# On the Noteworthy Properties of Tangentials in Cubic Structures

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## Abstract

**:**

## 1. Introduction

- C1.
- For any two points $a,b\in Q$, there is a unique point $c\in Q$ such that $[a,b,c]$ (i.e., $(a,b,c)\in \left[\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\right]$).
- C2.
- The relation $\left[\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\right]$ 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:

a | b | c | . |

d | e | f | |

g | h | i |

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Lemma**

**4.**

## 2. Some Properties of Tangentials in a General Cubic Structure

**Theorem**

**1.**

**Proof.**

a′ | r | p | . | □ |

a | q | p | ||

a | s | t |

**Theorem**

**2.**

**Proof.**

q | q | q′ |

a_{1} | a_{2} | r |

b_{1} | b_{2} | s |

q | c_{1} | d_{1} | . | □ |

q | c_{2} | d_{2} | ||

q′ | r | s |

**Theorem**

**3.**

**Proof.**

p | r | u | . | □ |

q | s | v | ||

a | a | a′ |

**Theorem**

**4.**

**Proof.**

t | a_{1} | a_{2} | t | c_{1} | c_{2} | |

t | b_{1} | b_{2} | t | d_{1} | d_{2} | |

t′ | e_{1} | e_{2} | t′ | e_{1} | e_{2} |

**Theorem**

**5.**

**Proof.**

a′ | a_{2} | a_{2} | . | □ |

a_{1} | b | c | ||

a_{1} | d | e |

**Theorem**

**6.**

**Proof.**

a | f | b | . |

a | e | c | |

h | g | d |

**Theorem**

**7.**

a_{1} | b_{1} | c_{1} |

a_{2} | b_{2} | c_{2} |

a_{3} | b_{3} | c_{3} |

**Proof.**

${b}_{1}^{\prime}$ | ${a}_{1}^{\prime}$ | ${c}_{1}^{\prime}$ | ${c}_{2}^{\prime}$ | ${b}_{2}^{\prime}$ | ${a}_{2}^{\prime}$ | |

${b}_{1}$ | ${b}_{2}$ | ${b}_{3}$ | ${c}_{2}$ | ${c}_{3}$ | ${c}_{1}$ | |

${b}_{1}$ | ${b}_{2}$ | ${b}_{3}$ | ${c}_{2}$ | ${c}_{3}$ | ${c}_{1}$ |

**Theorem**

**8.**

**Proof.**

a′ | a | a | . | □ |

b | b′ | b | ||

c | c | c′ |

**Theorem**

**9.**

**Proof.**

a | c | b | □ |

a′ | d | d | |

a | e | f |

**Theorem**

**10.**

**Proof.**

a′ | d | d | . |

a | c | e | |

a | b | f |

c | b | d | . | □ |

e | f | d | ||

a | a | a′ |

**Theorem**

**11.**

**Proof.**

q | d | b | , | p | d | c | , | p | c | d | . | ||

c | r | b | b | r | c | b | q | d | |||||

a | a | t | a | a | t | a | a | t |

a | b | p | , |

a | b | p | |

t | t | t′ |

**Theorem**

**12.**

**Proof.**

q | b | b | q | c | c | . | |

a | c | e | a | b | f | ||

a | d | p | a | d | p |

p | e | b | p | f | c | |

d | c | b | d | b | c | |

a | a | q | a | a | q |

**Theorem**

**13.**

**Proof.**

a | a | a′ | . | □ |

c | b | d | ||

e | f | p |

**Theorem**

**14.**

**Proof.**

e | f | d | , |

c | b | d | |

a | a | a′ |

**Theorem**

**15.**

**Proof.**

b | e | h | , | a | d | g | , | and | a | d | g | , | |

c | f | i | c | f | i | b | e | h | |||||

d | d | d′ | e | e | e′ | f | f | f′ |

**Theorem**

**16.**

**Proof.**

a_{k} | a_{j} | b_{i} | . | □ |

b_{j} | b_{k} | c_{i} | ||

a_{i} | a_{i} | ${a}_{i}^{\prime}$ |

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

**Theorem**

**17.**

**Proof.**

**Theorem**

**18.**

**Proof.**

a_{1} | a_{2} | c |

a_{1} | a_{2} | c |

a | a | a′ |

**Theorem**

**19.**

**Proof.**

**Theorem**

**20.**

**Proof.**

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

**Theorem**

**21.**

**Proof.**

**Theorem**

**22.**

**Proof.**

a′ | d′ | g′ |

b′ | e′ | h′ |

c′ | f′ | i′ |

d | g | a | , |

e | h | b | |

f | i | c |

**Theorem**

**23.**

**Proof.**

f | b | d |

e | c | d |

a | g | h |

d | c | e | . | □ |

b | d | f | ||

h | g | a |

**Theorem**

**24.**

**Proof.**

**Theorem**

**25.**

**Proof.**

f | d | b | , | e | d | c | , | and | e | c | d | . | |

f | g | e_{1} | e | g | f_{1} | e | f | g_{1} | |||||

p′ | a | q | p′ | a | q | p′ | a | q |

## 5. Hesse Configuration

$[{a}_{1},{b}_{1},{c}_{1}]$, | $[{a}_{2},{b}_{3},{c}_{4}]$, | $[{a}_{3},{b}_{4},{c}_{2}]$, | $[{a}_{4},{b}_{2},{c}_{3}]$, |

$[{a}_{1},{b}_{2},{c}_{2}]$, | $[{a}_{2},{b}_{4},{c}_{3}]$, | $[{a}_{3},{b}_{3},{c}_{1}]$, | $[{a}_{4},{b}_{1},{c}_{4}]$, |

$[{a}_{1},{b}_{3},{c}_{3}]$, | $[{a}_{2},{b}_{1},{c}_{2}]$, | $[{a}_{3},{b}_{2},{c}_{4}]$, | $[{a}_{4},{b}_{4},{c}_{1}]$, |

$[{a}_{1},{b}_{4},{c}_{4}]$, | $[{a}_{2},{b}_{2},{c}_{1}]$, | $[{a}_{3},{b}_{1},{c}_{3}]$, | $[{a}_{4},{b}_{3},{c}_{2}]$. |

**Theorem**

**26.**

**Proof.**

b_{1} | c_{j} | a_{j} |

c_{1} | b_{k} | a_{k} |

a_{1} | a_{i} | x_{i} |

**Theorem**

**27.**

**Proof.**

a_{1} | a_{i} | x_{i} | . | □ |

b_{1} | b_{j} | y_{j} | ||

c_{1} | c_{k} | z_{k} |

**Theorem**

**28.**

**Proof.**

**Theorem**

**29.**

**Proof.**

## 6. The de Vries Configuration

a_{01} | a_{0} | a_{1} | , | a_{01} | a_{0} | a_{1} | , | a_{01} | a_{3} | a_{3} | . | ||

a_{2} | c_{0} | b_{3} | a_{3} | c_{0} | b_{2} | a_{2} | b_{0} | c_{2} | |||||

a_{2} | b_{0} | c_{2} | a_{3} | b_{0} | c_{3} | a_{2} | c_{3} | b_{1} |

b_{1} | c_{3} | a_{2} | , | b_{1} | c_{2} | a_{3} | |

c_{0} | a_{3} | b_{2} | c_{0} | a_{2} | b_{3} | ||

a_{1} | b_{0} | c_{1} | a_{1} | b_{0} | c_{1} |

c_{0} | b_{0} | a_{0} | , | c_{0} | b_{0} | a_{0} | , | c_{0} | b_{0} | a_{0} | . | ||

a_{3} | c_{2} | b_{1} | a_{2} | c_{1} | b_{2} | a_{1} | c_{2} | b_{3} | |||||

b_{2} | a_{2} | c_{1} | b_{3} | a_{1} | c_{2} | b_{1} | a_{2} | c_{3} |

**Theorem**

**30.**

**Theorem**

**31.**

**Proof.**

b_{0} | c_{0} | a_{0} | , | b_{0} | c_{1} | a_{1} | , | b_{0} | c_{2} | a_{2} | , | b_{0} | c_{3} | a_{3} | . | |||

c_{2} | b_{2} | a_{0} | c_{2} | b_{3} | a_{1} | c_{0} | b_{3} | a_{2} | c_{0} | b_{2} | a_{3} | |||||||

a_{2} | a_{3} | a_{23} | a_{2} | a_{3} | a_{23} | a_{0} | a_{1} | a_{01} | a_{0} | a_{1} | a_{01} |

c_{0} | a_{0} | b_{0} | , | c_{0} | a_{1} | b_{1} | , | c_{0} | a_{3} | b_{2} | , | c_{0} | a_{2} | b_{3} | . | |||

a_{3} | c_{3} | b_{0} | a_{3} | c_{2} | b_{1} | a_{0} | c_{2} | b_{2} | a_{0} | c_{3} | b_{3} | |||||||

b_{2} | b_{3} | b_{23} | b_{2} | b_{3} | b_{23} | b_{0} | b_{1} | b_{01} | b_{0} | b_{1} | b_{01} |

a_{0} | b_{0} | c_{0} | , | a_{0} | b_{1} | c_{1} | , | a_{0} | b_{2} | c_{2} | , | a_{0} | b_{3} | c_{3} | . | |||

b_{2} | a_{3} | c_{0} | b_{2} | a_{3} | c_{1} | b_{0} | a_{2} | c_{2} | b_{0} | a_{3} | c_{3} | |||||||

c_{2} | c_{3} | c_{23} | c_{2} | c_{3} | c_{23} | c_{0} | c_{1} | c_{01} | c_{0} | c_{1} | c_{01} |

**Theorem**

**32.**

**Proof.**

**Theorem**

**33.**

**Proof.**

a_{3} | a_{2} | a_{23} | , | a_{3} | a_{2} | a_{23} | , | b_{3} | b_{2} | b_{23} | , | c_{3} | c_{2} | c_{23} | |||

b_{3} | b_{2} | b_{23} | b_{1} | b_{0} | b_{01} | c_{1} | c_{0} | c_{01} | a_{1} | a_{0} | a_{01} | ||||||

c_{1} | c_{1} | c_{23} | c_{2} | c_{2} | c_{01} | a_{3} | a_{3} | a_{01} | b_{2} | b_{2} | b_{01} |

**Corollary**

**1.**

**Theorem**

**34.**

**Proof.**

c_{0} | b_{1} | a_{1} | , | b_{0} | c_{1} | a_{1} | , | a_{0} | c_{1} | b_{1} | , | ||

b_{0} | c_{3} | a_{3} | c_{0} | b_{3} | a_{2} | c_{0} | a_{2} | b_{3} | |||||

a_{0} | a_{2} | x_{2} | a_{0} | a_{3} | x_{3} | b_{0} | b_{2} | y_{2} | |||||

a_{0} | c_{1} | b_{1} | , | a_{0} | b_{1} | c_{1} | , | a_{0} | b_{1} | c_{1} | , | ||

c_{0} | a_{3} | b_{2} | b_{0} | a_{3} | c_{3} | b_{0} | a_{2} | c_{2} | |||||

b_{0} | b_{3} | y_{3} | c_{0} | c_{2} | z_{2} | c_{0} | c_{3} | z_{3} |

**Theorem**

**35.**

**Proof.**

a_{0} | a_{1} | a_{01} | , | a_{0} | a_{1} | a_{01} | , | a_{2} | a_{3} | a_{23} | , | a_{2} | a_{3} | a_{23} | , | |||

b_{0} | b_{2} | y_{2} | b_{0} | b_{3} | y_{3} | b_{0} | b_{2} | y_{2} | b_{0} | b_{3} | y_{3} | |||||||

c_{0} | c_{3} | z_{3} | c_{0} | c_{2} | z_{2} | c_{2} | c_{0} | z_{2} | c_{2} | c_{1} | z_{3} | |||||||

a_{0} | a_{2} | x_{2} | , | a_{0} | a_{2} | x_{2} | , | a_{0} | a_{3} | x_{3} | , | a_{0} | a_{3} | x_{3} | , | |||

b_{0} | b_{1} | b_{01} | b_{2} | b_{3} | b_{23} | b_{0} | b_{1} | b_{01} | b_{2} | b_{3} | b_{23} | |||||||

c_{0} | c_{3} | z_{3} | c_{2} | c_{0} | z_{2} | c_{0} | c_{2} | z_{2} | c_{2} | c_{1} | z_{3} | |||||||

a_{0} | a_{2} | x_{2} | , | a_{0} | a_{2} | x_{2} | , | a_{0} | a_{3} | x_{3} | , | a_{0} | a_{3} | x_{3} | , | |||

b_{0} | b_{2} | y_{2} | b_{2} | b_{1} | y_{3} | b_{2} | b_{0} | y_{2} | b_{0} | b_{3} | y_{3} | |||||||

c_{0} | c_{1} | c_{01} | c_{2} | c_{3} | c_{23} | c_{2} | c_{3} | c_{23} | c_{0} | c_{1} | c_{01} |

## 7. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Hesse configuration $({12}_{4},{16}_{3})$ (of points ${a}_{i},{b}_{i},{c}_{i}\phantom{\rule{0.166667em}{0ex}}(i=0,1,2,3)$).

${b}_{0}$ | ${b}_{1}$ | ${b}_{2}$ | ${b}_{3}$ | |

${a}_{0}$ | ${c}_{0}$ | ${c}_{1}$ | ${c}_{2}$ | ${c}_{3}$ |

${a}_{1}$ | ${c}_{1}$ | ${c}_{0}$ | ${c}_{3}$ | ${c}_{2}$ |

${a}_{2}$ | ${c}_{2}$ | ${c}_{3}$ | ${c}_{0}$ | ${c}_{1}$ |

${a}_{3}$ | ${c}_{3}$ | ${c}_{2}$ | ${c}_{1}$ | ${c}_{0}$ |

**Table 2.**The de Vries configuration $({12}_{4},{16}_{3})$ (of points ${a}_{i},{b}_{i},{c}_{i}\phantom{\rule{0.166667em}{0ex}}(i=0,1,2,3)$).

${b}_{0}$ | ${b}_{1}$ | ${b}_{2}$ | ${b}_{3}$ | |

${a}_{0}$ | ${c}_{0}$ | ${c}_{1}$ | ${c}_{2}$ | ${c}_{3}$ |

${a}_{1}$ | ${c}_{1}$ | ${c}_{0}$ | ${c}_{3}$ | ${c}_{2}$ |

${a}_{2}$ | ${c}_{2}$ | ${c}_{3}$ | ${c}_{1}$ | ${c}_{0}$ |

${a}_{3}$ | ${c}_{3}$ | ${c}_{2}$ | ${c}_{0}$ | ${c}_{1}$ |

**Table 3.**The de Vries configuration $({12}_{4},{16}_{3})$ (of points ${a}_{01},{a}_{23},{x}_{2},{x}_{3},{b}_{01},{b}_{23},{y}_{2},{y}_{3},{c}_{01},{c}_{23},{z}_{2},{z}_{3}$).

${b}_{01}$ | ${b}_{23}$ | ${y}_{2}$ | ${y}_{3}$ | |

${a}_{01}$ | ${c}_{23}$ | ${c}_{01}$ | ${z}_{3}$ | ${z}_{2}$ |

${a}_{23}$ | ${c}_{01}$ | ${c}_{23}$ | ${z}_{2}$ | ${z}_{3}$ |

${x}_{2}$ | ${z}_{3}$ | ${z}_{2}$ | ${c}_{01}$ | ${c}_{23}$ |

${x}_{3}$ | ${z}_{2}$ | ${z}_{3}$ | ${c}_{23}$ | ${c}_{01}$ |

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**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