Rotation and pumping of nested Chinese puzzle balls as symbolizing worlds-within-worlds

Psychosocial Implication in Polyhedral Animations in 3D (Part #5)

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The animation above of the array of 12 Archimedean polyhedra (in its collapsed form) suggested the further possibility of emulating the classical Chinese puzzle balls, or mystery balls (hsiang ya ch'iu or hsiang ya qiu). As a traditional gift to the Emperor, these were carved out of a single piece of ivory, but now from synthetic ivory, resin, wood, jade, and other materials.

They consist of a number of concentric spheres -- typically from 3 to 7 -- which rotate freely with respect to one another. The sequence of balls is understood to represent the cosmos -- a symbolic reference to the sense of "worlds-within-worlds" as being the very nature of reality. Every sphere has distinctive symbolic carvings, usually of plants and animals. Most often, the outermost will either depict two dueling dragons, or hold a dragon (emperor/male), and a phoenix (empress/female), battling for hold upon the world and keeping it in balance, namely as representations of yin and yang. The most complex known is made of 42 spheres enclosing one another.

They are called "puzzle balls" due to the mystery and puzzling explanation behind their making. There is the possibility of manipulating the inner balls so that their holes align with the outer balls, thereby "solving" the puzzle in a technical sense.

Useful descriptions and illustrations, from various perspectives, are offered by the following:

• Stina Björkell: Chinese Puzzle Balls: a dazzling example of superior craftsmanship (gbtimes, 20 November 2013)
• Tibor Tarna: Ivory Polyhedra Turned on a Lathe (Bridges 2010: Mathematics, Music, Art, Architecture, Culture)
• Chinese Puzzle Balls: the Rubik's Cube of the Ancient World (Oddity Central, 5 July 2012
• Ivory Chinese puzzle ball - 14 Layers (YouTube, 2009)
• The Chinese Balls of Claude Lethiecq (YouTube, 2009)

Basis for exploring puzzle ball nesting using superposition of 12 Archimedean polyhedra (polyhedra rendered with same radius and common centre; animations with expansion/contraction of polyhedra at different rates)
Faces non-transparent Video animation (.mov); virtual reality (.wrl; x3d)	Wireframe version of image on left Video animation (.mov); virtual reality (.wrl; x3d)

As a proof-of-concept exercise, in considering interesting emulations of the Chinese puzzle balls, experiments were undertaken by virtual reality

• increasing significantly the diameter of the cylindrical edges of the polyhedra to correspond to the boundaries of the circular holes in the original balls. The "holes" would then be the polygonal faces of the polyhedra -- especially the most circular. This option was set aside.
• changing the transparency of the polyhedral faces to enable viewing through them. This was eventually reduced to 0.05, although recognizing that viewers may not render this degree of transparency, given that some (like H3DViewer) do not appear to render such transparency at all.
• modifying the nesting order of the polyhedra (by radius), from simple to complex, or vice versa. Exploring this possibility is left for future experiment. The option adopted was to switch from relative rotation of the polygons (in whatever nested order) to the "pumping" animation whereby the radius (scale) of each is changed according to a cycle, so that each in turn becomes prominent externally and is successively reduced to invisibility
• modifying the cycle of each polyhedron (relative to that of the others) so that numerous patterns of relative prominence become evident
• colouring the (cylindrical) edge lines of the different polyhedra distinctively. This was abandoned as contributing little to the animation at this stage -- especially given the effects of partial transparency
• colouring distinctively the transparent faces of the polyhedra to increase the aesthetic merits of the animation. Clearly distinctive colours could be given to triangular, square, pentagonal, etc faces. The default colouring was used, which is not consistent in this respect.
• inclusion or exclusion of the 13th Archimedean polyhedron which takes central position in the Archimedean array depicted above. Rather than exclude it, as in the 12-fold animation above, it was included on the same basis as the other 12 in the pumping animation.
• as originally envisaged, the 13th polyhedron could have been positioned at the centre -- specially lit or darkened, to become apparent only when the layers of holes are aligned. This option was not pursued.
• the possibility of building a rotation into the animation as a whole was abandoned, since this can be done interactively with some viewers. The further possibility of combining relative rotation with the pumping animation was also abandoned.

As it stands, some viewers, notably Cortona, enable navigation "into" the animated structure by a zooming process (although this appeared to be inhibited in Firefox, rather than Opera). The aesthetics could be improved to enhance the highly unusual experience of those dynamics, especially within the structure. Of some interest is the contrasting rendering of the Cortona and Xj3D viewers (of the same animation) of which screen shots are presented below -- of the .WRL and .X3D versions respectively.

Screen shots of the puzzle ball experiment with the 13 Archimedean polyhedra (NB: the cycle of "pumping" is lengthy and complex, so these views are of contrasting portions of the cycle; technical issues may constrain the rendering as shown: (.wrl; x3d)

Given Kepler's original solar system inspiration, there is a strong case for basing the distinctive rates, ordering and size on a cognitively meaningful mathematical series. Those understood to be associated with aesthetic enhancement were briefly considered (phi, Fibonacci, etc). This may call for greater sophistication, as with selection of any distinctive set of colours.

Following the Chinese tradition, also of interest is the capacity to associate (symbolic and mnemonic) images and textures with parts of the surface of each polyhedral layer -- leaving the most circular transparent to allow inner layers to be viewed. The pumping motion would notably ensure the emergence of each layer of images to the outermost position -- before it sank into the depths of the structure again.

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