8 May 2023 | Draft
Selected videos, animations and images on Laetus site
Comprehensible Global Packing of Complexity
- / -
Introduction
Polyhedra: Platonic, Archimedean and otherwise
Mapping onto symmetric polyhedra, Platonic, Archimedean and otherwise
Specific forms: dynamics and mappings with the drilled truncated cube
Specific forms: dynamics and mappings with the truncated icosahedron and balls
Specific forms: dynamics and mappings with the dodecahedron and cuboctahedron
Specific forms: torus, helix, spirals, sphere and knots
Symbols: representation of UN's Sustainable Development Goals and "global brain" in 3D
Symbols: representation of mandala in 3D
Symbols: dynamics of 1000-petal chakra rings in 3D
Symbols: patterns in 3D associated with Abrahamic religions and triadic symbols
Symbols: patterns associated with I Ching (Yi Jing), Ba Gua and Tao
Polyhedral Configuration of 384 Governance-relevant Yi Jing Transformations
Symbols: enneagram, ouroboros, lauburu and Zen ox-herding in 3D
Symbols: dynamics of Knight's move, magic squares and the swastika
Symbols: flag and laurel wreath dynamics in 2D and 3D
Symbols: dynamics of heart / temple patterns in 3D
Oppositional logic and geometry
Issues: environmental dynamics in 2D and 3D
Introduction
The following offers a form of index to representative animations and other images on the site Laetus-In-Praesens or in the associated segmentation of its documents into Kairos (a content management system based on Drupal 7). The primary concern is to explore visual renderings which may offer new means of mapping higher orders of complexity -- comprehensibly, as argued separately (Clarification of Thematic Focus; In Quest of Mnemonic Catalysts -- for comprehension of complex psychosocial dynamics, 2007).
Typically the animations are explained in a section of any document (to which a link is provided below). Within the document itself, the animation may be present as an Animated GIF. The animations accessible from the document may take a variety of forms of which the most readily accessible may be the YouTube variant:
- Videos: some animations are available as videos:
- MOV (Quicktime multimedia), WMV (Windows Media Video), or MP4, for example
- with some via a YouTube channel
- Interactive animations:
- Whether using the old virtual reality markup language VRML, (suffix .WRL) or the newer X3D (suffix .X3D).
- Some interactive X3D versions are also presented in X3DOM format, directly accessible in browsers via a complementary index page
- Other interactive animations take the form of force-directed layout graphs (using the d3.js protocol) -- presented within HTML files (Use of force directed layout to elicit memorable polyhedra)
- Older visualization experiments (now deactivated by changes to browser standards) have also been undertaken as:
- SVG animations (Gallery of Images: Virtual Reality, Scalable Vector Graphics, 2007).
- Interactive hypergraphs (Use of hypergraphs in presentation of the I Ching and the Tao te Ching, 2006)
The interactive variants (presented in an extensive complementary gallery) offer various advantages and disadvantages. The main advantage is of course their interactive nature, allowing the image to be variously oriented and explored. An additional advantage is that the source files in both cases can be modified with a simple text editor to allow change of colours, dimensions, speeds, transparency, etc. (according to skill). It is appropriate to note that the formats presented here can also be imported into a variety of applications for further modification to achieve additional effects, possibly with greater ease.
Many of the visualizations are in principle accessible via the associated Kairos database into which the Laetus documents are segmented and indexed to a greater degree. That database is however subject to upgrade and therefore some visual presentations may not be present or may not work for other reasons.
The main disadvantage of the VRML/WRL standard is its (in)compatibility with various browsers and the constraints associated with the requirement to install a plugin, of which there are a variety. The development of the animations here has been done primarily with the aid of the Cortona3D Viewer plugin. Movies have been made from the animated image using a screen recorder (typically into WMV or MOV formats) which can then be uploaded to YouTube (for example). Such formats can also be converted to animated GIF formats (notably via www.cloudconvert.com) with the further possibility of optimizing and resizing these (notably via www.gifgifs.com).
As with the VRML/WRL standard, the newer X3D format requires a special viewer of which there are a number. The H3DViewer has been the one used for most of the animations indicated here; also of value is Free WRL. These animations were developed using the X3D-Edit Authoring Tool, although the resulting source files can be easily modified with a text editor. This also allows animations to be saved into the older VRML format; the application also provides for importation of that format. It is appropriate to note the extensive assistance kindly provided by Sergey Bederov of Cortona3D, with respect to both VRML and X3D variants.
In working with polyhedra, extensive use was also made of the Stella Polyhedron Navigator which notably allows models to be exported into VRML format, from which they can be imported into X3D Edit to enable them to be modified and combined in a variety of ways. Animations of folding and morphing of polyyhedra can also be exported into GIF format; simulations can be made in 3D of rotations of 4D variants of polyhedra. It is appropriate to note the guidance occasionally provided by Robert Webb (as its developer) with respect to the extensive facilities offered by the Stella application.
Note that with respect to the form, the X3DOM variants offer a degree of interactivity through (many) web browsers, as with those using the force-directed layout protocol (dr.js).
A general problem with animations, which have worked in the past on some browsers, is that with the evolution of browser standards and web protocols these may no longer work, as discussed separately (Deactivation of Interactive and Animation Facilities on this Site: selective response to evolving web-related technology). Animations may therefore be "deactivated" at any time as a consequence of such developments. This has been the case with the hypergraph experiments, and some early animations using the VRML and SVG protocols. "Reactivating" them may require only a simple amendment, but resources have not necessarily been devoted to this.
It should be emphasized that the animations have all been developed in an experimental mode to discover whether they enabled more complex patterns of information to be comprehended otherwise -- in a more coherent manner. Many of the images and animations suggest the possibility of further design improvements with respect to colouring, relative sizes and rates of movement, and precision. The visuals presented here are a selection of others -- typically accessible from the explanatory documents in each case.
Polyhedra: Platonic, Archimedean and otherwise
Extensive focus has been placed on the coherence variously implied by the set of spherically symmetrical polyhedra -- as mnemonic aids -- and the transformations of their geometry inviting exploration. Use of these forms is variously made with respect to traditional symbols -- as noted in the following sections.
| Form | |||
 |
Closest packing configuration of 12 Archimedean polyhedra by Critchlow (enhanced with arrow animation indicating transformations) | animated gif | Cognitive Embodiment of Patterns of Governance of Higher Order (2022) |
 |
Relationships between Platonic and Archimdean polyhedra | image | Pathway "route maps" of potential psychosocial transformation? (2015) Polyhedral meta-patterns of relationships? (2015). |
 |
4 Platonic polyhedra rotating around tetrahedron | animated gif | Cyclic dynamic of emergent order versus States of emergency ordered spasmodically (2016) |
 |
12 Archimedean polyhedra configured around truncated tetrahedron | animated gif | as above |
 |
Rotation of cuboctahedral array of 12 Archimedean polyhedra (around an omitted 13th at the centre; totalling 984 edges, 558 vertices, 452 faces) |
animated gif interactive 3D |
Packing and unpacking of 12 semi-regular Archimedean polyhedra |
 |
Cuboctahedral array of 12 Archimedean polyhedra collapsing into centre (Cuboctahedron faces partially transparent) |
video (mov; YouTube) interactive 3D (wrl, x3d) video ( |
Packing and unpacking of 12 semi-regular Archimedean polyhedra |
 |
Cuboctahedral array of 12 Archimedean polyhedra collapsing into centre |
video (mov, YouTube, YouTube) interactive 3D (wrl, x3d ) |
as above |
 |
Nested Platonic polyhedra framed statically within rhombic triacontahedron | animated gif interactive 3D (wrl, x3d) |
Relative movement of nested Platonic polyhedra: pumping and rotation In: Psychosocial Implication in Polyhedral Animations in 3D: patterns of change suggested by nesting, packing, and transforming symmetrical polyhedra |
|
Nested Platonic polyhedra in "pumping" mode (within rhombic triacontahedron framework) |
animated gif video (YouTube YouTube) interactive 3D (wrl, x3d) |
as above |
|
Nested Platonic polyhedra in relative rotation |
interactive 3D (wrl , x3d) |
as above |
 |
Chinese "puzzle ball" nesting using superposition of 12 Archimedean polyhedra
(faces non-transparent) |
video (mov) interactive 3D (wrl, x3d) |
Rotation and pumping of nested Chinese "puzzle balls" as symbolizing "worlds-within-worlds" |
 |
Chinese "puzzle ball" nesting using superposition of 12 Archimedean polyhedra (wireframe version of above) |
video (mov) interactive 3D (wrl, x3d) |
as above |
 |
Puzzle ball experiment with the 13 Archimedean polyhedra (the "pumping" cycle is lengthy and complex) | interactive 3D (wrl, x3d) |
as above |
 |
Interactive force-directed layout to elicit cuboctahedron | interactive 3D (html/d3.js) |
Use of force directed layout to elicit memorable polyhedra |
 |
Interactive force-directed layout to elicit small rhombicuboctahedron | interactive 3D (html/d3.js) |
as above |
 |
Interactive force-directed layout to elicit truncated cuboctahedron | interactive 3D (html/d3.js) |
as above |
Mapping onto symmetric polyhedra, Platonic, Archimedean and otherwise
In contrast to the previous cluster, in the following case text has been experimentally associated with the vertexes, faces or edges. Note that with respect to the form, the X3DOM variants offer a degree of interactivity, as with those using the force-directed layout protocol (dr.js). Whilst this was the case with the hypergraph experiments, these have been deactivated. The arbitrary mapping is intended as a provocation to further reflection on how it should be done from a more systemic perspective -- possibly with the assistance of AI. An extensive list of accessible force-directed polyhedral presentations is available at (Use of force directed layout to elicit memorable polyhedra, 2015). Most include experimental mappings (accessible only by use of cursor on each node).
| Form | |||
|
Universal Declaration of Human Rights: 30 Articles displayed on 1 face-type of a rhombicosidodecahedron | image | Dynamic Exploration of Value Configurations: polyhedral animation of conventional value frameworks (2008) |
|
Arab Charter on Human Rights: Unfolded net of 53 Articles displayed on 2 face-types of a rhombicosidodecahedron | image | as above |
|
European Convention
on Human Rights: Partially folded polyhedral net of 18 Articles displayed on 2 face-types of a rhombicuboctahedron |
image | as above |
|
Universal Declaration of Human Rights: Selected "open" faces | image | as above |
 |
Arab Charter on Human Rights: Exploded augmentation | image | as above |
 |
European Convention on Human Rights: Faceting diagram |
image | as above |
 |
Universal Declaration of Human Rights: 4D projection into 3D | image | as above |
 |
Animation indicative of allocation of 36 dramatic situations onto 36 edges of Stella Octangula | animated gif | Thirty-six Dramatic Situations faced by Global Governance? Interrelating the array of narratives, plots, agendas, stories and conspiracy theories (2022) |
 |
Animation indicative of allocation of 36 dramatic situations onto 36 edges of dual of Stella Octangula | animated gif | as above |
 |
Animation indicative of allocation of 36 dramatic situations onto 36 vertices of Leonardo Octahedron | animated gif | Memorable systemic organization of dramatic situations and folk wisdom? (2022) |
 |
Indicative mapping of 40 Gilets Jaunes issues (from France) onto a dodecahedral compound | png video (YouTube) |
Systemic Function of Highly Unrepresentative Minorities (2018) Coherent visualization of social diversity (2018) Oppositional coherence implied by polyhedral mapping of debate (2019) |
 |
Experimental mapping of 40 "Gilets Jaunes" issues (from France) onto tetrahedral compound | png video (YouTube) |
as above |
 |
Selection of international nongovernmental groups | animated gif videos |
Configuring Global Governance Groups: Experimental animations and video sequences (2008) Configuring Global Governance Groups Experimental visualization of possible integrative relationships (2008) |
 |
as above | as above | as above |
 |
Selection of 12 UN Specialized Agencies | as above | as above |
 |
as above | as above | as above |
 |
20 members of the emergent L20 Group | as above | as above |
 |
as above | as above | as above |
 |
Configuration of Disparate Global Governance Initiatives | as above (video (YouTube) |
as above |
 |
Polyhedral Conference Representation as a Catalyst for Innovation | video (YouTube) |
Polyhedral Representation of Conference Dynamics integrative configuration of panelists, dialogues, topics and threads (2008) |
|
Specific forms: dynamics and mappings with the drilled truncated cube
Various experiments have been made with the 64-edged drilled truncated cube, and the possible dynamics of its elements ( (Proof of concept: use of drilled truncated cube as a mapping framework for 64 elements, 2015),
| Form | |||
 |
Cycles of movement by parallels of edges of the drilled truncated cube (see also individual component movements in x3d: inner cube; outer cube; partial framing; all framing) |
video (mov) interactive 3D (x3d, wrl) |
Decomposition and recomposition of a toroidal polyhedron -- towards vortex stabilization?In: Psychosocial Implication in Polyhedral Animations in 3D: patterns of change suggested by nesting, packing, and transforming symmetrical polyhedra |
 |
Animation of mapping with faces non-transparent of 64 genetic codons | animated gif | Implicate order through hypercube and drilled truncated cube? (2022) |
 |
Animation of mapping with faces transparent of 64 mathematical subjects |
animated gif | as above |
 |
Animation of mapping with faces transparent of 64 I Ching hexagrams |
animated gif | as above |
 |
Use of drilled truncated cube with rings of 64 hexagrams | video (mp4) interactive 3D (x3d, html/x3dom) |
3 mutually orthogonal rotating rings of 64 hexagrams Circles of hexagrams associated with faces and verticals of drilled truncated cube (video; x3d) |
Specific forms: dynamics and mappings with the truncated icosahedron and balls
Various experiments have been made with the 60-vertex truncated icosahedron, given its significance as the stitching pattern of the widely familiar ball of association football, especially given similar familiarity with the tennis ball stitching pattern and that of the baseball. The truncated icosahedron is notably significant as one of the most stable fullerenes -- buckminsterfullerene (C60), as dicussed separately (Sustainability through Global Patterns of 60-fold Organization, 2022).
| Form | |||
 |
Indicative mapping onto 60 vertices of a truncated icosahedron of 30 disabling trends with 30 remedial trends |
animated gif interactive 3D (hrml/d3.js) |
Nested and cage-like polyhedral configurations implying degrees of implicate order (2022) |
 |
Nesting of fullerenes as bucky-onions: C60 nested within C540 within C720 | image | Degrees of integration implied by nested fullerenes (bucky-onions) (2022) |
 |
Nesting of fullerenes as bucky-onions: C20 nested within C60 within C180 | image | as above |
 |
Nesting of fullerenes as bucky-onions: C180 nested within C60 within C20 | image | as above |
 |
4D fullerenes: Animation of truncated icosahedron projected into 3D with icosahedral symmetry | animated gif | Implications of higher dimensional fullerenes?(2022) |
| ** tennis |
Specific forms: dynamics and mappings with the dodecahedron and cuboctahedron
| Form | |||
 |
Successive addition of 36 great circles to dodecagonal-faced cubic framework |
interactive 3D (x3d, wrl/vrml) video (mp4, YouTube) |
Embedding twelve-fold, eleven-fold, nine-fold and seven-fold helices in appropriately encircled polyhedra (2017) |
 |
Embedding the triple helix in a spherical octahedron as a 3D animation | interactive 3D (x3d, wrl/vrml) video (mp4, YouTube) |
Embedding the triple helix in a spherical octahedron (2017) |
 |
Animation of quadrangle moving through a quadruple helix embedded in octahedral great circles | interactive 3D (x3d, wrl/vrml) video (mp4, YouTube) |
Embedding the quadruple helix in a spherical cube (2017 |
 |
Embedding the quintuple helix in a spherical dodecahedron and a pentagram | interactive 3D (x3d, wrl/vrml) video ( mp4, YouTube) |
Embedding the quintuple helix in a spherical dodecahedron and a Pentagramma Mirificum (2017) |
Specific forms: torus, helix, spirals, sphere and knots
Use of these forms is variously made with respect to traditional symbols -- as noted in the following sections.
| Form | |||
 |
Interlocking dynamics of two tori with complementary rotations | animated gif interactive 3D (x3d, wrl html/x3dom) |
Comprehension of Requisite Variety for Sustainable Psychosocial Dynamics: transforming a matrix classification onto intertwined tori (2006) |
 |
Pseudo-counter-coil with helixes out of phase, but spheres travelling
in opposite directions |
interactive 3D .x3d, wrl) |
Towards a 3D visualization of toroidal counter-coiling dynamics In: Visualization in 3D of Dynamics of Toroidal Helical Coils -- in quest of optimum designs for a Concordian Mandala |
 |
Experimental dynamics of 5 coil configuration | video (mov) interactive 3D (x3d, wrl) |
Dynamic possibilities of 3D visualizations of a helical toroid with multiple windings In: as above |
 |
Experimental configuration 5 9-sided helical toroids | video (YouTube) |
|
 |
Helical counter-rotation without counter-coiling (with multiple windings) | interactive 3D (x3d, wrl) |
Towards a 3D visualization of toroidal counter-coiling dynamics In: as above |
 |
Helical counter-rotation with counter-coiling (with multiple windings) |
interactive 3D (x3d, wrl) |
Towards a 3D visualization of toroidal counter-coiling dynamics In: as above |
 |
Helical counter-rotation with counter-coiling (with multiple windings; wireframe version) |
interactive 3D (x3d, wrl) |
Towards a 3D visualization of toroidal counter-coiling dynamics In: as above |
 |
Trefoil knot in 3D |
interactive 3D (x3d, wrl) |
Fearful attraction of a hole In: as above |
 |
Interactive display of three balls moving within Mereon Trefoil knot in 3D | interactive 3D (x3d, html/x3dom) |
Cognitive embodiment of knots: knotting and knitting processes (2021) |
 |
Mereon trefoil knot with nested polyhedra in 3D | interactive 3D (.x3d, html/x3dom |
Interactive display of nested Platonic polyhedra within Mereon Trefoil knot |
 |
Dynamic interlocking in a Borromean configuration of 3 Möbius strips in 3D (with Red surrounds blue; Blue surrounds green; Green surrounds red) |
interactive 3D (x3d; wrl) video (mp4, YouTube) |
Engaging globally with knots and riddles -- Gordian and otherwise (2018). |
 |
Interweaving of 3 mutually orthogonal tori in 3D | interactive 3D (x3d, wrl) video (mp4, YouTube) |
Engaging with Elusive Connectivity and Coherence: Global comprehension as a mistaken quest for closure (2018) Direction of propulsion in a global context -- as enabled by configuration of symbolic stars (2018) |
 |
Phases in various morphing methods between geodesic polyhedron and its dual | video (YouTube) |
Annexing the World as the Deal of the Century (2019) |
 |
Dynamics of axial cube movement within a cube | video (YouTube) |
Succinct mapping of multidimensional psychosocial dynamics? (2016) |
 |
Imagining the location of the metasphere in 3D | interactive 3D (html/x3dom) |
Memorable "location" of the metaphere? |
Symbols: representation of UN's Sustainable Development Goals and "global brain" in 3D
Various alternative approaches to representation of the SDGs have been explored (Imagining Partnership of the SDG Goals as Phases of the Cross, 2022; Eliciting Potential Patterns of Governance from 16 Sustainable Development Goals, 2022; Systemic Coherence of the UN's 17 SDGs as a Global Dream , 2021; Interplay of Sustainable Development Goals through Rubik Cube Variations, 2017; Refining the Value of Sustainable Development Goals, 2017; Memorable Packing of Global Strategies in a Polyhedral Rosetta Stone, 2023; Interrelationships between 64 Complementary Approaches to Sustainable Development, 2002)
| Form | |||
 |
Indicative allocation of a 4x4 pattern of 16 SDGs to a 4x4x4 cube | animated gif | Interplay of Sustainable Development Goals through Rubik Cube Variations (2017) |
 |
Represents of 16 SDGs on 32 faces drilled truncated cube | animated gif | Polyhedral representation of Sustainable Development Goals including "Own Goals"? (2022) |
 |
Indicative transformation of compound of 16 tetrahedra representing 16 SDGs | animated gif | Enabling pattern detection in a 16-tetrahedral compound(2022) |
 |
Animation of screen shots of alternative view of pathway dynamics of 16-tetrahedra complex |
animated gif interactive 3D (html/x3dom) |
Eliciting Potential Patterns of Governance from 16 Sustainable Development Goals (2022) |
 |
Envisaging global brain representation in 3D through use of animation | video (YouTube) |
Systemic feedback cycles of global brain interrelationships in 2D (2019) |
 |
Indicative pathways in a global brain -- "global brainwaves"? | video (YouTube) |
as above |
 |
Patterning and framing a global brain in the light of supercomputer memory organization | video (YouTube) |
Framing Cognitive Space for Higher Order Coherence (2019) |
Symbols: representation of mandala in 3D
With to the more generic understanding of "mandala:, this is a select list of what is presented more systematically as Eliciting Insight from Mandala-style Logos in 3D (2020)
| Form | |||
 |
3D configuration of nonagon mandala (Model A) |
interactive 3D (x3d, wrl) video (mov) |
Three-dimensional configuration of nonagon mandala In: Concordian Mandala as a Symbolic Nexus: Insights from dynamics of a pentagonal configuration of nonagons in 3D |
 |
3D configuration of nonagon mandala (Model B) | interactive 3D (x3d, wrl) video (mov) |
as above |
Symbols: dynamics of 1000-petal chakra rings in 3D
The complexity of the 1000-petal lotus chakra of Easter tradition, invites extensive exploration of how if might be visulized in 3D and dynamically.
| Form | |||
 |
Rotation of 1000-petal chakra rings Experimental rotating configuration of crown chakra of 20 50-petalled rings in 3D |
interactive 3D (x3d, html/x3dom) video (mp4, YouTube) |
Independent rotation of total set of 20 rings of 50 petals: |
 |
1000-petal chakra rings (stacked and mutually rotating) Experimental 3D configuration of crown chakra as a stack of 20 50-petalled rings |
interactive 3D (x3d, html/x3dom) video (mp4, YouTube) |
Stack of set of 20 moving rings of 50 petals - solid |
 |
Crown chakra with rotation of 20 rings of 50 petals ("1,000-petalled lotus") | video (YouTube) |
as above |
 |
Gyroscopic movement of selected rings of 1000-petal chakra Experimental gyroscopic rotation of 20 50-petalled rings of crown chakra in 3D |
interactive 3D (x3d, html/x3dom) video (mp4, YouTube) |
Gyroscopic movement of selected rings of 50 petals - outline |
 |
Movement of 20 rings of 1000-petal chakra through a torus | interactive 3D (x3d, html/x3dom) |
Movement of 20 rings through centre of a torus |
 |
Framing of 1000-petal lotus by dodecahedron | interactive 3D (x3d, html/x3dom) |
Crown Chakra framed by a Dodecahedron -- movement of 20 rings between dodecahedral vertices |
 |
Experimental 3D configuration of 20-ringed 50-petalled crown chakra with vajra symbols (Version II) | video (YouTube) |
as above |
 |
Presentation of 1000-petall chakra as twin tori | interactive 3D (html/x3dom) | Crown Chakra as Twin Tori: |
Symbols: patterns associated with triadic symbols and Abrahamic religions in 3D
Symbols: patterns associated with I Ching (Yi Jing), Ba Gua and Tao
| Form | |||
 |
Relating cultural symbols using dynamic I Ching configuration | image video (YouTube) |
Dynamic Exploration of Value Configurations Interrelating traditional cultural symbols through animation |
 |
Interactive visualization of 384 I Ching transformations using force-directed layout | interactive 3D (html/d3.js) | Enhancing Strategic Discourse Systematically using Climate Metaphors (2015) Four-dimensional requisite for a time-bound global civilization? (2015) |
 |
Relational Dynamics of the I Ching -- complementary pairs of hexagrams represented by 24 edges of a faceted cube | video (YouTube) |
Patterning psychosocial relationships in 3D or more? (2017) |
 |
Tao of "Us" and "Them" (animation of a 4D Klein bottle usefully explored as a 2D rendering) |
animated gif | Snoring of The Other: a politically relevant psycho-spiritual metaphor? 2006 |
 |
Tao (Yin-Yang) symbol in 3D | video (mp4) interactive 3D (x3d, wrl, html/x3dom) |
|
 |
Tao (Yin-Yang) symbol in 3D | video (mp4 (sold) (mp4 (wire)) interactive 3D (x3d, wrl, html/x3dom) |
Symbols: enneagram, ouroboros, lauburu and Zen ox-herding in 3D
Symbols: dynamics of Knight's move, magic squares and the swastika
| Form | |||
 |
Magic square animation | animated gif | Salvation Enabled by Systemic Comprehension: Via aesthetics of magic squares? (2015)
|
 |
Animation of 8 of the Knight's moves in relation to a magic square | animated gif | Predictability and pattern-breaking: the Knight's move (2011) |
 |
Animation of succession of Knight's moves across the BaGua engendering both forms of Swastika: left-facing (green) and right-facing (red) | animated gif | Alternative representations: Knight's move, Swastika and BaGua ? (2012) |
 |
Cyclic animation of Knight's move, Swastika and Ba Gua dynamics | animated gif | Cognitive implication in Merkabah as configuration of cycles essential to systemic viability (2017) |
Symbols: flag and laurel wreath dynamics in 2D and 3D
| Form | |||
 |
Rotating Borromean configuration of 28 rotating stars of a Flag of Europe | video (YouTube) |
Experimental Visualization of Dynamics of the European Parliament in 3D (2019) Coherent Representation of the European Union by Numbers and Geometry (2019) |
 |
as above | animated gif | Imagining the Flag of Europe otherwise? (2018) |
 |
Laurel wreath branches configured as spiral in 3D | video (YouTube) |
Requisite helical cognitive engagement within a global brain (2019) |
Symbols: dynamics of heart / temple patterns in 3D
| Form | |||
 |
2D animation of dynamics defining the 4 conditions of the heart-pattern (using juxtapositioned cross-sections of two 3D tori) |
animated gif | Cognitive heart dynamics framed by two tori in 3D In: Visualization in 3D of Dynamics of Toroidal Helical Coils -- in quest of optimum designs for a Concordian Mandala |
 |
As above (alternative) | animated gif | as above |
 |
Embedding of heart / temple pattern within contiguous tori (variants show one or both tori) | animated gif interactive 3D (x3d (no tori) x3d (with tori) wrl) |
as above |
 |
Superposition of 2 tori (in cross-section, outlining a 3D heart pattern) with circulating polyhedra | animated gif | Cyclic dynamic of emergent order versus States of emergency ordered spasmodically In: Global Challenge of the Global Challenge¿ In-quest of a decision-making framework appropriate to a world in crisis ? |
 |
Possible multi-level "stacking" of counter-rotating tori (multiple 3D heart patterns) | animated gif interactive 3D (x3d, wrl) |
Cyclic dynamic of emergent order versus States of emergency ordered spasmodically In: as above |
Oppositional logic and geometry
Oppositional Logic as Comprehensible Key to Sustainable Democracy (2018), Oppositional logic and its geometry -- 16 minus 2 connectives? (2021), Neglected recognition of logical patterns -- especially of opposition (2017)
| Form | |||
 |
Representations of arrays of logical connectives (Hasse diagram with binary codings added) |
,png | From 16 to 14 connectives -- precluding a logical meta-perspective? (2021) |
 |
Representations of arrays of logical connectives (Hasse rhombic dodecahedron with indicative Venn diagrams added) |
.png | as above |
 |
Representations of arrays of logical connectives (Aristotelian rhombic dodecahedron with indicative Venn diagrams added) |
,png | as above |
 |
Animation of 3D visualization of propositional connectives (Venn diagram of logical connectives mapped onto cuboctahedron) |
animated gif | as above |
 |
Animation of 3D visualization of propositional connectives (Mapped onto rhombic dodecahedron) |
animated gif | as above |
 |
Animation of 3D visualization of propositional connectives (Mapped onto cuboctahedron) |
animated gif | as above |
Issues: environmental dynamics in 2D and 3D
| Form | |||
 |
Mapping of strategic issues onto icosidodecahedron (folded) | image | Sustainable development issues fom Earth Summit (1992) |
 |
as above (unfolded) |
image | as above |
 |
9-fold pattern of planetary boundaries inspired by the "World Tree" (Yggdrasil) | video (YouTube) |
Axis Mundi, Yggdrasil, Omphalos and Sahasrara? (2020) |
 |
3D Configuration of "negative forces" as 72 "Demonic sigils" -- the spikes of the coronavirus | video (YouTube) |
Engaging with Hyperreality through Demonique and Angelique? (2016) Cognitive Engagement with Spike Dynamics of a Polyhedral Coronavirus (2020) |
 |
Experimental 3D configuration of "positive forces " as 72 "Angel names" | video (YouTube) |
as above |
 |
Spike dynamics on Pentakis-snub-dodecahedron with 72 vertices (wire frame version) | video (YouTube) |
as above |
 |
Coronavirus: spike dynamics on Pentakis-snub-dodecahedron with 72 vertices? | video (YouTube) |
as above |
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"Organizing" Global Governance: engaging playfully with coronavirus? | video (YouTube) |
Engaging Playfully with Coronavirus through "Organizing" Global Governance? (2020) |
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Animation in 3D of 120 geometrical transformations of the coronavirus | animated gif | as above |
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