### Abstract

Objectives: Many biological structures are products of repeated iteration functions. As such, they demonstrate characteristic, scale-invariant features. Fractal analysis of these features elucidates the mechanism of their formation. The objectives of this project were to determine whether human cranial sutures demonstrate self-similarity and measure their exponents of similarity (fractal dimensions). Design: One hundred three documented human skulls from the Terry Collection of the Smithsonian Institution were used. Their sagittal sutures were digitized and the data converted to bitmap images for analysis using box-counting method of fractal software. Results: The log-log plots of the number of boxes containing the sutural pattern, N_{r}, and the size of the boxes, r, were all linear, indicating that human sagittal sutures possess scale-invariant features and thus are fractals. The linear portion of these log-log plots has limits because of the finite resolution used for data acquisition. The mean box dimension, D_{b}, was 1.29289 ± 0.078457 with a 95% confidence interval of 1.27634 to 1.30944. Conclusions: Human sagittal sutures are self-similar and have a fractal dimension of 1.29 by the box-counting method. The significance of these findings includes: sutural morphogenesis can be described as a repeated iteration function, and mathematical models can be constructed to produce self-similar curves with such D_{b}. This elucidates the mechanism of actual pattern formation. Whatever the mechanisms at the cellular and molecular levels, human sagittal suture follows the equation log N_{r} = 1.29 log 1/r, where Nr is the number of square boxes with sides r that are needed to contain the sutural pattern and r equals the length of the sides of the boxes.

Original language | English (US) |
---|---|

Pages (from-to) | 409-415 |

Number of pages | 7 |

Journal | Cleft Palate-Craniofacial Journal |

Volume | 40 |

Issue number | 4 |

DOIs | |

State | Published - Jul 1 2003 |

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### Keywords

- Fractal analysis
- Fractal dimension
- Human sagittal suture

### ASJC Scopus subject areas

- Oral Surgery
- Otorhinolaryngology

### Cite this

*Cleft Palate-Craniofacial Journal*,

*40*(4), 409-415. https://doi.org/10.1597/1545-1569(2003)040<0409:AFAOHC>2.0.CO;2

**A fractal analysis of human cranial sutures.** / Yu, Jack C.; Wright, Ronald L.; Williamson, Matthew A.; Braselton, James P.; Abell, Martha L.

Research output: Contribution to journal › Article

*Cleft Palate-Craniofacial Journal*, vol. 40, no. 4, pp. 409-415. https://doi.org/10.1597/1545-1569(2003)040<0409:AFAOHC>2.0.CO;2

}

TY - JOUR

T1 - A fractal analysis of human cranial sutures

AU - Yu, Jack C.

AU - Wright, Ronald L.

AU - Williamson, Matthew A.

AU - Braselton, James P.

AU - Abell, Martha L.

PY - 2003/7/1

Y1 - 2003/7/1

N2 - Objectives: Many biological structures are products of repeated iteration functions. As such, they demonstrate characteristic, scale-invariant features. Fractal analysis of these features elucidates the mechanism of their formation. The objectives of this project were to determine whether human cranial sutures demonstrate self-similarity and measure their exponents of similarity (fractal dimensions). Design: One hundred three documented human skulls from the Terry Collection of the Smithsonian Institution were used. Their sagittal sutures were digitized and the data converted to bitmap images for analysis using box-counting method of fractal software. Results: The log-log plots of the number of boxes containing the sutural pattern, Nr, and the size of the boxes, r, were all linear, indicating that human sagittal sutures possess scale-invariant features and thus are fractals. The linear portion of these log-log plots has limits because of the finite resolution used for data acquisition. The mean box dimension, Db, was 1.29289 ± 0.078457 with a 95% confidence interval of 1.27634 to 1.30944. Conclusions: Human sagittal sutures are self-similar and have a fractal dimension of 1.29 by the box-counting method. The significance of these findings includes: sutural morphogenesis can be described as a repeated iteration function, and mathematical models can be constructed to produce self-similar curves with such Db. This elucidates the mechanism of actual pattern formation. Whatever the mechanisms at the cellular and molecular levels, human sagittal suture follows the equation log Nr = 1.29 log 1/r, where Nr is the number of square boxes with sides r that are needed to contain the sutural pattern and r equals the length of the sides of the boxes.

AB - Objectives: Many biological structures are products of repeated iteration functions. As such, they demonstrate characteristic, scale-invariant features. Fractal analysis of these features elucidates the mechanism of their formation. The objectives of this project were to determine whether human cranial sutures demonstrate self-similarity and measure their exponents of similarity (fractal dimensions). Design: One hundred three documented human skulls from the Terry Collection of the Smithsonian Institution were used. Their sagittal sutures were digitized and the data converted to bitmap images for analysis using box-counting method of fractal software. Results: The log-log plots of the number of boxes containing the sutural pattern, Nr, and the size of the boxes, r, were all linear, indicating that human sagittal sutures possess scale-invariant features and thus are fractals. The linear portion of these log-log plots has limits because of the finite resolution used for data acquisition. The mean box dimension, Db, was 1.29289 ± 0.078457 with a 95% confidence interval of 1.27634 to 1.30944. Conclusions: Human sagittal sutures are self-similar and have a fractal dimension of 1.29 by the box-counting method. The significance of these findings includes: sutural morphogenesis can be described as a repeated iteration function, and mathematical models can be constructed to produce self-similar curves with such Db. This elucidates the mechanism of actual pattern formation. Whatever the mechanisms at the cellular and molecular levels, human sagittal suture follows the equation log Nr = 1.29 log 1/r, where Nr is the number of square boxes with sides r that are needed to contain the sutural pattern and r equals the length of the sides of the boxes.

KW - Fractal analysis

KW - Fractal dimension

KW - Human sagittal suture

UR - http://www.scopus.com/inward/record.url?scp=0038783312&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038783312&partnerID=8YFLogxK

U2 - 10.1597/1545-1569(2003)040<0409:AFAOHC>2.0.CO;2

DO - 10.1597/1545-1569(2003)040<0409:AFAOHC>2.0.CO;2

M3 - Article

C2 - 12846606

AN - SCOPUS:0038783312

VL - 40

SP - 409

EP - 415

JO - Cleft Palate-Craniofacial Journal

JF - Cleft Palate-Craniofacial Journal

SN - 1055-6656

IS - 4

ER -