The Animated Geometry and Topologic Situation Modelling
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A new use of topology is presented here, the non-algorithmic Topologic Situation Modelling© method, which enables to connect global and local situations. The dimensional-geometric language of topology (a discipline of mathematics) and its non-local nature are alien to most people, especially in the capacity to deal with deploying geometric dimensions – a shaping –, rather than edges and corners of fixed shapes. This page is an attempt to communicate the shifts from one dimension to another (the ‘animation’ aspect of shaping) by using fragmentary videos to give a few hints of what this language is and can be used for.
The most striking finding is that, as an intellectual tool, this method can be used for reasoning about various logics, especially a dimensional logic of stages and states of deployment, and about orientation and limits. This is consistent with developmental, evolutionary and historical models based on producing an end shape. However it can model simultaneously an origination process that has specific, albeit simple properties of geometry, and so it does not require explanations in terms of spatial system-environment, time, or in-out processes such as dynamics or adaptation.
Dr Bouchon’s research in cognitive anthropology revealed the existence of an Animated Geometry that can be modelled with the above method, and which lies at the root of cultural developments and symbols tat govern action, such as the medical caduceus.
- In humans, it is a « Geometry of Mind » , which a few philosophers have sought as the origin of thought and abstract geometry (and perception in the brain). It is useful to understand general developments of human thinking and culture, and to understand or create those of appearance or physical existence. It gives rise to some forms of scientific insight.
- In parallel, it is also a « Geometry of Sensation », an innate awareness that humans can access through their biology, simple enough to be expressed in gesture. It is particularly useful to describe one’s general state, in terms of health, activation, strain or attention effort, or of ease. Its topologic nature makes it challenging to formulate in words.
Topology and geometry
The geometric discipline of mathematics called topology fundamentally studies small distortions that do not change the properties of a geometric figure, and it involves the geometric dimensions (1D line, 2D plane or surface, 3D volume, the hyperbolic 4D, and the 0D point). Using a 2D surface to simplify, it is described as ‘a rubber sheet geometry’ of stretching. This is useful to understand behaviours of spreading and swelling.
However, in applied mathematics, topology has mostly come to be used primarily to describe fixed shapes and complicated patterns, to make theoretical models and categorical cartographies that are topographic rather than topologic, and to describe large transformations at points of no-return, which trans-form, radically changing form and properties (e.g. appearance of a more complex system) — numbers have taken precedence over geometry, and theorems are taught without their geometry (Villani).
In Bouchon’s geometric method, the topology used remains geometric, quite simple, and refers to ‘shaping’ (small change or stretch) rather than measured shapes, staying closer to the original name of topology, ‘geometria situs’.
The Topologic Situation Modelling© method involves a dimensional logic that describes states or stages, and orders of logic, levels of abstraction, orders of complexity (e.g. cell and organism), etc. It can account for both small and large changes in space, and can work with timing, whether progressive, simultaneous or sudden. The modelling geometry itself is non-algorithmic, not mathematised with numbers – the geometric figures are abstract figures but without metric, not measured or based on rules (as in school learning). Instead, a figure distorts in the same way as the presenting situation being modelled alters. This does not produce a mathematical model, a naturalistic analogy to an object or realistic process, but a similarity or ‘likeness’ of how the situation works, changes, how it is deploying, and its properties, just as we can gesture or draw stages of the shaping of an embryo, without details.
As a tool to extract understanding from a situation, it can give rise to a mathematician’s insight, for example this one from Roger Penrose, who expresses his topologic ideas with gestures.
Roger Penrose – Gestures (53s)
Roger Penrose – Insight & Awareness (3mn36)
Here is an example, rendered in computer animation, of this topologic ‘thinking space’ (neither physical nor real, nor unreal), a typical imaging in dimensional geometry in its raw form of distorting shape, without numbers, measures, words, or valuations. It apprehends a situation without fragmenting it. Distorting shape is useful to describe technically what the humanities disciplines call ‘change’.
Bouchon – Thinking in Geometric Imaging (1mn)
The topologic nature of this kind of imaging is apparent in a simple demonstration by French mathematician Cédric Villani while explaining the pragmatic reasons for the developments of higher geometric dimensions through history. Teaching theorems without their geometric grounding allows to invent new mathematics, but they also loose something.
Cédric Villani – Paper Sheet Bend (25s)
How useful is this topologic method?
The dimensions of geometry (1D, 2D, 3D, 4D, 0D) can translate into different logics useful for the understanding: geometric dimensions in flat graphs or 3D maps, logical orders, orders of abstraction, orders of complexity, orders of activation, orders of gravity, and different scales (e.g. human vs planet scales, but not ‘scaling up’ in just numbers). This helps find similarity in representations of the same thing with different words in different disciplines or contexts (e.g. depending on the framework of representation, agitation in an animal is interpreted, in human behaviour, anxiety in the mind, or tension in the body, or as neurological activation). This approach could reduce the field specific biases and the fragmentation of knowledge.
The dimensions can be used to map out stages of deployment going through states. This is particularly useful in medicine to compare orders of illness gravity. For example the striking similarity of diffuse autonomic symptoms in end stage diabetes or cancer to those in the flaring sub-clinical syndromes of homeostatic instability, helps detect an inversion of one or two characteristics; this facilitates differential diagnosis connectively, operationally, or through distribution (e.g. presence or absence of measurable organic tissue destruction), without reducing the sub-clinical symptoms to non-existence in medical-material terms, or confusing irreversibility with a still present functional capacity to reverse the deployment of the state of health. Such a topologic mapping of health states helps predict the adequacy or effectiveness of treatments. It also provides a means to actually prevent the deployment of a degenerative disease in a more fundamental way than by stimulating counter-measures, which only prevent the symptomatic expression of the (unchanged) state. This could also be a global strategy to reduce, in the population, the susceptibility to other diseases (e.g. covid).
This topologic modelling also provides a means of understanding how a topologically ‘oriented’ situation deploys into both productive and counter-productive effects simultaneously, although one may be more obvious and the other hidden for a time or in another space. Re-presentations (of the presenting situation) that use space-time parameters conceive them as two opposite directions (e.g. up and down), and in physical-human terms, often as evaluations (e.g. constructive & destructive, or good & bad, positive & negative, light & dark). In the same way, we intuitively know that to achieve a ‘high’ goal (highly deployed), we are stretching ourselves to limits – one does not come without the other. Understanding that both occur simultaneously at limits permits a new understanding — it is a symmetry at surface. This geometric apprehension then allows to use the Topologic Situation Modelling to apprehend what occurs away from this symmetry, away from limits. This new method would be particularly useful to find new strategies to face the current state of human affairs, at limits in many domains, often with productive and counter-productive processes in conflict.
A topologic thinking space
The video to the right is an example of using the topologic method to reason in a topologic thinking space, a generic intellectual tool to understand, in this case, understand human evaluations of people. The human mind associates a vertical axis called ‘Up’ (or ‘Ex-‘ for philosophical extension, or topologic deployment) with individual development and population-level evolution (axis of the cone, not on the surface itself); but a mind without topologic thinking does not differentiate this from a horizontal axis of expansion (arrow on the surface of the cone, and of the square). In this example, the uni-direction (the arrow of developmental or evolutionary explanations) appears vertical (up) like the iconic tree of evolution, but it is biased, and thus produces an ‘off track’ bias, and damaging de-valuations and sur-valuations in the human domain, which plague people’s lives, unnecessarily.
‘Figuring’ the situation too, not just ‘Figuring it out’
The general methods of knowledge produce specific information, data, facts, and tactics under overall strategies, but they leave anomalies and what can only apprehended as ‘chance’ or some mysterious intent. Taking perspective to find specifics and make generalisations has limits. It does not explain why limits are both good and bad, and ‘where’ the logic of deploying them is taking us all. The result is now clear: the fragmentation of knowledge. All these forms of knowledge have become too complex to integrate or unify, or to really resolve fundamental problems.
Figuring out problems and solutions also keeps producing new problems, and the solutions ultimately all go in the same direction – the linguistic expression itself of figuring ‘out’ in detail contains this very direction of deployment, the ‘out’ or ‘ex-‘ or ‘up’ –, in a circular merry go round which is becoming a problem, and is going right back to the original problem, a seeminly unsolvable human problem of being subject to and acting out the breaching of boundaries.
Zooming in and out to take perspective
Perspective based on specifics and generalisations does not restore the topologic animation of oriented deployment or non-orientating.
The methods and animated geometry presented here constitute a complementary approach. It is way of by passing the limitations of perspective, fragmentation and holistic re-integration. The less differentiate, but also more global nature of ‘basic’ geometric topology provides a way of looking at the undivided situation as it is, as it presents, without the re-presentations, a way of directly ‘figuring’ the situation.
Here is another use for this method of thinking. We know that we need to not go past the limits of both the planet and people’s capacity to cope — to ‘not push too far’. On the other hand, developing new technologies might help. So where is that ‘too far’? It changes with contexts, and with individuals. How can we enable this in our many local situations? How to reach collective agreement on the direction to take? How to know which action will take ‘us all’ in that direction? The limits and boundaries are not just physical/material, but also human, this we know. However they often clash, with usually opposite perspectives. Conventional frameworks have not resolved this; the topologic method for generic gauging offered here might, by removing contextual parameters and biases.
The COP26 meeting has shown that humanity as a whole does not actually behave as a ‘whole’ because of varied needs, and is not managing to reach collective agreement on what to do and how to do it. It remains bogged in endless negotiations and clashes of general perspectives. Humanity is in a post-modern paralysis, torn on one hand, between exponential growths good and bad (poverty, population, economic, technological) and getting back to basics (leaving noone behind, enabling all life, human included, to meet it most basic biological needs, and for humans the need for dignity). On the other hand, global economy, governments, intitutions, businesses, technologists are torn between the need to provide people for their survival in society because they arise from people, and the necessity to restore physical nature, because we all arise from it, and bioological life requires this ground.
Topology, as used in this research, gets around the clashes of material and human perspectives, of advancements and basics of life, because it is generic, indepedent of these contexts, perspectives, and representations, of their inherent reference biases, and of valuings. The simple geometric topology of Topologic Situation Modelling© shows with just a few animated geometric figures the connection between the ‘ground’ and the deployments (the ‘up’ direction of advancement / the ‘down’ direction of ‘return to’ nature or basics). It does not rely on perspective, on valuing numbers for the material or physical, or evaluations and self/human-centered views or motivations – it just models and places relatively to see overall workings rather than distinguish and separate elements, categories or types on flat maps or models. Further, instead of localising problems and solutions, and re-presenting them with parameters of convention (e.g. ‘human’ and ‘nature’, words, numbers…), it provides a non-fragmented, non-local view (simultaneous local and global, if you will, without parameters). It describes generically the sitution that presents, free of complexity and detail complications. It is then easier to see what to do locally and be clearly aware of global implications and local consequences, thus leaving noone behind and nothing in the dark. The many limits, borders and boundaries are expressions of surface phenomena,of Boundary, a simple topologic parameter.
Topology has another advantage. It lumps both operational limits and connective or dis-connective borders under the term ‘Boundary’, modelling them altogether, generically. It models this in the form of surfaces that stretch but keep the integrity of shape and properties, and does this independently of whether the limits/borders are detected in physical or human domains, or not yet detected through these limiting conventions of representation. This parameter of topologic ‘Boundary’ (singular word, it is generic) is hidden from the conventionalised re-presentations because their languages are numbers or words describing many fragmented aspects, even if reintegrated; they do not use the generic language of dimensional-geometric topology.
The topologic space is capable of modelling Boundary globally and simultaneously in the local situation, and can be both abstract and concrete, so it can be confusing. Localising can also cause confusion, but can show a broader picture. For example, actions that are good for the maintenance of body or planet may be counter-productive for the life in society or economy – but this modelling shows there is an other option. Confusion can be deep: if re-presented in physical or human terms (or material and mental), what is done here may look contrary to what is done elsewhere or globally, or can appear to not participate at all, yet still have the same orienting towards Boundary, to extremes, despite a fragmentary or local appearance of retreating from physical or human limits, or that pushing it is making improvements – localised in time or space. Going where we want to go, staying away from topologic Boundary also manifests this way, but the Animated Geometry removes the confusion.
This ‘where’ is a very ancient question (at least 3000 years); it relates to a topologic space rather than human or physical, extension or return, real or spiritual — Boundary is in a different world or space or dimension. This modulates differently what needs to be done locally to not ‘tear the fabric’, and not make a topologic hole by breaking boundary. This is one reason why the Foraging Station Experiment, despite its unusual approach, contributes to current actions to stay ‘within the safe zone’.
Boundaries of the physical earth… or topologic Boundary working through wild biology
The Boundary parameter is inherent in biology, its surfaces, tubes and phylogenetic shaping of a living organism. Again, biology is not just a matter of physical and material bodies, environments, blueprints, survival and brain faculties. Therefore learning to understand how this Boundary parameter works in wild biology, learning to use it through health, from human to planet, will at least benefit not just humans, including those who may not feel directly affected yet, but also human diversity in physiological operations, biodiversity, and the integrity of the biosphere and biological life. Even more significant, it is at work not just through extreme achievements and emergencies, but also in daily life.
These explanations and video clips may seem rather enigmatic, but if you have an abstract mind, more computer animations later will nourish your curiosity about this modelling method.
If you are a person of action, you will see how the Foraging Station Experiment explores the practical implications of this modelling, and the effects that will begin to occur as soon as it is set up.
I am passionate about making more accessible the thinking space of this simple geometric-dimensional topology, both as an intellectual tool to de-fragment and de-bias our re-presentations, and as a very practical means of understanding biological health integrity at any scale. It supports our common desire to resolve the human-planet situation, both globally and for all individuals to be Safe & Sound, sans frontières, of space(s) or time, or species.
If this page encourages you to share the website to your network, please mention the Penrose Institute. Have in mind that a specialist of topologic animation software (the kind used in mathematical physics) is required to communicate further this most basic topology. I invite you to partner with aspects of the Station, or simply support this research by gifting specialised expertise or resources.
YouTube channel:
Animated Geometry
Kindly support this research and the Foraging Station Experiment
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Sources
Roger Penrose. Gestures, extract from: https://www.youtube.com/watch?v=9Gl8pwY2kW8&t=32s
Roger Penrose. Mathematical insight and non-computable awareness, extract from https://www.youtube.com/watch?v=6dHQ5Kno4MY and from interview by Lex Fridman https://www.youtube.com/watch?v=hXgqik6HXc0&t=1141s
Penrose Institute: https://www.facebook.com/penroseinstitute/
Bouchon. Thinking in imaging (the raw animated geometry). https://www.youtube.com/watch?v=d5BqwxK-gOI
Cédric Villani. Extract from https://www.youtube.com/watch?v=iHKa8F-RsEM&list=PLVKDOliPZC9BDfC2UJXr_cJjF_Zlrf2s0&index=11&t=896s
Bouchon. Topologic thinking space to understand. https://www.youtube.com/watch?v=5hiQ28M-HIE&t=341s
David Attenborough. Breaking boundaries of earth [preview] https://www.youtube.com/watch?v=2Jq23mSDh9U
David Attenborough. Address to COP26. https://www.youtube.com/watch?v=qjq4VWdZhq8&ab_channel=COP26