"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

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.

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.

Logarithmic spiral images

Logaritmic spiral animated tiles and tilings

A 3-D tiling based on a logarithmic spiral

Orthologia, a similarity tiling based on the Thue-Morse sequence

Matlab code for generating the basic shape, any pitch

"Golden" logarithmic spirals

Other spiral resources

Logaritmic spiral animated tiles and tilings

A 3-D tiling based on a logarithmic spiral

Orthologia, a similarity tiling based on the Thue-Morse sequence

Matlab code for generating the basic shape, any pitch

"Golden" logarithmic spirals

Other spiral resources

Spiral Mandala flickr group,
many photographs of a wide range of spiral forms.

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.

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.

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. |

A logarithmic wave animation combined into tiles with various
spatio-temporal symmetries:

small tiling |
small tiling |
small tiling |
small tiling |
small tiling |
small tiling |

large tiling |
large tiling | large tiling | large tiling | large tiling | large tiling |

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 |

All rendering done with Space
Software.

Orthologia

Thue-Morse in rabbit land |
Orthologia
twist animation Ortho_twist_640_640.swf 10 MB, 640 x 640 px. Ortho_twist_640_480.wmv 7 MB, 640 x 480 px. |
Othologia
trispiralis |

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 (2

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

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

The
Geometry Junkyard: Spirals, a list of many resources

Spiral
tilings, similarity tilings

Spiral Mandala flickr group,
many photographs of a wide range of spiral forms.

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.

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 required.

Comments are welcome (dow[at]uoregon.edu).