Mark
Dow
Geek
art
"But
the shell retains its unchanging
form in spite of its
assymetrical growth; it grows at one end only ... . And this
remarkable property of increasing by terminal growth, but
nevertheless retaining unchanged the form of the entire figure, is
characteristic of the equiangular spiral, and of no other mathematical
curve." Sir D'Arcy
Wentworth Thompson in On Growth and Form,
1942 edition
Logarithmic spirals,
waves, and tilings
These images and
animations are variations on a single theme, a binary division of the
plane by two
anti-symmetric
(flipped and mirrored, or rotated 180 degrees) logarithmic spirals. The
basic figure has the same symmetry as a "yin-yang" image.
The images are also examples of binary fractals,
divisions of
the plane into two equal fractal (symmetric and self-similar) parts.
See
binary
fractals for other examples of binary fractal images.
Why
logarithmic spirals are interesting
The logarithmic
spiral, or Spira
mirabilis (miraculus spiral),
is remarkable because of its unique self-similarity; it is invariant
after a similarity transform. After any scaling (uniformly
increasing
or decreasing the size), logarithmic spirals can be rotated such that
they match the original figure:
The spiral boundaries can have any rate of twist, or
pitch. The pitch is defined as the angle between a tangent to the curve
and the tangent to the circle at that radius. The pitch is the same
everywhere on a logarithmic spiral. [To Do: geometry diagram]
Several simple algorithms result in logarithmic
spirals. For example,
{
pick a center and a point
repeat indefinitely (iterate):
scale the
point's coordinates proportionally (a constant fraction) and rotate a
constant amount
}
Any such linear iterated
function system
(IFS) that includes scaling and rotation (an affine
transformation) will
contain logarithmic spiral patterns. Naturally occuring agorithms
(e.g. the expression of a genetic code, hurricane dynamics, galaxy
formation) commonly result in this shape.
Because of it's fundamental simplicity, this figure
is a good basis pattern. Like other simple geometric shapes (circle,
squares, etc.), logarithmic spirals are a logical elemental pattern; a
2-D
basis vector.
The shape also seems to be appealing to the
eye, perhaps because our visual perception is tuned for interpreting
similarity (scale and rotational invariance) of known objects. For
example we commonly recognize the actual size and orientation of
objects no matter their size or orientation on the retina. [To Do:
Scale invariance of vision examples.] All fractals have scale and
rotational invariances, and any fractal formed from a process that has
a non-zero rotational component will include logarithmic spiral forms.
From "Chaos, Fractals, Nature: a New
look at Jackson Pollock" (
Richard P.
Taylor, 2006, Fractals Research, Eugene OR):
"It is interesting to speculate that,
just as there are regions of the brain where the neural cells are
responsible for processing the colour and form of an observed object,
there may be another region geared to detecting the fractal content.
This concept, which suggests an instinctive appreciation of fractals,
has also been discussed by Mandelbrot: “We know the brain has
cells which handle shapes, and other cells handle the colours. Does the
brain have also cells which handle fractals? Well, we don’t know.
It is a purely hypothetical question” [24]. How would such cells
respond to the visual stimulus of a fractal pattern? Neuroscientists
using electroencephalographic (E.E.G.) data have shown that the
electrical activity of brain cells is fundamentally chaotic and that
the level of chaos depends on the observer’s familiarity to a
stimulus[25]. It is not known, however, if there is any link between
the brain’s naturally chaotic state and its ability to recognise
the chaos-induced fractals of Nature’s scenery. What is known is
that the human visual system is particularly well suited to the
detection of fractal objects. The spatial information in a scene is
thought to be processed within a ‘multi-resolution’
framework where the cells in the visual cortex are grouped into
so-called ‘channels’ according to the spatial frequency
they detect. The way these ‘channels’ are distributed in
spatial frequency parallels the scaling relationship of the fractal
patterns in the observed scenery [1]. It has been speculated that this
is not coincidental but the consequence of the visual system’s
adaptation to the fractal character of the natural environment during
evolution — in order to be efficient “the human visual
system should be tuned to the ensemble of images that it sees”
[27]. If the human visual system has evolved to detect fractals then it
seems reasonable that the ability to recognise fractals might also be a
product of evolution."
1. Ellen G. Landau, Jackson Pollock,
London, Thames and Hudson, 1988, 12.
24. Sam Hunter, ‘Among The New Shows’, New York Times,
30/1/1949, 2. See also Landau [1], 12.
25. Robert Coates, New Yorker Magazine, 17/1/1948.
Note that a similar logarithmic mapping (scaling and
rotation) is embodied
in the primate visual system, in the spatial mapping between the retina
and primary visual cortex. It has been suggested (Schwartz 1981) that this mapping is a an
example of computational anatomy in vision.
Logarithmic spiral images
 |
 |
|
| The top and bottom portions can
be viewed as inversions or
half rotations of each other. What is figure and ground? there are four
similar spiral regions, but I can only hold any two "in mind" at one
time. |
Undersampled, very tight (low
pitch) logarithmic spirals. Best at full view. Op-art, but also fractal.
The image structure can be understood as a moire pattern due to
interference between the spiral and the rectangular (pixel) sampling
grid.
The top half is slightly undersampled, and the bottom is highly
undersampled.
Scrolling produces an interesting twist, on most monitors at full view. |
Each plane of this figure is
like the image at the far right, but with a pitch that smoothly varies
with depth.
Figure and ground are made explicit by volume rendering. The top and
bottom portions are similar and conjugate; they could be interlocked to
fill space.
Rendered with Space
Software. |
Logaritmic spiral
animated tiles and tilings
A logarithmic wave animation combined into tiles with various
spatio-temporal symmetries:
A 3-D tiling based on a
logarithmic spiral
 |
 |
 |
 |
A 3-D tiling, having a
self-conjugate surface, based on logarithmic spirals.
(stereo pair touching at center, cross-view)
|
Same surface form, with
different coloring, as the tiling to the right.
(stereo pair, cross-view) |
Disected
element of the tiling to the right. This shape, forms a (monohedral)
tiling that fills space with an octahedral symmetry.
(stereo pair, cross-view)
Rotation movie:
Vortex_apple_320x240.wmv
Vortex_apple_640x480.wmv |
Three tiles (see image at left)
interlinked to show how they are rotated and translated to fill space.
(stereo pair, cross-view)
Rotation movie:
Vortex_apple_triple_320x240.wmv
Vortex_apple_triple_640x480.wmv |
Orthologia
Imaginary beast, a bugit, based on a similarity tiling of logarithmic
spirals and colored by the
Thue-Morse
sequence.
An imaginary beast, a bugit, based on a logarithmic
spiral similarity tiling and the Thue-Morse sequence. [To Do: Context
of a square tesselation and Thue-Morse coloring.]
The basis pattern (at left) is an infinite division
of the plane by logarithmic spirals, a binary fractal.
It's a checkerboard division (8x2) of the
plane by logarithmic spirals with a pitch of +/- pi/4 radians. The
checkerboard is colored (orthogonal to the contours) by repetition of
the third generation
(23 = 8 element) Thue-Morse sequence
[0 1 1 0 1 0 0 1], where 1 and 0 represent black and white. The
relative phase between the two sets of opposite
chirality spirals rotates through pi/2 radians; in this example the
phase of one set is fixed.
There is a nice bi-stable perception illusion in the
animation (above left): when I
first see it, I percieve a wholistic pattern rotating CW and shrinking.
But after a short time, I see half the contours as fixed, and the other
half rotating (with no shrinking). Try fixating on a single edge point
to experience the second percept.
Also see description,
notes and code for the Orthologia images
Matlab code for
generating the basic shape, any pitch
"Golden" logarithmic
spirals
The pitch of logarithmic spirals is arbitrary
and is directly related to the ratio of radii at which the spiral
crosses any radial line. For logarithmic spirals this ratio is the
constant, for any radial and any succesive crossing. [To Do: geometry
diagram]
There is a unique ratio, called the golden ratio,
for which the ratio of the whole to the large part is equal to the
ratio of the parts. [To Do: diagram] In this sense the golden ratio is
self-similar, much like a logarithmic spiral. A golden spiral is
a logarithmic spiral for which the sequential radii ratios are related
to the golden ratio.
[To Do: golden ratio spiral as a special case]
[To Do: another special case, the "square" logarithmic spiral]
[To Do: About anti-symmetry. The black/white spatial symmetry of these
images is a fundamental notion in particle physics, embodied by the Pauli
exclusion principle: for two identical fermions, the total
wave function is anti-symmetric,
so no two identical fermions may occupy the same quantum state.
Ordinary matter is composed of fermions (electrons, protons, and
neutrons), and consequently matter exhibits space-occupying behavior.]
Other spiral resources
References
Schwartz E
L, 1981, "Cortical anatomy, size invariance, and spatial
frequency analysis" Perception 10(4)
455 – 468
Abstract. In a recent
application of an algorithm
developed in computer and optical pattern recognition, Cavanagh has
suggested that a composite of spatial frequency mapping and complex
logarithmic mapping would provide a translationally, rotationally, and
size-invariant mechanism for human vision. In this work, Cavanagh has
not made explicit the fact that this transformation is composite, that
is, that the first step (global Fourier analysis) is perceptually,
anatomically, and physiologically inconsistent with primate vision, but
that the second step (complex logarithmic mapping) is actually embodied
in the anatomy of the primate retinostriate projection. Moreover, it is
the complex logarithmic remapping step which is entirely responsible
for the computational simplification of the symmetries of size and
rotation invariance. These facts, which have been extensively discussed
in a recent series of papers, are briefly reviewed and illustrated.
Furthermore, it is shown that the architecture of the retinostriate map
may provide an example of computational anatomy in vision, such that
the spatial representation of a stimulus in the brain may be of direct
functional significance to perception, and to the nature of certain
visual illusions.
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There are no restrictions on use of the images or code
on this page. Claiming to be the originator of the material,
explicitly or implicitly, is bad karma. A link (if appropriate), a note to dow[at]uoregon.edu, and credit are appreciated but not
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Comments are welcome (dow[at]uoregon.edu).
