Cognitive operations potentially analogous to generation of tiling patterns

Systemic Coherence of the UN's 17 SDGs as a Global Dream (Part #6)

[Parts: First | Prev | Next | Last | All] [Visuals] [Links: To-K | From-K | From-Kx | Refs ]

"Cognitive tiling" or "conceptual tiling"? As noted above, the set of 17 tiling patterns is generated by distinctive combinations of so-called Euclidean plane isometries. With respect to the geometry of the patterns, these operations are described in terms of translation, rotation, reflection and glide reflection. The question is to what degree these operations bear significant resemblance to cognitive operations, most obviously as might be perceptible in the discourse by which strategies are engendered.

Ironically, it is appropriate to note that "tiling" may itself be used in a way which is relevant to this argument. Most specific are quotations by Bernardo Kastrup:

• I invite you to try and briefly remove the conceptual tiling that blocks your view of reality, so you can see the raw chunks of perceptual pixels in the medium of mind.
• But Zen is largely an attempt to, during meditation, transcend thought (i.e. the intellect) and the "conceptual tiling" we live in. And, insofar as our whole civilization an culture are based and driven by "intellectual tilings", Zen does try to transcend that too, doesn't it? Naturally, one always returns to the ego state, and so do Zen practitioners. (The Biggest Error Ever Made in the Name of Science, Metaphysical Speculations, October 2013)

Related understandings are offered by:

• Francis Rousseaux: Like the sacred place, the conceptual model is intended to reassure and protect us against the thought and experience of a singular situation. Can all situations be adapted to such an automatic prescription? Obviously, never, because the conceptual tiling is an integral part of the situation, given in this case as a different and overloaded form, but always allowing the possibility of a singular interpretation. (Knowledge acquisition or manifestation of the thought? Knowledge Management and Philosophy, 2003)
• Martin J Pickering and Simon Garrod: However, in my view, the particular integration of production and comprehension processes that the authors propose is unrealistic in consisting of a complex cognitive tiling of predictive modeling processes... (An integrated view of language production and comprehension, Behavioral and Brain Sciences, 36, 2013, 4)

Geometrical metaphors in discourse: There is an extensive literature on the role of geometrical metaphors in discourse, although this tends to focus on the use of metaphor in mathematical education. This includes:

• Mathieu Aubry: Metaphors in Mathematics (2009)
• Maciej Rosinski: Metaphor Activation in Multimodal Discourse: case studies on the emergence of geometrical concepts. (University of Warsaw Repository, 2018)
• Gloriana González: A geometry teacher's use of a metaphor in relation to a prototypical image to help students remember a set of theorems (The Journal of Mathematical Behavior, 32, 2013, 3)
• Wajeeh Daher: Students' Positioning and Emotions in Learning Geometric Definition (Journal on Mathematics Education, 11, 2020, 1)
• Carlos Cornejo, et al: The physiognomic and the geometrical apprehensions of metaphor (Culture and Psychology, 19, 2013, 4)
• J. Rees: Geometry and Rhetoric: Thinking about Thinking in Pictures (Nexus Network Journal, 12, 2010, 3)
• Nathalie Sinclair et al: To be or to become: how dynamic geometry changes discourse (Research in Mathematics Education, 10, 2008, 2)
• Barry Stocker: Pascal and Derrida: Geometry, Origin and Discourse (Symposium, IV, 2000, 1)

For Patricio Herbst:

The present study, close to Otte's [1983[approach, investigated a way to carry out an internal critique of a text that would allow one to address questions like: how docs textuality as a linear temporal process create its own mathematic? And what are the textual meanings of the mathematical notions which are developed inside a textbook and along that temporal axis? The main theoretical influences on my efforts to conceptualize and analyze text have been Foucault's (1972) archaeology of knowledge, Chevallard's (1991) theory of didactic transposition and Eco's (1979) notion. of model reader and open-closed work. However, their presence is mostly' silent; my understanding of their works guides my way of interrogating and asserting., but they are not responsible for my actual questions and conclusions.( The Number-Line Metaphor in the Discourse of a Textbook Series (For the Learning of Mathematics, 17, 1997, 3)

The argument can be developed with respect to the simplest features of geometry, namely points, lines, volumes and holes (Metaphorical Geometry in Quest of Globality -- in response to global governance challenges, 2009),

Tiling as an instance of cognitive closure? What could be recognized as the central mystery of any recognized tiling pattern is how a sense of satisfaction is associated with the closure of pattern recognition. More generally this is the satisfaction of resolving a riddle or completing, any puzzle (crossword, sudoku, Rubik cube, etc), or solving a problem of mathematics or design. How is such closure most fruitfully understood in generic terms (Hilary Lawson: Closure: A Story of Everything, 2001).

Mathematics frames the matter in terms of symmetry in relation to closure, with an extensive literature on closure in set theory and on symmetric closure -- expressed formally through requisite abstractions (Marcin J. Schroeder, Concept of Symmetry in Closure Spaces as a Tool for Naturalization of Information 2008).  In general, the closure of a relation is the smallest extension of the relation that has a certain specific property such as the reflexivity, symmetry or transitivity. Distinctions are made between difunctional closure, contact closure,  reflexive transitive closure,  reflexive transitive symmetric closure,  equivalence closure and congruence closure.

A seemingly quite unrelated approach to the experience of closure has been developed by Gestalt psychology which offers a numbetr of principles (termed "laws") that now influential in design, including: Law of Closure, Law of Symmetry, Law of Similarity, Law of Proximity. Thus the Law of Symmetry is the gestalt grouping law that states that elements that are symmetrical to each other tend to be perceived as a unified group; with closure, parts are combined to form a simpler whole (Steven Bradley, Design Principles: Visual Perception and The Principles of Gestalt, Smashing Magazine, 29 March 2014). These insights have been developed in relation to problem solving.

Metaphorical implications from tiling operations: It is curious to note, although consistent with the possibility argued here, that the geometrical terms by which the tiling patterns are distinguished are far from unrelated to those descriptive of discourse and debate -- if only metaphorically and/or via synonyms. Potentially more relevant, however, is the manner in which metaphorical uses of the geometrical terms are employed far more loosely in discourse than they are with respect to the transformations associated with those operations. This may indeed have implications for distinctive formulation of strategies.

Given the common interpretations of the terms describing the operations in the Euclidean plane, it is to be expected that the metaphors would be suggestive of dynamics that are well recognized in discourse, group dynamics, negotiation and politics. It is however possible that their generic significance would be even more familiar as articulated in dance -- notably through the manner in which dialogue and group dynamics are frequently framed in musical terms or as a "dance":

• Dana Mills: Dance and Politics: moving beyond boundaries (2016)
• Anca Giurchescu: The Power of Dance and Its Social and Political Uses (Yearbook for Traditional Music, 33, 2001)
• Susan A. Reed: The Politics and Poetics of Dance (Annual Review of Anthropology, 27, 1998),
• Siobhan Burke: Making (and Seeing) Dance in the Politicized World (The New York Times, 17 November 2017)

Do the dynamics of pattern transformation between "left" and "right" in politics call for reframing in terms of dance? More fundamental are then the cognitive implications of any such dance, as suggested by the work of Mark Johnson (The Meaning of the Body: aesthetics of human understanding, 2007) and Maxine Sheets-Johnstone (The Primacy of Movement, 1999). One exploration of potential relevance is that of Paris Arnopoulos (Natural Energy and Social Power: metaphors from physics to politics, Gamma Institute, 1985).

Cognitive implications of operational modification of polyhedra: Distinctive polyhedra may be created  through modification of a seed polyhedron by various prefix operations, as described by the Conway polyhedron notation, and discussed separately (Topological operations on polyhedra as indicative of cognitive operations, 2021). The following example shows how 11 new forms can be derived from the cube using 3 operations (named dual, ambo and kis). The new polyhedra are shown as maps on the surface of the cube so the topological changes are more apparent.

Conway relational chart Showing 12 forms created by 3 operations on the cube

Tomruen at English Wikipedia, Public domain, via Wikimedia Commons

The three basic operations which are sufficient for generation of the Platonic and Archimedean polyhedra are:

• dual: replaces each face with a vertex, and each vertex with a face
• ambo: creates degree-4 vertices (otherwise kn own as rectification)
• kis: raises a pyramid on each face

Other operations have been distinguished, together constituting a more conventional total of 13; these have been extended in the Antiprism application of Adrian Rossiter to a further set of 18 -- although how they might together be understood as a set remains unclear (Conway Notation Transformation, Antiprism; Wythoff-style constructions, Antiprism).

As developed by Rossiter to prodiuce weave patterns on tiled surfaces, the "Wythoff constructive notation" can be used to define "Conway" operations more precisely. According to Rossiter, the number of Conway operator stype patterns is unlimited. raising the question as to whether they can be meaningfully ordered. This would probably depend on the definition of "Conway operation" pattern, and in particular the definition of when the pattern tiles close.

Of primary relevance to this argument is how such operations might be distinguished in cognitive terms with regard to the transformation of any ordered memeplex in the course of discussion -- or in the development of a strategy. What indeed are the distinctive "operations" in discourse? Again some terms identified with respect to transformation of polyhedra offer clues when metaphorically understood: truncate, expand, join. Valuable indications of potential relevance to discourse analysis are to be found in the work of B. R. S. Recht (Operations on Polyhedra, GitHub, 15 July 2019).

Morphogenetic implications: The generation of a tiling pattern could potentially be understood in terms of morphogenesis as notably articulated by the topologist René Thom (Structural Stability and Morphogenesis, 1972), especially given his interest in dance, as noted below..

There is however little explicit trace of this connection in the literature, with the apparent exception of a study by Mostafa Alani on the hexagonal patterns in Islamic architecture (Algorithmic investigation of the actual and virtual design space of historic hexagonal-based Islamic patterns, International Journal of Architectural Computing, 16, 2018,1). This found that such designs correlate to each other beyond just the formal dimension and that deep, morphological connections exist between them. The study identifies these connections and presents a categorization system that groups designs together based on their "morphogenetic" characteristics. As indicated there, determining the wallpaper group is important for identifying the "fundamental unit" representing the minimum geometric composition that is being systematically replicated. 

From this perspective it is interesting to recognize the "suggestive" nature of the following sequence of 16 images -- possibly to be understood as "visual primitives" characteristic of the attraction dynamics of intercourse, in its specific and general senses. From a cognitive perspective, the question might be asked how Thom's 16-fold identification relates both to the set of Conway operations (above) and to the 16+1 set of SDGs.

Changes: the locus of principal changes of topological type reproduced from René Thom, Structural Stability and Morphogenesis, 1972

1. curve with cusp pointing downward 2. appearance of new point at origin, where lip formation begins -- 3. this grows... . 4. pierces the cusp.. 5. and crosses it ...to form the phallic mushroom... characteristic of the parabolic umbilic... 6. the cusp meets the lower branch of the lip in a hyperbolic umbilic... 7. and then the two branches cross to form a curvilnear triangle piercing laterally a convex curve 8. the triangle shrinks, first touching the curve 9. and then shrinking inside it 10. to form a hypercycloid with three cusps, and finally vanishes in an elliptic umbilic.. 11. reappearing immediately with the same orientation 12. its lower cusp meets the curve 13. and pierces it 14. the curve and upper edge of the triangle touch in beak-to-beak singularity, which separates 15. producing two symmetric swallowtails, reabsorbed into the curve 16. leading to the original configuration

[Parts: First | Prev | Next | Last | All] [Visuals] [Links: To-K | From-K | From-Kx | Refs ]