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Rendering a 48-fold set of insights memorable through visualization
AI Commentary on an Envisaged Synthesis Dialogue (Part #8)
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Whilst the AI suggestions for mappings constitute a valuable provocation to further reflection, no attempt was made in the following animations to attribute the koans according to those suggestions. These appear to call for much further consideration and insight, notably from polyhedral combinatorics. The challenge for any potential integrative insight, whether for an individual or a society, can however be framed as how to render a 48-fold pattern comprehensible and memorable. The set of koans of the Gateless Gate is one representation of that challenge to comprehension -- as it can be presented through various polyhedra.
| Indicative mapping of 48 koans and micronutrients onto Archimedean polyhedra | |||
| 48-edged Rhombicubocthedron mapped with 48 koans | 48-vertex Truncated cuboctahedron mapped with 48 koans | ||
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| Produced with the aid of Stella4D | |||
An obvious question is how the arbitrary mappings above might be transformed into a more orderly memorable pattern indicative of related cognitive processes. In the following configuration -- rather than use the edges (above left) or vertices (above right), the faces of a polyhedron are used. A 3-fold colouring of the 48 elements of the set of koan-micronutrients can then be used to frame the question of how the "vitamins" (green), "amino acids" (cyan), and "minerals" might be ordered on the polyhedron. The challenge is somewhat reminiscent of the widely recognized challenge of appropriately ordering Rubik's "magic cube"as previously discussed (Harmonizing Dissonance via a Rubik's Cube Framework, 2024; Interplay of Sustainable Development Goals through Rubik Cube Variations, 2017; Rethinking Rubik's Cube: a mnemonic device for ways of knowing and engagement? 2009).
| Indicative mapping of 48 koans and corresponding micronutrients on 48-faced dual of Truncated cuboctahedron ("Disdyakisdodecaheron") | |||
| Dual of Truncated cuboctahedron | Unfolded network | (Un)folding animation | |
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| Produced with the aid of Stella4D | |||
Of potential interest to recognition of the challenge of higher dimensionality to integrative comprehension -- in the light of the set of insights of the 48 koans -- is a configuration in 4D. Ths is known variously as the tetracontoctachoron, commonly called the 48-cell -- consisting of 48 truncated cubes whose facets are termed "cells". As a bitruncated 24-cell, it is a convex noble uniform polychoron (one of 64). Four cells join at each vertex. It is the medial stage of the truncation series between a regular icositetrachoron and its dual. Alternatively, it is also the stellation core of the compound of two opposite icositetrachora, the stellated tetracontoctachoron. Its configuration in 4D can only be indicated through various approaches to its projection into 3D. Such images frame the question of the cognitive correspondence to "truncation" with respect to comprehension.
| Alternative (necessarily distorted) representations in 3D of 4D 48-cell (bitruncated 24-cell) configuration | ||
| Orthographic projection of the bitruncated 24-cell into 3D | Tetracontoctachoron | Animation of cross-sections of tetracontoctachoron |
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| Produced with the aid of Stella4D | ||
| By Tetracube at en.wikipedia, CC BY-SA 3.0, Link | Tomruen at English Wikipedia | User:Galoomba |
The animation on the right above is suggestive of the manner in which the cognitive relationship between the insights of the sets of koans is dynamic rather than static. More generally this reinforces the argument that integrative synthesis is itself dynamic rather than statically configured.
The above visualizations were preceded by earlier exercises in formatting the koans of the Gateless Gate for Twitter (Configuring a Set of Zen Koan as a Wisdom Container, 2012). Those exercises were further developed at that time with concluding arguments and a complete set of references presented in a later paper (Enabling Wisdom Dynamically within Intertwined Tori: requisite resonance in global knowledge architecture, 2012).
Question to AI: If the 48 koans are mapped onto edges of a rhombicuboctahedron of 48 edges, it seems that one preliminary method of distribution is by pairs -- since there are 24 pairs forming 24 great circles. Can you suggest how the set of 48 might be split into opposing pairs
| ChatGPT 4 | Claude 3 |
More meaningful pairings could potentially be derived from thematic or conceptual relationships between the koans rather than their sequence numbers. If we aim for deeper thematic connections, we can reconsider the pairings to reflect more meaningful relationships. Here’s a revised attempt to pair the koans based on thematic oppositions or complements:
This thematic pairing aims to provide more meaningful connections between the koans, enhancing their pedagogical value by emphasizing their complementary or contrasting insights. | Certainly! Let's explore a possible way to split the 48 koans into 24 opposing pairs, which could then be mapped onto the 24 pairs of edges forming 24 great circles on the rhombicuboctahedron. This pairing could be based on contrasting themes, complementary insights, or dialectical relationships between the koans. Here's one potential set of pairings:
These pairings are just one possible way to create contrasting or complementary relationships between the koans. The specific pairs could be adjusted based on different interpretations and understandings of the koans' themes and insights. When mapped onto the rhombicuboctahedron, each pair of koans would be placed on opposite edges of a great circle, creating a sense of tension, balance, or dialectical relationship between the two. This arrangement could highlight the way in which seemingly contradictory or paradoxical teachings can ultimately point towards a deeper, non-dual understanding of reality. For example, the pairing of "Joshu's Dog" (which asks whether a dog has Buddha-nature) and "Nansen Cuts the Cat in Two" (which presents a shocking act of violence) could be seen as representing the contrast between intellectual understanding and direct, embodied realization. By placing these koans on opposite sides of a great circle, we create a visual and spatial metaphor for the way in which these two aspects of practice can inform and illuminate each other. Similarly, the pairing of "This Mind Is Buddha" and "This Mind Is Not Buddha" could be seen as representing the dialectical relationship between affirmation and negation in Zen practice, and the way in which both perspectives can be simultaneously true and false depending on the context and level of understanding. By engaging with these paired koans in the context of the rhombicuboctahedron's geometry, we may discover new insights and connections that deepen our appreciation for the multi-dimensional and paradoxical nature of Zen teachings. This approach can also serve as a tool for contemplation and reflection, inviting us to explore the ways in which seemingly opposed or contradictory ideas can be integrated and transcended in the pursuit of awakening. The numbering also highlights the way in which this pairing arrangement creates a new sequence and structure for engaging with the koans, distinct from the linear progression of the original text. This reordering can invite fresh perspectives and insights, as familiar koans are encountered in new contexts and juxtapositions. When mapped onto the rhombicuboctahedron, the koan numbers could be used to label the corresponding edges, creating a visual and spatial representation of the interconnections and resonances between the teachings. This could serve as a powerful tool for contemplation and reflection, allowing practitioners to explore the koans in a more dynamic and multidimensional way. By engaging with the koans through this geometric framework, we may discover new layers of meaning and insight that deepen our understanding and appreciation of these profound teachings. The pairing and mapping of the koans can also serve as a reminder of the ultimately non-dual and interdependent nature of reality, as seemingly separate and contrasting ideas are revealed to be intimately connected and mutually illuminating. |
Highlighting a pattern of 24 -- as with pairs of koans -- through "bi-truncation" of that pattern invites a distinctive approach to visualization through a 3D rendering of the 4-fold lauburu symbol central to Basque culture (24-fold Pattern Implied by Dynamics of the Lauburu in 3D, 2016). This was explored as the visualization of the interplay of sets of voices in discourse, as discussed separately (Improvisation in Multivocal Poetic Discourse: Basque lauburu and bertsolaritza as catalysts of global significance, (2016).
Question to AI: Whilst those pairings can be arbitrarily mapped onto the polyhedron, the obvious challenge is how the 4 great circles intersecting at a vertex are distinguished -- especially since "connecting" co-linear edges through each vertex form "sub-circles". Can you comment on this
| ChatGPT 4 | Claude 3 |
Mapping the 48 koans onto a rhombicuboctahedron, where each edge corresponds to a koan and forms part of intersecting great circles, presents an interesting challenge in distinguishing and interpreting the intersections and connections: Intersecting Great Circles
Interpretation of Intersections
Thematic Distinction of Great Circles: To distinguish thematically the four great circles intersecting at a vertex from the two sub-circles in the rhombicuboctahedron, consider the following approach:
Visualization Techniques
Interpretation and Reflection
This approach helps to distinguish and interpret the thematic pathways of great circles and sub-circles within the rhombicuboctahedron, facilitating a deeper understanding of their interconnectedness and enhancing the learning experience. Conclusion: The rhombicuboctahedron serves as a powerful metaphor for the interconnectedness of Zen teachings, with the great circles and their intersections representing thematic pathways and points of integration. This structure not only aids in visualizing the relationships between the koans but also enhances the understanding of their collective insights, encouraging a holistic approach to learning and practice. | You raise an important point about the complexity of mapping the koan pairs onto the rhombicuboctahedron. The challenge of distinguishing the four great circles intersecting at each vertex, as well as the "sub-circles" formed by connecting co-linear edges, highlights the need for a more nuanced and multidimensional approach to arranging the koans. One way to think about this is to consider the different levels of relationship and interconnection between the koans. At the most basic level, we have the pairing of koans based on contrasting themes, complementary insights, or dialectical relationships. These pairs can be mapped onto opposite edges of a great circle, creating a sense of tension, balance, or resonance between the two teachings. However, as you point out, each vertex is also the intersection point of four great circles, which means that each koan is not only related to its paired opposite but also to the other koans that share the same vertex. This creates a more complex web of connections and associations, as each koan is situated within a network of related teachings and insights. Furthermore, the "sub-circles" formed by connecting co-linear edges through each vertex add another layer of complexity to the arrangement. These sub-circles could be seen as representing more subtle or implicit relationships between the koans, perhaps based on shared imagery, linguistic parallels, or underlying philosophical principles. To address this challenge, we might need to develop a more sophisticated system of mapping and labeling the koans that takes into account these multiple levels of relationship and interconnection. This could involve using different colors, symbols, or annotations to distinguish between the different types of connections and associations. For example, we could use solid lines to represent the primary pairings between koans, dotted lines to represent the connections between koans sharing the same vertex, and dashed lines to represent the sub-circles formed by co-linear edges. We could also use different colors to represent different thematic clusters or categories of koans, helping to highlight patterns and commonalities across the collection. Another approach could be to create a three-dimensional, interactive model of the rhombicuboctahedron that allows users to explore the connections and associations between the koans in a more dynamic and intuitive way. By clicking on individual vertices or edges, users could reveal the corresponding koans and their relationships, as well as access additional commentary, translations, or multimedia resources. Ultimately, the challenge of mapping the koans onto the rhombicuboctahedron in a way that captures their complex web of connections and associations is an invitation to engage with these teachings in a more creative, exploratory, and multidimensional way. By embracing the ambiguity and complexity of the geometric arrangement, we can develop new tools and approaches for studying and practicing with the koans that deepen our understanding and appreciation of their profound insights. This process of mapping and arranging the koans can itself become a form of spiritual practice, as we grapple with the paradoxes and contradictions of the teachings and strive to integrate them into a coherent and comprehensive view of reality. By engaging with the koans in this way, we may discover new possibilities for awakening and transformation that transcend any fixed or linear understanding of the path. |
From the perspective of environmental design, an interpretation of the "gate" meme used by Zen is offered by Christopher Alexander in The Timeless Way of Building (1979), as discussed separately (Beyond Edge-bound Comprehension and Modal Impotence: combining q-holes through a pattern language, 1981). For Alexander:
- To seek the timeless way we must first know the quality without a name.
- To reach the quality without a name we must then build a living pattern language as a gate.
- Once we have built the gate, we can pass through it to the practice of the timeless way.
- And yet the timeless way is not complete, and will not fully generate the quality without a name, until we leave the gate behind.
A configuration of insights can be undersood as a "cognitive gate". Alexander and colleagues presented a preliminary list of 253 patterns in A Pattern Language (1977). These can be seen as focusing on the physical environment of towns, buildings, and construction. The 48-fold pattern might be explored as clustering the 253 in some manner. As a systemic gate, the 253 also invite experimental interpretation, implying sets of four "corresponding" patterns (5-fold Pattern Language, 1984):
- Template: This endeavours to describe the pattern in content-free terms as pure relationship. As such it is a guideline for the elaboration of patterns for other arenas.
- Physical environment: This is an adaptation of Alexander's own pattern description
- Socio-organizational environment: This describes the pattern as it applies to the organization of social groups, organizations and networks.
- Conceptual environment: This describes the pattern as it applies to the organization of a conceptual framework or a body of knowledge.
- Intra-personal environment: This describes the pattern as it applies to the organization of modes of awareness adopted by a person.
Alexander's subsequent magnum opus (The Nature of Order: an essay on the art of building and the nature of the universe. Center for Environmental Structure, 2003-4) inspired his further reflection on patterns (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; 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). These in turn invite further reflection (Harmony-Comprehension and Wholeness-Engendering: eliciting psychosocial transformational principles from design, 2010).
Whilst Alexander's design patterns are indeed individually comprehensible, the set as a whole poses a major challenge -- despite their focus on the physical environment. Given their abstract nature, this is potentially even more so in the case of the psychosocial pattern of 48 koans of the Gateless Gate. It is therefore intriguing to consider the pattern of some 47 patterns of micronutrients with which many are variously familiar as vital to healthy nutrition. Surprisingly it would appear that these micronutrients are also a challenge to memorability and comprehension. There are many references to nutrition mnemonics and diet mnemonics -- typically referring only to subsets of the 47. These may even emphasize those that are "essential" -- with the implication that they cannot be generated by the body and are dependent on environmentally supplied nutrition.
The challenge of memorable comprehensibility is especially relevant with respect to the UN's set of 17 inherently unmemorable Sustainable Development Goals -- or the efforts to promote a set of equally unmemorable Inner Development Goals (as discussed below). With respect to the sustainable health of the body -- and seeing things whole -- there is however considerable irony to the equivalent challenges of memorability of the set of micronutrients (vitamins, amino acids and minerals).
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