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.

Logarithmic spiral affine invariance, link to Logarithmic spiral images, link to Logarithmic spiral Logarithmic spiral wave animation 3-D tiling with logarithmic spirals, link to Orthologia, link to
Why logarithmic spirals are interesting
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
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.
Logaritmic spiral illusions
Equiangular Spiral (or Logarithmic Spiral) and Its Related Curves
Logarithmic spiral animation and sound sculpture
References

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:
logarithmic spiral rotational invariance logarithmic spiral scale invariance

    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

 logarithmic spiral mirrored logarithmic spiral moire logarithmic spiral antisymmetric volume
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:
logarithmic spiral animated tile 1
small tiling
logarithmic spiral animated tile 2
small tiling
logarithmic spiral animated tile 3
small tiling
logarithmic spiral animated tile 5
small tiling
logarithmic spiral animated tile 6
small tiling
logarithmic spiral animated tile 7
small tiling
large tiling
large tiling large tiling large tiling large tiling large tiling


A 3-D tiling based on a logarithmic spiral

 3-D logarithmic spiral violin tiling, link to 3-D logarithmic spiral vortex tiling, link to 3-D logarithmic spiral tile 3 interlinked 3-D logarithmic spiral tiles
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
Thue-Morse in rabbit land


Orthologia twist animation
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 trispiralisOrthologia trispiralis, link to

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

logarithmic_spiral_fixed_edge.m

"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

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.
Logaritmic spiral illusions
Equiangular Spiral (or Logarithmic Spiral) and Its Related Curves
Logarithmic spiral animation and sound sculpture
The Mathematical Study of Mollusk Shells


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 required.

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