12 October 2015 | Draft
Psychosocial Implication in Polyhedral Animations in 3D
Patterns of change suggested by nesting, packing, and transforming symmetrical polyhedra
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Introduction
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
References
Introduction
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. |
Relative movement of nested Platonic polyhedra: pumping and rotation
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 |
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| 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) |
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| 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 |
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The earlier document was especially focused on a response to the current incoherence of global discourse with respect to governance, as has been remarkably demonstrated with respect to a variety of issues (climate change, resource overshoot, refugees, environmental degradation, nuclear power, and the like). As argued there, oversimplistic understandings of coherence and consensus may well call for more complex patterns, given the implications for governability (The Consensus Delusion: mysterious attractor undermining global civilization as currently imagined, 2011; Ungovernability of Sustainable Global Democracy? 2010).
| Exploratory animations of tetrahedral morphing -- whether to be understood as distinctively 2-fold, 4-fold or 8-fold | |||
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| 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) |
Decomposition and recomposition of a toroidal polyhedron -- towards vortex stabilization?
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) |
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| Virtual reality variant | Virtual reality variant |
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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 |
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| Framed inner cube movements (#1/2) Access X3D variant |
Framed outer cube movements (#3/4) Access X3D variant |
Access X3D variant |
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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 many movements can a given line make to parallel positions. Does each line have six parallels to which it can move on the array?
- how many kinds of moves are then to be recognized
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) |
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| Drilled truncated cube coloured by parallels (with indication of edge type, numbered 0-8; generated by Stella Polyhedron Navigator) |
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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) |
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| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||
| 0 | outer cube face | parallels (skipping) |
x | x | 1 | |||||
| 1 | inner slant. cols |
orthogs | corners | corners | 2 | |||||
| 2 | orthogs | slant. cols |
corners | corners | 3 | |||||
| 3 | parallels (skipping) |
inner cube | x | x | 4 | |||||
| 4 | octa corners | 5 | ||||||||
| 5 | x | inner cube | parallels (skipping) |
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| 6 | opp. corners | corners | octa corners | corners | 7 | |||||
| 7 | corners | corners | corners | oct corners | 8 | |||||
| 8 | x | x | parallels (skipping) |
outer cube face | 9 | |||||
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 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) |
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| inner cube | 3-3-3-(3) | 5-5-5-(5) | |||
| outer cube | 0-0-0-(0) | 8-8-8-(8) | |||
| edges of octa faces | 2-1-6-7-(2) |
2-1-2-7-(2) | 6-1-6-1-(6) | 6-1-2-1-(6) | 7-2-7-2-(7) |
| 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) |
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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) |
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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) ?
Packing and unpacking of 12 semi-regular Archimedean polyhedra
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)..
| Archimedean
polyhedra (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) |
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| Successive truncations of octahedron 2, 3, 4-fold symmetry |
Successive truncations of icosahedron 2, 3, 5-fold symmetry |
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| truncated tetrahedron (8 polygons: 3 / 6 sided) | |
| Arrangement of the 12 Archimedean polyhedra in their most regular pattern, a cuboctahedron, around a truncated tetrahedron | |
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| 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) |
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| Animation prepared with the aid of Stella Polyhedron Navigator Interactive 3D version |
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) |
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| 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) |
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| 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.
| Edges | Vertices | Faces | Totals | |
| Platonic polyhedra | 90 | 50 | 50 | 190 |
| Archimedean polyhedra | 984 | 558 | 452 | 1994 |
| Totals | 1074 | 608 | 502 | 2184 |
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).
Rotation and pumping of nested Chinese "puzzle balls" as symbolizing "worlds-within-worlds"
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) |
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| Faces non-transparent Video animation (.mov); virtual reality (.wrl; x3d) |
Wireframe version of image on left Video animation (.mov); virtual reality (.wrl; x3d) |
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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) |
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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.
Technical note on interactive animation in virtual reality
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:
- order: present any of the nested polyhedra in a different order, whether the simpler are containers for the more complex, or vice versa. This is achieved by increasing their scale and axial separation relative to those considered less evident
- scale: modify the change of scale so that a given polyhedral pattern shifts between larger or smaller scales in the cycle
- rates/frequency: modify the cycle to ensure that changes are made with greater or lesser frequency relative to the other polyhedral patterns (rate of movement, rate of "pumping", rate of rotation), especially where this is done to explore various forms of memorable synchronization between the dynamic cycles
- size of polyhedral edges
- colour: background, lines, vertices, faces (including transparency)
- rendering, most notably in wireframe mode
- other possibilities (for which no provision has been made) include
- change of colour of a polyhedral pattern during movement
- change of change of transparency of a polyhedral pattern during movement
- change of edge thickness of a polyhedral pattern during movement
- interactive modification of the above (without any requirement for modification of the file)
- association of sound files
- imposition of images on polyhedral surfaces
- provision of more insightful viewpoints
- user interactivity facilities
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.
- Norms: A point of departure since the 1990s has been the virtual reality (VRML) standard in its various incarnations. This is in process of being superceded by the X3D standard of the Web3D Consortium. Of particular relevance is the forthcoming mass market release of Oculus Rift, a virtual reality headset.
- Browsers and Viewers:
There is a wide range of viewers for virtual reality displays over the web, many freely available (see checklist). These include:
- Cortona 3D Viewer: As a plugin for browsers based on a Microsoft platform, its recent upgrades have the advantage of also being able to process legacy VRML files (.WRL) as well as those exported as .WRL files from more recent editing software. It does not however render X3D files.
- Xj3D: This is specially designed for X3D files, most notably as an adjunct to the X3D Edit software. However its state of development and challenging installation make its use problematic in any other mode.
- H3DViewer: As a standalone application this has the advantage of being able to render both .WRL animations and those based on the .X3D norm. This is the least problematic since (without care) the Cortona and FreeWRL viewers block each other. However H3DViewer does not appear to allow for transparency effects.
- FreeWRL: This is a convenient browser plugin for certain purposes, with advantages and disadvantages in comparison with the others.
- Facilitating access: Of some relevance is the challenge of rendering accessible any (interactive) animation to those who are introduced to it via a 2D document (as with this one). Screen shots necessarily offer little sense of the dynamic. Links to .WRL or .X3D files will not necessarily be followed, especially if viewers are not enabled with plugins or otherwise. However, suggestive GIF animations may be inadequate and access to video variants raises other issues. A variety of approaches to these dilemmas have been adopted above
- Configuration of polyhedra: The principal challenge in exploring polyhedra in 3D relates to the coordinates by which they are defined -- typically requiring the greatest of precision. The above displays would not have been possible without the use of Stella Polyhedron Navigator. Whilst this offers considerable facility in the manipulation of a very wide variety of polyhedra, it also has the advantage of exporting such polyhedra into the VRML format to enable further modification and animations of polyhedral elements.
- Editing:
There is a range of editing software for 3D scenes (see checklist), but of particular interest is that which is freely available:
- X3D Edit: With the increasing options for displays in 3D, and their animation, the X3D Editor is particularly useful, notably for validating complex displays. It offers the advantage of being able to import .WRL files as well as exporting into that format. The environment allows projects to be reviewed in an extensive array of 3D viewers.
- Text editor: A major merit of both the .WRL (VRML) and the .X3D files in the ease with which they can be created and modified using any standard text editor. There is of course no validation available in this mode.
- Validation: This process is especially important, given the potential complexity of models and the possibility of error. The X3D Edit application has usefully integrated this feature. An online validator is also available. The H3DViewer also provides useful feedback on errors; the Cortona viewer to a lesser extent.
- Code generation: For convenience, use was made of a DOS-based program for repetitive adaptation of the X3D file in order to generate the many X3D variants explored in the case of the drilled truncated cube
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.
For the more sophisticated, the files might be imported into MeshLab and Blender (both freely available), possibly to enable 3D printing -- as was explored with the "Kepler" model.
References
Christopher Alexander:
- The Nature of Order: an essay on the art of building and the nature of the universe. Center for Environmental Structure, 2003-4. [summary]
- New Concepts in Complexity Theory: an overview of the four books of the Nature of Order with emphasis on the scientific problems which are raised. 2003 [text]
- Harmony-Seeking Computations: a science of non-classical dynamics based on the progressive evolution of the larger whole. International Journal for Unconventional Computing (IJUC), 5, 2009 [text]
Keith Critchlow:
- The Hidden Geometry of Flowers: living rhythms, form and number. Floris Books, 2011
- The Mystery of Flowers: the sacred geometry of plants. Resurgence, 272, May/June 2012
- Order in Space: a design source book. Thames and Hudson, 2000
- Islamic Patterns: an analytical and cosmological approach. Inner Traditions, 1999
R. Buckminster Fuller with E. J. Applewhite:
- Synergetics: explorations in the geometry of thinking. Macmillan, 1975 [text]
- Synergetics 2: explorations in the geometry of thinking. Macmillan, 1979 [text]
- Operating Manual for Spaceship Earth. 1968 [summary]
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:
- The Geometry of Meaning. Delacorte Press, 1976
- The Reflexive Universe: Evolution of Consciousness. Delacorte Press, 1976
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