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Here is a proposed alternative opening to the lede, that might be better for nonspecialists, and can readily be integrated to the rest of the lede. Note most of the references are to those already in the lede (no removal of that content, unless shown), but the early new "references" are actually notes, drawn from Michael Frame's course at Yale, [1], to be wikilinked and sourced if this edit is accepted:
The term fractal (L.frāctus, broken or fractured) was coined by mathematician Benoît Mandelbrot in 1975 and is used both to describe smaller scale patterns in natural phenomena—e.g., of branching (as in fern fronds and ice crystals),^{[1]} formed boundaries (such as in coastlines),^{[2]} and other patterns of growing structures (as in eddies in mass fluids such as hurricanes, and compartments in the Nautilus shell)^{[3]}—that exhibit repeating two- and three-dimensional patterns at different magnifications (scales), but also, importantly, to describe the mathematical sets and functions that model them, or on graphing and analysis, that otherwise exhibit repeating patterns remaining constant across varying scales. When such repeating patterns in natural phenomena and in mathematical sets remain precisely the same at every scale, they are termed self-similar patterns. An example of a mathematical set designed to mimic a pattern seen in nature is the Barnsley fern representation of the natural black spleenwort fern, shown in the image. A further example that will appear later is the two-dimensional Sierpinski carpet, and the three-dimensional Menger Sponge that derives from it. Fractals can also be nearly the same at different levels.Language too non-specific, redundant. An example of the invariance of pattern over large changes of scale (magnification) is shown in a set of figures from Benoît Mandelbrot, the founder of the modern field who extended the concept of theoretical fractional dimensions to geometric patterns in nature.^{[4]}^{: 405 }^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}
Include the following earlier lede statement only if it is given in quotation marks, and/or further significant explanation given, because as it stands the statement is so vague it seems to encompass all manners of phenomena and maths:
"Fractals also includes the idea of a detailed pattern that repeats itself.^{[4]}^{: 166, 18 }^{[6]}^{[5]}"
^Note: These include cases such as dendritic crystals and mineral deposits, sectored plate ice crystals, and plant foliage and canopies, etc.
^Note: These include further cases such as the cranial sutures between skull plates, and boundaries to geologic formations caused by weathering, etc.
^Note: These include further cases such as the extraplanetary meteorologic red spot of Jupiter, coiled fern fronds, plasma loops in solar prominences, inflorescences (buds) of Romanesco broccoli, and telescopic structures of nebulae.
^ ^{a}^{b}Albers, Donald J.; Alexanderson, Gerald L. (2008). "Benoît Mandelbrot: In his own words". Mathematical people : profiles and interviews. Wellesley, MA: AK Peters. p. 214. ISBN9781568813400.
Black spleenwort fern frond representation, computed as a Barnsley fern, showing the whole blade of the frond in multicolor, and two pinnae (leaflets) in dark blue and red, wherein can be seen the number of pinnules (subleaflets) composing the pinnae. Here, the ideas of expanding scale while maintaining self-similarity can be seen.
The Koch curve, a classic iterated fractal curve. It is made by iteratively scaling a starting line segment. In each iteration, the new construction is composed of 4 new pieces laid end to end, each scaled to 1/3 of the original segment length. (The construction can be envisioned as dividing the original segment into three equal subsegments, composing an outward-pointing equilateral triangle atop the middle third, then erasing the base of that triangle to give the new 4 segments of 1/3 length.) The new segment created by the iteration fits across the traditionally measured length between the endpoints of the previous segment. This animation shows 5 iterations of a process that can continue infinitely (though after ~5 iterations on a small image, detail is lost).
The last sentence of paragraph 3 of the introduction requires explanation, preferably with a diagram. Here is the sentence:
The fractal curve divided into parts 1/3 the length of the original line becomes 4 pieces rearranged to repeat the original detail.