Talk:fractal

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It should say that it looks the same at all scales of length AND that the parts of it seem identical to the whole picture, don't you think?

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fractal

Rfv-sense: A geometric figure that repeats itself under several levels of magnification, and that shows self-similarity on all scales.

What I want to see demonstrated is use of "fractal" by which Mandelbrot set is not a fractal, as it is not perfectly self-similar, per the challenged definition. There is another definition which remains unchallenged: "A geometric figure that appears irregular at all scales of length, e.g. a fern." This unchallenged definition is probably intended to be coextensive with "A geometric figure which has a Hausdorff dimension which is greater than its topological dimension", a definition that I have just removed bz reverting back, in order to enable challenging the definition with perfect self-similarity. --Dan Polansky (talk) 21:12, 16 November 2012 (UTC)

Do we really need an RFV for this? The accuracy is debatable but it's perfectly real isn't it? If it's deleted we'll need another definition to cover the English usage of 'fractal' anyway, so this RFV seems like a bad idea to me. Move to Talk:fractal, doesn't seem to be bad enough to merit WT:RFC. Mglovesfun (talk) 21:25, 17 November 2012 (UTC)
Please look again. There are two definitions now. The one constrains "fractal" to figures that are perfectly self-similar. The other one includes figures that are perfectly self-similar, but also includes figures that are not, such as Mandelbrot set. I have sent the first definition to RFV. What evidence do you have that "fractal" is ever used in a way that excludes Mandelbrot set? --Dan Polansky (talk) 13:51, 18 November 2012 (UTC)
No it doesn't. You've added the word 'perfectly', the entry itself says 'self-similar' not 'perfectly self-similar'. Are we RFVing the definition in the entry or your definition? Mglovesfun (talk) 14:03, 18 November 2012 (UTC)
I admit that I read "self-similar" as "perfectly self-similar", and that I read "similar" as "differing only by scaling, rotation and translation", that is, in a mathematical way. If what the definition intends by "self-similar" is "approximately self-similar", it remains to clarify in what way is the second definition ("A geometric figure that appears irregular at all scales of length, e.g. a fern") intended to cover a different class of things from the first definition, that is, what are the examples of geometrical figures such that they satisfy definition 2 ("irregular at all scales") but not definition 1 ("showing self-similarity at all scales"). In any case, I still do not see that Mandelbrot set is "a geometric figure that repeats itself under several levels of magnification"; that is, I have hard time reading your "approximately" into the definition as it stands. What I want to see attested is that there are two uses of "fractal" that do not apply to the same class of geometric figures. --Dan Polansky (talk) 14:23, 18 November 2012 (UTC)
I would remove the "irregular" definition since there are many irregular shapes that would not be considered fractals. We could include the word irregular in the main single definition, since not all self-similar shapes are fractals. I don't think we should have two definitions. Dbfirs 16:52, 18 November 2012 (UTC)
I think that the wrong sense has been RfVed. Self-similarity (not necessarily perfect (a word that is not in our definition)) at different scales is the most important part of a fractal, as far as I can tell. The OED defines this as "A mathematically conceived curve such that any small part of it, enlarged, has the same statistical character as the original." - statistical similarity, not perfect. The second definition (that I added many years ago) may, indeed be wrong (I'll investigate). SemperBlotto (talk) 17:42, 18 November 2012 (UTC)
I've added some citations and split them between "self-similar" and "irregular" to the best of my ability. It isn't always obvious which sense is meant. Feel free to add more. SemperBlotto (talk) 18:00, 18 November 2012 (UTC)
p.s. The definition of fractal from mathworld is, in part, "A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales. A plot of the quantity on a log-log graph versus scale then gives a straight line, whose slope is said to be the fractal dimension." - this is our sense #1. SemperBlotto (talk) 18:15, 18 November 2012 (UTC)
I have now added rfv-sense to the second sense. For the purpose of existence of the term, "fractal" is doubtless sufficiently cited at Citations:fractal; thank you. For the purpose of showing there is more than one sense of "fractal", the quotations at Citations:fractal do not do the job for me. It is not clear how I should evaluate the quotations. Like, for the C.W. Ormel quotation from 2006, should I look up what "PCA/CCA fractal model" refers to? Without doing that, how am I to know what sense of "fractal" the quotation uses?
Note that the first definition currently says "that repeats itself under several levels of magnification", which seems much stronger than the notion of self-similarity defined as having the "same statistical character". --Dan Polansky (talk) 21:13, 18 November 2012 (UTC)
What about having this single definition: "A geometric figure that shows self-similarity on all scales; technically, a geometric figure which has a Hausdorff dimension that is greater than its topological dimension."? --Dan Polansky (talk) 21:17, 18 November 2012 (UTC)
I liked your suggestion until I remembered that an ordinary parabola (and even a straight line) satisfies the first part. I agree that we ought to have a non-technical definition, but how do we word it so that it includes only those patterns that most people call fractals? We need to include some concept of irregularity to eliminate the trivial geometric figures. ("Fractus" did mean "broken" or "shattered" in Latin.) Dbfirs 14:24, 23 November 2012 (UTC)
@Dbfirs: what about this: "A geometric figure that shows self-similarity at all scales and that, unlike a line segment, shows an ever-expanding detail of shape at all scales; technically, a geometric figure which has a Hausdorff dimension that is greater than its topological dimension." --Dan Polansky (talk) 19:23, 23 November 2012 (UTC)
Something like "that repeats itself under several levels of magnification" or "that is (exactly, approximately, or statistically) self-similar on all scales" seems to fit common usage of the term, though it may need to be reworded to be technically accurate. Merriam-Webster defines it as "any of various extremely irregular curves or shapes for which any suitably chosen part is similar in shape to a given larger or smaller part when magnified or reduced to the same size" (emphasis mine). Dictionary.com has "a geometrical or physical structure having an irregular or fragmented shape at all scales of measurement between a greatest and smallest scale such that certain mathematical or physical properties of the structure, as the perimeter of a curve or the flow rate in a porous medium, behave as if the dimensions of the structure (fractal dimensions) are greater than the spatial dimensions", which seems too technical (jargon-y) without necessarily satisfying all mathematicians. Since mathematical authorities themselves are said to disagree on the definition, perhaps we should have more than one definition? Alternatively, we could have a technical definition and then explain in a usage note what characteristics are associated with fractals in the popular imagination and/or differing mathematical definitions. Wikipedia has an entire section, w:Fractal#Characteristics, devoted to various definitions. - -sche (discuss) 17:36, 23 November 2012 (UTC)
@-sche: I think we should have more than one definition only if the definitions are not coextensive, that is, if there is at least one geometric figure that is a "fractal" per one of the definitions but not per the other definition. --Dan Polansky (talk) 19:28, 23 November 2012 (UTC)
It seems to me that the term is commonly used in math with a relatively narrow meaning, but sometimes also used with a broader meaning that includes all instances of the narrow meaning but also other cases which are not otherwise/often regarded as fractals. Wikipedia has examples. - -sche (discuss) 03:53, 29 November 2012 (UTC)
What are the examples--whether from Wikipedia or elsewhere--of things that are not fractals per "relatively narrow meaning" but are fractals per "broader meaning"? --Dan Polansky (talk) 08:19, 1 December 2012 (UTC)
From WP: "Mandelbrot [] illustrated his mathematical definition with striking computer-constructed visualizations. These images, such as of his canonical Mandelbrot set pictured in Figure 1, captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal"." However, Kenneth Falconer argues for a broad definition of "fractal", which would include things like this strange attractor. (Oddly, I found a use of "fractal" in that specific "attractor" sense in Chaos, Criminology, and Social Justice by Dragan Milovanovic, a non-mathematician: he writes of corporations which "produce a fractal basin of outcomes with any number of attractors".) - -sche (discuss) 16:09, 1 December 2012 (UTC)
An extremely common example given of fractals is coastline [1]. It should be fairly obvious that this is not mathematically truly a fractal, it does not self-repeat forever. At a small enough scale we come down to individual grains of sand: inside the grain the atoms are arranged in a regular crystal structure - very non-fractal. Likewise, at a large enough scale we are looking at the whole globe, again not particularly fractal. Any physical object is only ever going to be only approximately fractal within a limited range. If nothing else, a smaller and smaller scale will eventually reach the Planck length beyond which all known physics fails and it will be impossible to say whether or not the item is still fractal. SpinningSpark 09:44, 12 July 2013 (UTC)
Can we fix this by saying that the mathematical fractal "shows approximate self-similarity" or "shows some degree of self-similarity" at all scales, and that the naturally occurring things informally called fractal show self-similarity across a broad range of scales, rather than all scales? bd2412 T 01:01, 21 July 2013 (UTC)
I'm not sure that that is a useful thing to do. After all, we would not put a clarification on hexagon to say that real-world hexagons (eg honeycombe) cannot be perfect, nor would we distinguish geometric and real-world senses. Sorry, I was merely responding to a question, not suggesting a change. The real question being raised, I think, is whether the informal use of fractal ever diverges from the mathematical definition? Well, that depends on whose definition one uses, but I don't think there can be much doubt the word is used metaphorically nowadays. See for instance The Fractal Company. What I do think needs changing is we need a clear definition for the mathematical sense of fractal. This needs reference to a technical term like Hausdorff dimension (which was removed, quite perversely, in order to raise this RfV!!!). One way to define a fractal is as an object which has a fractional (ie non-integer) Hausdorff dimension constant over all scales. Mathworld gives an easily understood informal definition of the Hausdorff dimension and uses this to define self-similarity which may help us with our entry. SpinningSpark 12:15, 21 July 2013 (UTC)
The current second sense specifies irregularity "at all scales of length", which seems to be the equivalent of defining a hexagon as having its properties "at all scales of length". Obviously, a fern (the example provided in the definition) does not conform to this definition, because, as you noted above, it will cease to be fractal at the Planck scale. If a fern is an example, then the definition should at least be fixed to indicate "many" scales instead of "all" scales. bd2412 T 17:03, 26 July 2013 (UTC)
A hexagon is not self-similar at all scales of length. If one zooms in to a small piece of the hexagon all one sees is a straight line. If one zooms out one sees a small dot. Changing "all" to "many" would be incorrect. A circle is a plane figure with a constant distance from its centre. A perfect circle is impossible in the real world - any construction will always have some small flaw. However, we do not change the definition of the circle to a place figure that is "mostly" or "approximately" a constant distance from its centre. Likewise with the fractal, we should define it as an abstract figure, but it can be taken as read that any real-world example will not be perfect. SpinningSpark 21:36, 26 July 2013 (UTC)
Of course a hexagon is not self-similar at all scales of length. No one is saying that it is. However, a mathematical hexagon is made of lines, and these lines do not become jagged or wavy when you zoom in. The lines of a real-world hexagon might, which makes it inappropriate to define, for example, a hexagon in the structure of a beehive as an object that is made of straight lines, no matter how much you magnify the lines. If it can indeed be "taken as read that any real-world example will not be perfect", then what use is it to have two definitions for fractal, one in the mathematical realm and another in the real world? bd2412 T 21:58, 26 July 2013 (UTC)

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I agree that the two definitions we have right now are essentially "self-similar" (excuse pun). I noted above that prior to this RFV the first definition was mathematical in terms of the Hausdorff dimension, and the second was a looser self-similar definition, but they could still both sensibly be on the same line. For instance,

A mathematical set that has a non-integer and constant Hausdorff dimension; a geometric figure that is self-similar at all scales.

I think however that there is a second metaphorical meaning. The fractal company link I gave above is one example. Here are two more cites,

I can see several more example uses in gbooks, but perhaps not so neatly quotable as these. SpinningSpark 08:25, 27 July 2013 (UTC)

The first example seems to be using fractal as a synonym for microcosm. The second seems to be using it as shorthand for an unpredictable outcome, perhaps something like the butterfly effect. bd2412 T 01:33, 28 July 2013 (UTC)
I agree with your assessment on both counts. Both of those properties are properties of fractals. A definition that captures both is what is required. SpinningSpark 20:47, 28 July 2013 (UTC)
In reviewing this discussion, I agree with the initial comment by Mglovesfun: this is not an RfV issue, but a question of how the definition(s) should be worded. I therefore think that this discussion should be moved to Talk:fractal, to resolve these issues there. bd2412 T 23:30, 28 July 2013 (UTC)
I agree that this is not apprpriate for RFV. I would also like to suggest this for the second definition;
  • (figuratively) An object, system, or idea that exhibits a fractal-like property.
SpinningSpark 13:02, 29 July 2013 (UTC)
I think that is an excellent solution, and conforms with our general treatment of terms having a technical meaning, when used by extension to refer to things sharing properties with the technical sense. If there is no objection, I will move this discussion to Talk:fractal at the end of the day. bd2412 T 13:32, 29 July 2013 (UTC)

Striking and inserting suggested defs in entry. SpinningSpark 16:50, 29 July 2013 (UTC)