# Spectra of Reduced Fractals and Their Applications in Biology

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## Abstract

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## 1. Introduction

## 2. Reduced Fractals in Biology

## 3. Applications to Biology

#### 3.1. Selected Biological Systems and Their Self-Similarity

#### 3.2. Fractal Dimension and Box-Counting Method

#### 3.3. Fractal Dimension for Selected Algae

#### 3.4. Spectra of Reduced Fractals for Selected Green Algae

#### 3.5. Discussion of the Obtained Results

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**DIC microscopy images of unicellular algae of the class Zygnematophyceae. (

**a**) Euastrum bidentatum ${F}_{d}=1.8408$, (

**b**) Euastrum oblongum ${F}_{d}=1.8598$, (

**c**) Euastrum verrucosum ${F}_{d}=1.8739$, (

**d**) Euastrum ansatum ${F}_{d}=1.8801$, (

**e**) Euastrum humerosum ${F}_{d}=1.8907$, (

**f**) Euastrum crassum ${F}_{d}=1.8948$, (

**g**) Micrasterias americana ${F}_{d}=1.8117$, (

**h**) Micrasterias truncata ${F}_{d}=1.8703$ and (

**i**) Micrasterias rotata ${F}_{d}=1.8749$.

**Figure 2.**DIC microscopy images of algae of class Chlorophyceae. (

**a**) Volvox globator ${F}_{d}=1.2288$, (

**b**) Volvox aureus ${F}_{d}=1.3701$, (

**c**) Eudorina elegans ${F}_{d}=1.6975$, (

**d**) Scenedesmus granlulatas ${F}_{d}=1.7097$, (

**e**) Pediastrum clothratum ${F}_{d}=1.7182$, (

**f**) Pediastrum angulosum ${F}_{d}=1.7806$, (

**g**) Desmodesmus magnus ${F}_{d}=1.7447$ and (

**h**) Tetraedron minimum ${F}_{d}=1.7087$.

**Figure 3.**Spectra of reduced fractals for algae of different form (panel (

**A**)), class (panel (

**B**)), shape (panel (

**C**)), and family (panel (

**D**)).

**Table 1.**Fractal dimension of unicellular algae of the class Zygnematophyceae and its family Desmidiaceae.

Genus sp. | Colony Size | Magnification |
---|---|---|

Eudorina elegans | 16, 32 | 2 |

Desmodesmus magnus | 4, 8, 16 | 3 |

Scenedesmus granlulatas | 2, 4, 8, 16, 32 | 5 |

Pediastrum angulosum | 4, 8, 32, 64, 128 | 5 |

Pediastrum clothratum | 8, 16, 32, 64 | 4 |

Tetraedron minimum | 2, 4, 8, 16 | 4 |

**Table 2.**Fractal dimension of unicellular algae of the class Zygnematophyceae and its family Desmidiaceae.

Genus Species | Cell Shape | Fractal Dimension |
---|---|---|

Euastrum oblongum | Ellipsoid | $1.8598$ |

Euastrum verrucosum | Ellipsoid | $1.8739$ |

Euastrum ansatum | Ellipsoid | $1.8801$ |

Euastrum humerosum | Ellipsoid | $1.8907$ |

Euastrum crissum | Ellipsoid | $1.8948$ |

Micrasterias americana | Spherical | $1.8117$ |

Micrasterias truncata | Spherical | $1.8703$ |

Micrasterias rotata | Spherical | $1.8749$ |

**Table 3.**Fractal dimension of colonial and multicellular algae of the class Chlorophyceae and its three different families.

Family | Genus sp. | Form | Cell Shape | Fractal Dimension |
---|---|---|---|---|

Volvocaceae | Volvox globator | Multicellular | Spherical | 1.2288 |

Volvocaceae | Volvox aureus | Multicellular | Spherical | 1.3701 |

Volvocaceae | Eudorina elegans | Colonial | Spherical | 1.6975 |

Scenedesmaceae | Scenedesmus granulatus | Colonial | Ellipsoid | 1.7097 |

Crescent | ||||

Scenedesmaceae | Desmodesmus magnus | Colonial | Ellipsoid | 1.7447 |

Hydrodictyaceae | Tetraedron minimum | Colonial | Ellipsoid | 1.7087 |

Spherical | ||||

Hydrodictyaceae | Pediastrum clothratum | Colonial | Oval | 1.7182 |

Hydrodictyaceae | Pediastrum angulosum | Colonial | Cylindrical | 1.7806 |

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**MDPI and ACS Style**

Pham, D.T.; Musielak, Z.E.
Spectra of Reduced Fractals and Their Applications in Biology. *Fractal Fract.* **2023**, *7*, 28.
https://doi.org/10.3390/fractalfract7010028

**AMA Style**

Pham DT, Musielak ZE.
Spectra of Reduced Fractals and Their Applications in Biology. *Fractal and Fractional*. 2023; 7(1):28.
https://doi.org/10.3390/fractalfract7010028

**Chicago/Turabian Style**

Pham, Diana T., and Zdzislaw E. Musielak.
2023. "Spectra of Reduced Fractals and Their Applications in Biology" *Fractal and Fractional* 7, no. 1: 28.
https://doi.org/10.3390/fractalfract7010028