.; Kolar-Šuper, R. Tangentials in cubic structures. Glas. Mat. Ser. III 2020, 55, 337–349. [Google Scholar] [CrossRef]
  6. Cremona, L. Einleitung in Eine Geometrische Theorie der Ebenen Kurven; Hansebooks: Greifswald, Germany, 1865. [Google Scholar]
  7. Durège, H. Die Ebenen Curven Dritter Ordnung; Teubner: Leipzig, Germany, 1871. [Google Scholar]
  8. Schroeter, H. Die Theorie der Ebenen Kurven Dritter Ordnung. Auf Synthetisch-Geometrischem Wege; Teubner: Leipzig, Germany, 1888. [Google Scholar]
  9. Smogorzhevski, A.S.; Stolova, E.S. Handbook in the Theory of Plane Curves of the Third Order; State Publishing House of Physical and Mathematical Literature: Moscow, Russia, 1961. (In Russian) [Google Scholar]
  10. Green, H.G.; Prior, L.E. Systems of triadic points on a cubic. Amer. Math. Mon. 1934, 41, 253–255. [Google Scholar] [CrossRef]
  11. de Vries, J. Über gewisse Configurationen auf ebenen kubischen Curven. Sitzungsber. Akad. Wiss. Wien 1889, 98, 1290–1298. [Google Scholar]
  12. de Vries, J. Over vlakke configurations. Versl. Kon. Akad. Wetensch. 1889, 5, 105–120. [Google Scholar]
  13. Schroeter, H. Die Hessesche Configuration (124,163). J. Reine Angew. Math. 1891, 108, 269–312. [Google Scholar] [CrossRef]
  14. Bydžovský, B. Poznámky k theorii konfigurace (124,163). Čas. Pěst. Mat. Fys. 1949, 74, 249–251. [Google Scholar]
  15. Martinetti, V. Sopra un gruppo di configurazioni contenute nell’ esagramma di Pascal. Atti Accad. Gioenia Sci. Nat. Catania 1890, 3, 31–50. [Google Scholar]
  16. Zacharias, M. Untersuchungen über ebene Konfigurationen (124,163). Deutsche Math. 1941, 6, 147–170. [Google Scholar]
Table 1. Hesse configuration ( 12 4 , 16 3 ) (of points a i , b i , c i ( i = 0 , 1 , 2 , 3 ) ).
Table 1. Hesse configuration ( 12 4 , 16 3 ) (of points a i , b i , c i ( i = 0 , 1 , 2 , 3 ) ).
b 0 b 1 b 2 b 3
a 0 c 0 c 1 c 2 c 3
a 1 c 1 c 0 c 3 c 2
a 2 c 2 c 3 c 0 c 1
a 3 c 3 c 2 c 1 c 0
Table 2. The de Vries configuration ( 12 4 , 16 3 ) (of points a i , b i , c i ( i = 0 , 1 , 2 , 3 ) ).
Table 2. The de Vries configuration ( 12 4 , 16 3 ) (of points a i , b i , c i ( i = 0 , 1 , 2 , 3 ) ).
b 0 b 1 b 2 b 3
a 0 c 0 c 1 c 2 c 3
a 1 c 1 c 0 c 3 c 2
a 2 c 2 c 3 c 1 c 0
a 3 c 3 c 2 c 0 c 1
Table 3. The de Vries configuration ( 12 4 , 16 3 ) (of points a 01 , a 23 , x 2 , x 3 , b 01 , b 23 , y 2 , y 3 , c 01 , c 23 , z 2 , z 3 ).
Table 3. The de Vries configuration ( 12 4 , 16 3 ) (of points a 01 , a 23 , x 2 , x 3 , b 01 , b 23 , y 2 , y 3 , c 01 , c 23 , z 2 , z 3 ).
b 01 b 23 y 2 y 3
a 01 c 23 c 01 z 3 z 2
a 23 c 01 c 23 z 2 z 3
x 2 z 3 z 2 c 01 c 23
x 3 z 2 z 3 c 23 c 01
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Volenec, V.; Kolar-Šuper, R. On the Noteworthy Properties of Tangentials in Cubic Structures. Axioms 2024, 13, 122. https://doi.org/10.3390/axioms13020122

AMA Style

Volenec V, Kolar-Šuper R. On the Noteworthy Properties of Tangentials in Cubic Structures. Axioms. 2024; 13(2):122. https://doi.org/10.3390/axioms13020122

Chicago/Turabian Style

Volenec, Vladimir, and Ružica Kolar-Šuper. 2024. "On the Noteworthy Properties of Tangentials in Cubic Structures" Axioms 13, no. 2: 122. https://doi.org/10.3390/axioms13020122

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop