Re: [Fractint] FOTD 23-05-03 (Isolated Minibrot [8])
At 09:45 AM 5/27/03 -0500, Prog. Dude wrote: <sippped>
Doesn't "fractal" have a *mathematical* definition?
Yes. I now quote from page 1101 of the second edition of the CRC Concise Encyclopedia of Mathematics: [A fractal is] "An object or quantity which 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." IMO, words such as 'somewhat' and 'not exactly' and 'same type' in this definition leave it open to individual interpretation. Jim M.
Jim Muth wrote:
Doesn't "fractal" have a *mathematical* definition?
Yes. I now quote from page 1101 of the second edition of the CRC Concise Encyclopedia of Mathematics:
[A fractal is] "An object or quantity which 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."
IMO, words such as 'somewhat' and 'not exactly' and 'same type' in this definition leave it open to individual interpretation.
*Interesting*!! I had (mis)understood that a "true" fractal exhibited genuine self-similarity whereas "fractal-like" objects have approximate self-similarity. That would, for example, make a Julia a true fractal and the Mbrot a fractal-like object. Guess not, eh! ;-) -- |_ CJSonnack <Chris@Sonnack.com> _____________| How's my programming? | |_ http://www.Sonnack.com/ ___________________| Call: 1-800-DEV-NULL | |_____________________________________________|_______________________|
Jim Muth wrote:
At 09:45 AM 5/27/03 -0500, Prog. Dude wrote:
<sippped>
Doesn't "fractal" have a *mathematical* definition?
Yes. I now quote from page 1101 of the second edition of the CRC Concise Encyclopedia of Mathematics:
[A fractal is] "An object or quantity which 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."
IMO, words such as 'somewhat' and 'not exactly' and 'same type' in this definition leave it open to individual interpretation.
In a fight over whether or not some specific object is a fractal, yes, but in a broader sense, the definition is recognizing that we haven't yet formalized the description of the kinds of similarities fractals display. Fortunately, we've got a terrific pattern-recognizing computer that doesn't need to have things like "the same 'type' of structures" formalized. It is, of course, the human brain. Mike
participants (3)
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Jim Muth -
Mike Traynor -
Programmer Dude