ABOUT THE COVER ©
by Tira Brandon-Evans & Jude Forese
ABOUT THE PICTURE
This beautiful spiralling image is a Julia set fractal. Fractals seem to describe the order at the heart of all creation. See Spiral Within in this issue, which relates to the spiral patterns found in nature and in Celtic art.
"A fractal is "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,"[1] a property called self-similarity. Roots of mathematical interest in fractals can be traced back to the late 19th Century; however, the term "fractal" was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.[2]
A fractal often has the following features:[3]
It has a fine structure at arbitrarily small scales.
- It is too irregular to be easily described in traditional Euclidean geometric language.
- It is self-similar (at least approximately or stochastically).
- It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).[4]
- It has a simple and recursive definition.
Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, snow flakes, various vegetables (cauliflower and broccoli), and animal coloration patterns. However, not all self-similar objects are fractals—for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics; for instance, it is regular enough to be described in Euclidean terms.
Images of fractals can be created using fractal-generating software. Images produced by such software are normally referred to as being fractals even if they do not have the above characteristics, as it is possible to zoom into a region of the image that does not exhibit any fractal properties. It should also be borne in mind that, in common with other software, fractal generating programs have bugs, so some of the images produced by these programs may exhibit properties that could be termed software artifacts rather than characteristics of true fractals.
In nature
Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, snow flakes, crystals, mountain ranges, lightning, river networks, cauliflower or broccoli, and systems of blood vessels and pulmonary vessels. Coastlines may be loosely considered fractal in nature.
Trees and ferns are fractal in nature and can be modeled on a computer by using a recursive algorithm. This recursive nature is obvious in these examples—a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. The connection between fractals and leaves are currently being used to determine how much carbon is contained in trees.[5]"
References
1. Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman and Company.. ISBN 0-7167-1186-9.
2. Briggs, John (1992). Fractals:The Patterns of Chaos. pp. 148. ISBN 0500276935, 0500276935.
3. Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Ltd.. xxv. ISBN 0-470-84862-6.
4. The Hilbert curve map is not a homeomorhpism, so it does not preserve topological dimension. The topological dimension and Hausdorff dimension of the image of the Hilbert map in R2 are both 2. Note, however, that the topological dimension of the graph of the Hilbert map (a set in R3) is 1.
5. "Hunting the Hidden Dimension." Nova. PBS. WPMB-Maryland. 28 October 2008.
(Excerpted from: http://en.wikipedia.org/wiki/Fractals)
ABOUT THE WORDS
dream scene (n) ...
the spiral staircase
in my dream, a giant spiral staircase
rises into the heavens
from within the heart of an iceberg
it’s on fire
the melodic force of the wind
howls across its sparkling structure
and frozen night air
releases waves of arctic warmth
bursts of light
erupt across moonlit clouds,
as if the universe is signaling
a secret code …
i climb this spiraling staircase
engulfed in flames,
consumed by the passion of ascension
(neither fear nor apprehension
prevent my dream from thawing)
as i reach the top platform,
the spiral staircase collapses
but i have already crossed over its surface
without being burned
i will dream another dream tomorrow
when the horizon reawakens
and the spiral staircase appears once again.
(dream scene (n) ... the spiral staircase
jude forese ©2005)
Jude Forese resides in The Bronx, NY. He grew up in a terrain where poets are born in streets of raw transcendent thought. His poems have appeared in Ginkgo, Magpie, Through the Looking Glass, Dream Network, Phantasm, as well as various e-zines. He has one collection of poetry published entitled "Moods in Motion," and is presently compiling a second collection of poems tentatively entitled “Acts of Flight.” One critique on Jude’s poetry states, "Jude’s style is musical and airy, and breathes naturally with an assumed ease and quietude. Always rich in metaphor, he articulates profound thought with unique expression using strikingly fresh images."
Jude endeavors to explore the dreamscape via poetic expressionism. "Dreams are a portal to the oversoul , an avenue to the universal imagination, and a natural viaduct exploring the landscape of waking experience." Contact Jude at: judeace@aol.com
About the Cover:About the Words copyright © 2009 by Jude Forese, all rights reserved. Used with permission. Top of Page
Earthsongs: International Journal of the Society of Celtic Shamans copyright © 2009 by Elder Grove Press and content providers. All rights reserved. International copyright laws prohibit reproduction of or distribution of this page by any means whatsoever, electronic or otherwise, without first obtaining the written permissions of the copyright holders. We retain legal counsel to protect our copyrights.