-- / --
Relative movement of nested Platonic polyhedra: pumping and rotation
Decomposition and recomposition of a toroidal polyhedron -- towards vortex stabilization?
Packing and unpacking of 12 semi-regular Archimedean polyhedra
Rotation and pumping of nested Chinese "puzzle balls" as symbolizing "worlds-within-worlds"
Technical note on interactive animation in virtual reality
The focus here is the demonstration in three dimensions of movements relative to one another of geometrical objects, as indicated in an earlier document (Enhancing Strategic Discourse Systematically using Climate Metaphors: widespread comprehension of system dynamics in weather patterns as a resource, 2015). As that title indicates, the objects in question are understood as carriers for distinctions in discourse expressed metaphorically, notably with respect to climate. Climate itself is understood there as encompassing its tangible experience in nature as well as the widespread use of climate and weather metaphors to distinguish intangible psychosocial phenomena. The point is usefully made in the wordplay between weather and whether, given the manner in which much decision-making is determined by "weather" -- notably with respect to climatic conditions framed metaphorically.
The primary concern here is proof-of-concept in employing readily available software facilities to enable web access to relatively complex structures using animations that render their unusual nature comprehensible through their pattern dynamics. The further objective is to enable access to those animations in a form such that many others may experiment with them, notably with those aspects which are potentially of psychosocial significance.
A technical note is included to clarify such possibilities. A commentary on the implications of the approach is presented separately in a document of which this is effectively an annex (Weather Metaphors as Whether Metaphors: transcending solar illusion via a Galilean-style cognitive revolution? 2015). This notably discusses the interrelationship and potential significance of the seemingly disparate four models.
Work-in-progress: Although the animations presented constitute a successful proof-of-concept experiment, this document can usefully be considered as a work-in-progress -- subject to further modification. As partially indicated in the appended technical note, there remain many issues of how best to engage with the constraints imposed by the variety of browsers, platforms, plugins, and proprietary approaches to virtual reality -- many of which are in process of active evolution. In bypassing some of these through presentations in video format, there is the issue of how lengthy to make any movie, whether to embed it in the document, or whether only to enable optional access to it (on YouTube, for example). Various options are presented to enable experiment by others.
In an effort to communicate quickly the possibilities suggested by some of the more striking animations, to those with only a passing interest, there is the further concern of what to present in what variants and in what quantity. Some of these considerations may involve aesthetic choices. Many of them call into question the competence and comprehension of the author, notably with respect to efficient coding. Meta-tags have yet to be rendered consistent.
The following two images were presented together in the earlier document as a means of drawing attention to the possibility of new insight into systemic relationships -- as proved so inspiring with the early image of Johannes Kepler, and continues to do so. The pattern indicated by him is now recognized to be fundamentally incorrect from a variety of perspectives. The point to be made, however, is that there are few patterns of that degree of integrity -- of relevance to current global psychosocial organization. Considered as a transitional exercise, the question is whether analogues to Kepler's model could be usefully explored at this time. Arguably the image that has recently offered some degree of equivalent symbolic inspiration is that of the Earth, articulated geographically, as photographed from the Moon.
|Nesting 5 Platonic polyhedra:
octahedron, icosahedron, dodecahedron, tetrahedron, cube
|Rhombic Triacontahedron (green) as a nesting framework
(virtual reality variants static: vrml or x3d;
mutual rotation: vrml or x3d; "pumping": vrml or x3d;
videos: "pumping" mp4; "rotation" mp4)
|Polyhedral model of solar system of Johannes Kepler
on Mysterium Cosmographicum (1596)
|Developed with X3D Edit and Stella Polyhedron Navigator||Reproduced from Wikipedia entry|
The significance of the distinct elements of the image on the left (above) were discussed as systemic holding patterns in the earlier document. Links are provided to facilities with which that pattern can be interactively explored in virtual reality as a potential source of inspiration.
Kepler found that each of the five Platonic solids could be uniquely inscribed and circumscribed by spherical orbs; nesting these solids, each encased in a sphere, within one another, thereby producing six layers, corresponding to the six known planets known at that time -- Mercury, Venus, Earth, Mars, Jupiter, and Saturn. By ordering the solids correctly--octahedron, icosahedron, dodecahedron, tetrahedron, cube.The spheres could then be placed at intervals corresponding (within the accuracy limits of available astronomical observations) to the relative sizes of each planet's path, assuming the planets circle the Sun. Kepler also found a formula relating the size of each planet's orb to the length of its orbital period: from inner to outer planets, the ratio of increase in orbital period is twice the difference in orb radius. However, Kepler later rejected this formula, because it was not precise enough.
This is recognized as a beautiful astronomical model. For example, explaining why there were only six planets: How could there be a seventh planet, when Euclid proved that there are only five Platonic solids! Of course, the model is completely false, the interplanetary distances it predicts are not sufficiently accurate, and Kepler was scientist enough to accept this eventually. But it is considered an excellent example of how truth and beauty are not always equivalent.
The potential significance can be further enhanced by introducing a dynamic into the image itself, moving beyond reliance on static configurations, as criticized in the earlier document -- whatever the degree to which dynamics may be implied (as in Kepler's model). As argued there, the dynamics which might be associated with the distinctive polyhedra -- especially relative to one another -- are then suggestive as templates of new levels of systemic insight.
|Screen shots of a virtual reality rendering of an expansion/contraction ("pumping") dynamic
Version as video animation (.mov); access to X3D variant
|Exploratory animations of tetrahedral morphing -- whether to be understood as distinctively 2-fold, 4-fold or 8-fold|
|Animations prepared using Stella Polyhedron Navigator|
Further insights can be suggested by rotating the polyhedra around the common axis at different rates, rather than using that axis to guide the "pumping" motion indicated above
|Relative rotation of nested Platonic polyhedra around a common axis|
|Version as video animation (.mov); version in virtual reality (.wrl)|
The earlier document focused extensively on the value of the drilled truncated cube as a mapping surface, given its relatively unique characteristic amongst regular polyhedra of having 64 edges. As a mapping template, these could then be associated with the 64 conditions of change of the I Ching -- encoded in its 64 hexagrams and rendered memorable both by the notation and by distinctive metaphors. as separately discussed.
As a trigger to further reflection, the challenge presented was that of rendering memorable the pattern of 384 transformations between those conditions, as described separately (Transformation Metaphors derived experimentally from the Chinese Book of Changes (I Ching) -- for sustainable dialogue, vision, conferencing, policy, network, community and lifestyle, 1997).
Such a visual rendering can at least be partially achieved by allowing the edges of the polyhedron -- understood as encoding conditions of potential change -- to move across the polyhedral template to other positions. This would then be indicative of one condition transforming into another -- as encoded in the pattern of that classical "Book of Changes". In the process, the integrity of the polyhedral pattern as a template is both decomposed and recomposed -- as indicated by the animations below in virtual reality.
The diagram on the left (below) was the basis for the distinctive colouring of the edges of the polyhedron in diagonally opposed clusters (as on the right). This gave rise to the following virtual reality representation
|Drilled truncated cube -- a Stewart toroid with 64 edges
(prepared using Stella Polyhedron Navigator)
|Virtual reality variant||Virtual reality variant|
Especially interesting is the manner in which the appropriateness (or viability) of transformations can be distinguished in terms of the parallelism between the source edge and the destination edge of a given movement. The parallelism is especially relevant to perception of the set of movements and their memorability.
|Transformations distinguished in terms of parallelism in a cubic context|
|Inner cube movements (#1/2)||Outer cube movements (#3/4)||Framed movements|
|Access X3D variant||Access X3D variant||Access X3D variant|
|Framed inner cube movements (#1/2)
Access X3D variant
|Framed outer cube movements (#3/4)
Access X3D variant
|Access X3D variant|
Perspectival parallelism was provisionally used to limit transformations to patterns of lines between which this obtained -- recognizing the manner in which the transformations then followed a distinctive cycle according to the type of transformation within the polyhedron. The following gives some indication of the range of lines moving to parallel positions. The exercise focused on those which do not move via the centre, or with respect to the implied diagonals of the inner cube. Whether or not they should be considered, the concerns are:
How might these be detected systematically by appropriate maths? How do these relate to the 9 types of lines distinguished in the profile sheet of Stella Polyhedron Navigator from which the model was exported?
|Drilled truncated cube coloured by edge type
(numbered 0-8, but excluding reflections; generated by Stella Polyhedron Navigator)
|Drilled truncated cube coloured by parallels
(with indication of edge type, numbered 0-8; generated by Stella Polyhedron Navigator)
The following table identifies parallels of various edge types between which two-way movements could be assumed to occur.
|Bidirectional movements between parallels of drilled truncated cube based on edge type
Edge type numbered 0 to 8 (top and left) or as 1 to 9 (right and bottom
(the 0 to 8 convention follows the numbering in the schematics above)
|0||outer cube face||parallels
|1||inner slant. cols
|6||opp. corners||corners||octa corners||corners||7|
|outer cube face||9|
The above table focuses on bidirectional movement between two parallels of the same or dissimilar types. Of potentially much greater interest, notably in terms of memorability, is the movement between a greater number of particular parallels -- thereby taking the form of a cycle of parallels. Selected cycles are presented below as they may effectively define the polyhedral form -- usefully understood as loops.
Of further relevance is that the direction of movement in each cycle may also be reversed. The table is selective because it raises the question of the possibility of a more systematic analysis of parallels to enable a complete set of cycles to be identified -- given the significance that might then be attributed to cycles of different types, and greater complexity, notably with respect to issues relating to chirality.
|Selected cycles of parallel line movement within a drilled truncated cube
(in terms of edge types numbered 0 to 8; the last in any loop is the first, and is therefore in parenthesis)
|edges of octa faces||2-1-6-7-(2)
|diagonals across faces||7-1-6-2-(7)||6-6-7-7-(6)||2-2-1-1-(2)||7-7-2-6-(7)|
The following is one example combining cycles from the above table.
|Alternative views of selected cycles of movement of parallels along edges of the drilled truncated cube
Video version (.mp4); virtual reality (.x3d; .wrl)
Arguably the drilled truncated cube structure is the most compact (succinct) representation of a comprehensive set of patterns of change. Through the focus on parallelism as a symmetry effect, such effects serve to highlight comprehensibility and memorability, and to suggest correspondences of systemic significance (Theories of Correspondences -- and potential equivalences between them in correlative thinking, 2007). Given the objective of using the structure as a mapping device for the 64 conditions of change of the I Ching, to what extent can their attribution to the structure be rendered consistent with the transformations they respectively encode -- from one line position to another (Proof of concept: use of drilled truncated cube as a mapping framework for 64 elements, 2015; Relating configurative mappings of 64 I Ching conditions and 48 koans, 2012 ).
The issue is then to use visual triggers, most notably colour, to render such dynamic patterns comprehensible. The earlier document referred to previous analysis of the hexagram pattern in the light of the unit cube (below left).
|Association of Ba Gua trigrams with unit cube and drilled truncated cube|
|Association of Ba Gua trigrams with unit cube
(reproduced from Z. D. Sung, Symbols of Yi King, 1934)
|Adaptation to drilled truncated cube
of unit cube encoding (on left)
The patterns indicated are a step in the investigation of how each line in the drilled truncated cube -- with which a hexagram can be associated -- might be recognized as transforming into 6 other conditions indicated by lines parallel to it. How might the structure then be understood as encoding 6x64 transformations, namely 384 (or 2 6, or 3 x 2 7) ?
In contrast with the 5 regular Platonic polyhedra, of similar interest is the set of 13 semi-regular polyhedra -- of which 12 can be uniquely configured around the 13th. This configuration is notably described in detail by Keith Critchlow (Order in Space: a design source book, 1969) where it is illustrated as follows. Critchlow specifically cites the inspiration of Buckminster Fuller (noted above)..
(reproduced from Towards Polyhedral Global Governance: complexifying oversimplistic strategic metaphors, 2008, and from Union of Intelligible Associations: remembering dynamic identity through a dodecameral mind, 2005)
|Successive truncations of octahedron
2, 3, 4-fold symmetry
|Successive truncations of icosahedron
2, 3, 5-fold symmetry
|truncated tetrahedron (8 polygons: 3 / 6 sided)|
|Arrangement of the 12 Archimedean polyhedra in their most regular pattern, a cuboctahedron, around a truncated tetrahedron|
|Rotation of cuboctahedral array of 12 Archimedean polyhedra
(around an omitted 13th at the centre; totalling 984 edges, 558 vertices, 452 faces)
Virtual reality variant (.wrl)
|Animation prepared with the aid of Stella Polyhedron Navigator|
The configuration immediately suggests possibilities of animating the relationships between the polyhedra in the array, most notably by animating them in a "pumping" motion of contraction to the common centre and expansion from it, as is evident from the following.
|Screen shots of animation of cuboctahedral array of 12 Archimedean polyhedra collapsing into centre
(without indication of the 13th at the centre: the truncated tetrahedron)
|Contextual cuboctahedron rendered partially transparent
Video animation (.mov); virtual reality (.wrl; .x3d)
|Wireframe version with all faces transparent
Video animation (.mov); virtual reality (.wrl; .x3d)
|Animations prepared with the aid of Stella Polyhedron Navigator|
This cuboctahedral configuration is especially significant given the importance associated with it by Buckminister Fuller (Synergetics: explorations in the geometry of thinking, 1975/1979). He variously renamed it vector equilibirum and dymaxion, associating the expansion and contraction of the configuration with a fundamental jitterbug movement. This is comprehensively summarized by Fuller (Jitterbug: Symmetrical Contraction of Vector Equilibrium). [see videos: Vector Equilibrium: R. Buckminster Fuller; Buckminster Fuller's Jitterbug; Bucky's "Jitterbug": Vector Equilibrium].
The most comprehensive video is that presented to the American Mathematical Society by Joseph Clinton (R. Buckminster Fuller's Jitterbug: its fascination and some challenges, Synergetics Collaborative, 2006). A summary of the associated movements is provided by Robert W. Gray (The "Jitterbug" And Its Motion, 2001; The Jitterbug Motion, 2002). An earlier exercise discussed the tranaformations in some detail, with a mapping of many of them (Vector Equilibrium and its Transformation Pathways, 1980):
The cuboctahedron is the polyhedron obtained by bisecting the 12 edges and truncating the eight corners of the cube. It can also be developed, however, from the omnidirectional closest packing of spheres around one nuclear sphere. The centres of 12 such spheres define the 12 nodes of the cuboctahedron. As all spheres are the same size it can be seen that the length of the cuboctahedron's edges equal the distance from its centre to its 12 nodes. Thus the form can be considered to be a system of equal vectors which are in equilibrium -- a vector equilibirum -- where the outward radial thrust of the vectors from the centre is balanced by the circumferentially restraining chordal vectors. The explosive forces perfectly balance the implosive forces.
As explained, variants of the jitterbug are also distributed as an educational toy. As indicated by the videos, models of it have been presented as virtual reality animations, most notably by Bob Burkhardt (Jitterbug, 2008). The development of 3D animation now enables presentation of the dynamics of the jitterbug transformation of the cuboctahedral configuration of Archimedean polyhedra to be explored otherwise. This would offering a stimulus to the imagination which is otherwise constrained by the 2D representation (above) and access to physical models.
There is the further possibility of animating the lines of the cuboctahedral (jitterbug) array of Platonic polyhedra (as was done with the lines of the drilled truncated cube above). Given the use of the cuboctahedron to configure the Archimedean polyhedra, an even further possibility is to consider animating relations between those polyhedra and the Platonic polyhedra. This would offer a mapping facility composed of the following elements.
The question would then be what significance might be assoctiated with the transformations possible within such a complex, as partially discussed separately in relation to such patterns (Memetic Analogue to the 20 Amino Acids as vital to Psychosocial Life? 2015).
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:
|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)
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
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.
Design adjustments to the animations: Whilst the interactive possibilities could be considerably extended (given a higher degree of software and aesthetic competence), in its present form the animation parameters in the virtual reality file can be readily modified with respect to:
Software: The possibility of generating and displaying animations (such as those above( via the web has evolved considerably over the last two decades. Whilst relatively straight forward, many aspects of the process are rendered problematic by changing norms and the capacity of particular web browsers (and plugins) to handle particular aspects of animations -- to whatever norms they correspond.
Animation preparation: The point of departure was the export of polyhedra into the VRML format from the Stella application, after pre-configuration -- most notably in the case of the relatively complex cuboctahedral array of Archimedean polyhedra. The file was then imported into X3D Edit, experimenting with various approaches.
Considerable difficulty was experienced with the the 600k Archimedean array (984 edges, 558 vertices, 452 faces) due to overloading of computer system resources during some forms of minor editing (occasionally confirmed by messages that the application had been rendered unstable). Consideration was given to reducing precision of the coordinates.
The import into X3D Edit was however successfully done by module, with final editing in a text editor (saving with an X3D extension). Validation continued to be successfully performed within X3D Edit. This enabled the export to the WRL format. The process was a learning exercise in its own right due to a combination of ignorance and incompetence, compounded by the confusing documentation and examples available on the web regarding X3D (at least for novices).
Despite the interest in promoting X3D, and the existence of a response X3D Community, there are challenges which could be avoided otherwise, notably with a more useful range of examples comprehensible to novices (with only the faintest recollection of spherical geometry). Of interest in this respect is the contrast between learning processes focused on acquiring comprehensive mastery of program "grammar", the presentation of detailed technical "explanations" of that grammar, and those emphasizing "examples". The latter may be preferable for those seeking early closure on a concrete result ("getting a cup of coffee"), rather than acquiring long-term proficiency. A notable frustration was determining how to indicate interesting viewpoints, involving a combination of positional coordinates and orientation. The effort was initially abandoned (despite a multiplicity of examples) and left to the interactive navigation of users. Appreciation is due to Roy Walmsley for vital support in overcoming these constraints.
Throughout the process, progress was reviewed using both the H3DViewer and FreeWRL. The Xj3D application integrated into X3D Edit appeared to be a major contributor to system overload on the 600k Archimedean array. Its use was therefore avoided, whatever its advantages for smaller projects (although useful for reviewing models subsequent to the editing process). The plugin version of FreeWRL was uninstalled in favour of the Cortona browser plugin for preparation of the final screen shots because of some navigation and rendering advantages. It is however noteworthy that Cortona wireframe renderings impose triangulation of polygons in contrast with H3DViewer and FreeWRL. The standalone version of FreeWRL later proved to be of value because of its ability to render both X3D and WRL.
At each stage of the process it remained unclear whether any failure was due to coding errors, ignorance, or exceeding unstated software limitations. It was therefore finally somewhat of a surprise that the relative complexity of the 600k project (requiring over 12,000 lines of code) could be readily rendered -- with animations -- in H3DViewer and Cortona 3D.
For those interested, both the X3D and WRL variants (above) can be downloaded as straightforward (and relatively simple) examples -- readily modified experimentally with a text editor (as noted above). Of some relevance, was the discovery of the need to define multiple copies of structural elements (like lines), as when required to move in distinct directions. The files therefore have "ghost" copies of such lines to which distinct animation instructions are applied.
R. Buckminster Fuller with E. J. Applewhite:
Yasushi Kajikawa. New Models of Synergetics Topology and their Reciprocal Space-filling Transformations. Science on Form: Proceedings of the First International Symposium for Science on Form, KTK Publishers, 1986 [text]
George Lakoff and Rafael Núñez. Where Mathematics Comes From: how the embodied mind brings mathematics into being. Basic Books, 2000 [summary]
Arthur M. Young:
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