The DOI matches its Crossref record; 21 of the 21 references it cites were found in a registry or an index.
A generic geometric transformation that unifies a wide range of natural and abstract shapes
Abstract
To study forms in plants and other living organisms, several mathematical tools are available, most of which are general tools that do not take into account valuable biological information. In this report I present a new geometrical approach for modeling and understanding various abstract, natural, and man‐made shapes. Starting from the concept of the circle, I show that a large variety of shapes can be described by a single and simple geometrical equation, the Superformula. Modification of the parameters permits the generation of various natural polygons. For example, applying the equation to logarithmic or trigonometric functions modifies the metrics of these functions and all associated graphs. As a unifying framework, all these shapes are proven to be circles in their internal metrics, and the Superformula provides the precise mathematical relation between Euclidean measurements and the internal non‐Euclidean metrics of shapes. Looking beyond Euclidean circles and Pythagorean measures reveals a novel and powerful way to study natural forms and phenomena.
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Cites
The Algorithmic Beauty of Plants
Prusinkiewicz, P., & Lindenmayer, A. (1990) · The Virtual Laboratory · doi:10.1007/978-1-4613-8476-2 · Cited by 2,823
Thompson, D. W. (1917) · University Press eBooks · doi:10.5962/bhl.title.11332 · Cited by 1,913
Elliptic Fourier features of a closed contour
Kuhl, F. P., & Giardina, C. R. (1982) · Computer Graphics and Image Processing · doi:10.1016/0146-664x(82)90034-x · Cited by 1,839
Superquadrics and Angle-Preserving Transformations
Barr (1981) · IEEE Computer Graphics and Applications · doi:10.1109/mcg.1981.1673799 · Cited by 1,100
Weyl, H. (1952) · Princeton University Press eBooks · doi:10.1515/9781400874347 · Cited by 767
Biometrics, biomathematics and the morphometric synthesis
Bookstein, F. L. (1996) · Bulletin of Mathematical Biology · doi:10.1007/bf02458311 · Cited by 456
Hersh, R. (1998) · Mitteilungen der Deutschen Mathematiker-Vereinigung · doi:10.1515/dmvm-1998-0205 · Cited by 410
The Algorithmic Beauty of Sea Shells
Meinhardt, H. (1998) · The Virtual Laboratory · doi:10.1007/978-3-662-03617-4 · Cited by 405
(1998) · Choice Reviews Online · doi:10.5860/choice.35-5132 · Cited by 374
Radical alterations in the roles of homeobox genes during echinoderm evolution
Lowe, C. J., & Wray, G. A. (1997) · Nature · doi:10.1038/39580 · Cited by 301
Cited by
Motion Silences Awareness of Visual Change
Suchow, J. W., & Alvarez, G. A. (2011) · Current Biology · doi:10.1016/j.cub.2010.12.019 · Cited by 121
Calculation of the radiative properties of photosynthetic microorganisms
Dauchet, J., Blanco, S., Cornet, J.-F., & Fournier, R. (2015) · Journal of Quantitative Spectroscopy and Radiative Transfer · doi:10.1016/j.jqsrt.2015.03.025 · Cited by 80
Metadata
- Authors
- Gielis, J.
- Venue
- American Journal of Botany
- Publisher
- Wiley
- Published
- 2003-03
- Volume
- 90
- Issue
- 3
- Pages
- 333-338
- ISSN
- 0002-9122 (ISSN-L)
- 1537-2197
- OpenAlex
- W2141270723
- Cited by
- 585
- References
- 21
- FWCI
- 16.47
- Type
- journal article
- Language
- en
- Open access
- closed
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Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
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