Logarithmic spirals, waves and tilings

Logarithmic spirals, waves, and tilings

These images and animations are variations on a single theme, a binary division of the plane by two antisymmetric (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 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

3-D tiling with logarithmic spirals, 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

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

    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.]

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:

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 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

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 (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]

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

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