# Inflection Points in Cubic Structures

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction and Motivation

- 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 shown in the following table:$$\begin{array}{c}a\\ d\\ g\end{array}\begin{array}{c}b\\ e\\ h\end{array}\overline{)\begin{array}{c}c\\ f\\ i\end{array}}$$

## 2. Inflection Points

**Lemma**

**1.**

**Proof.**

**Example**

**1.**

a | b | c | |

a | a | c | b |

b | c | b | a |

c | b | a | c |

**Lemma**

**2.**

**Proof.**

**Example**

**2.**

a | b | c | d | |

a | a | b | d | c |

b | b | a | c | d |

c | d | c | b | a |

d | c | d | a | b |

**Proposition**

**1.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Example**

**3.**

a | b | c | d | e | |

a | d | c | b | a | e |

b | c | e | a | d | b |

c | b | a | c | e | d |

d | a | d | e | b | c |

e | e | b | d | c | a |

**Lemma**

**4.**

**Proof.**

**Example**

**4.**

a | b | c | d | e | f | g | h | |

a | e | d | g | b | a | h | c | f |

b | d | f | h | a | g | b | e | c |

c | g | h | c | d | f | e | a | b |

d | b | a | d | c | e | f | h | g |

e | a | g | f | e | d | c | b | h |

f | h | b | e | f | c | d | g | a |

g | c | e | a | h | b | g | f | d |

h | f | c | b | g | h | a | d | e |

**Lemma**

**5.**

**Proof.**

**Example**

**5.**

${a}_{1}$ | ${a}_{2}$ | ${a}_{3}$ | ${a}_{4}$ | |

${a}_{1}$ | ${a}_{3}$ | ${a}_{4}$ | ${a}_{1}$ | ${a}_{2}$ |

${a}_{2}$ | ${a}_{4}$ | ${a}_{3}$ | ${a}_{2}$ | ${a}_{1}$ |

${a}_{3}$ | ${a}_{1}$ | ${a}_{2}$ | ${a}_{4}$ | ${a}_{3}$ |

${a}_{4}$ | ${a}_{2}$ | ${a}_{1}$ | ${a}_{3}$ | ${a}_{4}$ |

**Lemma**

**6.**

**Proof.**

**Example**

**6.**

${a}_{1}$ | ${a}_{2}$ | ${a}_{3}$ | ${a}_{4}$ | |

${a}_{1}$ | ${a}_{4}$ | ${a}_{3}$ | ${a}_{2}$ | ${a}_{1}$ |

${a}_{2}$ | ${a}_{3}$ | ${a}_{4}$ | ${a}_{1}$ | ${a}_{2}$ |

${a}_{3}$ | ${a}_{2}$ | ${a}_{1}$ | ${a}_{4}$ | ${a}_{3}$ |

${a}_{4}$ | ${a}_{1}$ | ${a}_{2}$ | ${a}_{3}$ | ${a}_{4}$ |

**Corollary**

**1.**

**Lemma**

**7.**

**Proof.**

**Example**

**7.**

a | b | c | d | e | f | g | |

a | a | e | f | g | b | c | d |

b | e | f | d | c | a | b | g |

c | f | d | g | b | e | a | c |

d | g | c | b | e | d | f | a |

e | b | a | e | d | c | g | f |

f | c | b | a | f | g | d | e |

g | d | g | c | a | f | e | b |

## 3. Inflection Points in Cubic Structures of Rank 2

**Lemma**

**8.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Example**

**8.**

${a}_{1}$ | ${a}_{2}$ | ${a}_{3}$ | ${a}_{4}$ | ${a}_{5}$ | ${a}_{6}$ | ${a}_{7}$ | ${a}_{8}$ | |

${a}_{1}$ | ${a}_{5}$ | ${a}_{6}$ | ${a}_{7}$ | ${a}_{8}$ | ${a}_{1}$ | ${a}_{2}$ | ${a}_{3}$ | ${a}_{4}$ |

${a}_{2}$ | ${a}_{6}$ | ${a}_{5}$ | ${a}_{8}$ | ${a}_{7}$ | ${a}_{2}$ | ${a}_{1}$ | ${a}_{4}$ | ${a}_{3}$ |

${a}_{3}$ | ${a}_{7}$ | ${a}_{8}$ | ${a}_{5}$ | ${a}_{6}$ | ${a}_{3}$ | ${a}_{4}$ | ${a}_{1}$ | ${a}_{2}$ |

${a}_{4}$ | ${a}_{8}$ | ${a}_{7}$ | ${a}_{6}$ | ${a}_{5}$ | ${a}_{4}$ | ${a}_{3}$ | ${a}_{2}$ | ${a}_{1}$ |

${a}_{5}$ | ${a}_{1}$ | ${a}_{2}$ | ${a}_{3}$ | ${a}_{4}$ | ${a}_{8}$ | ${a}_{7}$ | ${a}_{6}$ | ${a}_{5}$ |

${a}_{6}$ | ${a}_{2}$ | ${a}_{1}$ | ${a}_{4}$ | ${a}_{3}$ | ${a}_{7}$ | ${a}_{8}$ | ${a}_{5}$ | ${a}_{6}$ |

${a}_{7}$ | ${a}_{3}$ | ${a}_{4}$ | ${a}_{1}$ | ${a}_{2}$ | ${a}_{6}$ | ${a}_{5}$ | ${a}_{8}$ | ${a}_{7}$ |

${a}_{8}$ | ${a}_{4}$ | ${a}_{3}$ | ${a}_{2}$ | ${a}_{1}$ | ${a}_{5}$ | ${a}_{6}$ | ${a}_{7}$ | ${a}_{8}$ |

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Volenec, V.; Kolar-Begović, Z.; Kolar-Šuper, R.
Inflection Points in Cubic Structures. *Mathematics* **2021**, *9*, 2819.
https://doi.org/10.3390/math9212819

**AMA Style**

Volenec V, Kolar-Begović Z, Kolar-Šuper R.
Inflection Points in Cubic Structures. *Mathematics*. 2021; 9(21):2819.
https://doi.org/10.3390/math9212819

**Chicago/Turabian Style**

Volenec, Vladimir, Zdenka Kolar-Begović, and Ružica Kolar-Šuper.
2021. "Inflection Points in Cubic Structures" *Mathematics* 9, no. 21: 2819.
https://doi.org/10.3390/math9212819