In geometry two objects are said to be similar if they have the same shape --- which happens if their angles are the same size and occur in the same sequence. For example, any pair of triangles with angles 30, 60, and 90 degrees are similar; also, the lengths of pairs of corresponding sides of these triangles have the same ratio.
A term used in the terminology of fractals is self-similarity: a self-similar object has exactly (or approximately) the same shape as a part of
itself.
A variety of objects in the real world, such as ferns and coastlines, are approximately
self-similar: parts of them show the same statistical properties at many
scales. At the end of this post are a couple of diagrams that illustrate how a fractal may be developed. But first, experience the generative beauty of self-similarity via a poem by Maryland poet Greg McBride. Mathematician Benoit Mandelbrot (1924-2010), quoted in McBride's epigraph, often is nicknamed "the father of fractals."
Showing posts with label Benoit Mandelbrot. Show all posts
Showing posts with label Benoit Mandelbrot. Show all posts
Tuesday, March 10, 2015
Thursday, April 10, 2014
Fractal Geometry
Lee Felice Pinkas is one of the founding editors of cellpoems -- a poetry journal distributed via text message. I found her poem,"The Fractal Geometry of Nature" in the Winter/Spring 2009 Issue (vol.14, no 1) of Crab Orchard Review.
The Fractal Geometry of Nature by Lee Felice Pinkas
Most emphatically, I do not consider
the fractal point of view as a panacea. . .
--Benoit Mandelbrot (1924-2010)
Father of fractals, we were foolish
to expect a light-show from you,
hoping your speech would fold upon itself
and mimic patterns too complex for Euclid.
The Fractal Geometry of Nature by Lee Felice Pinkas
Most emphatically, I do not consider
the fractal point of view as a panacea. . .
--Benoit Mandelbrot (1924-2010)
Father of fractals, we were foolish
to expect a light-show from you,
hoping your speech would fold upon itself
and mimic patterns too complex for Euclid.
Labels:
Benoit Mandelbrot,
complex,
dimension,
Euclid,
fractal,
geometry,
Lee Felice Pinkas,
pattern,
repeated,
roughness,
self-similarity,
simple,
snowflake
Tuesday, October 29, 2013
From order to chaos -- a sonnet
Fractals by Diana Der-Hovanessian
Euclid alone has looked on beauty bare
--Edna St. Vincent Millay
Euclid alone began to formulate
the relation of circle, plane and sphere
in equations making it quite clear
that symmetry is what we celebrate.
Euclid alone has looked on beauty bare
--Edna St. Vincent Millay
Euclid alone began to formulate
the relation of circle, plane and sphere
in equations making it quite clear
that symmetry is what we celebrate.
Labels:
Benoit Mandelbrot,
chaos,
Diana Der-Hovanessian,
Euclid,
fractal,
symmetry,
turbulence
Sunday, October 17, 2010
The Length of a Coastline
In the nineties, fifteen or so years ago, when I began posting mathematical poems on the Internet, two of my earliest connections were Ken Stange, a poet and polymath and professor of psychology at Ontario's Nipissing University, and his daughter Kate, then a teen. Kate publicized her love of mathematics and poetry by creating an online collection,"Mathematical Poetry: A Small Anthology" which she has continued to maintain for many years--during which she has completed undergraduate and graduate studies in mathematics.
Labels:
anthology,
Benoit Mandelbrot,
coastline,
distance,
Euclid,
fractal,
function,
infinite,
Kate Stange,
Ken Stange,
mathematical,
million,
poetry,
ruler,
significant digit
Friday, May 14, 2010
Poems starring mathematicians - 6 (Mandelbrot)
More familiar than the name Benoit Mandelbrot are images, like the one to the left, of the fractal that bears his name. Born in Poland (1924) and educated in France, Mandelbrot moved to the US in 1958 to join the research staff at IBM. A fractal is a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole, a property called self-similarity.
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