Andrius:
- I am learning Active Inference from the textbook along with Cohort 8.
- I am investigating triangle centers.
- What are the probability distributions for three positive real numbers that add up to 1?
Active Inference Project: Language of Verbalization: Triangle Geometry
I want to learn by investigating a project of my own. I am interested to model a language of verbalization (how meaning arises), a language for the consciously questioning mind, the second of the three minds. But this may also mean modeling a language of argumentation (how issues come to matter) for the unconsciously answering mind, the first of the three minds.
The free energy principle can then be understood as the means by which the third mind relates the first two minds. This may be what Jere calls the Goldilocks maximum entropy principle, seeking a middle way.
Data to Study
Andrius: Key to a successful investigation is a well chosen data source to study.
I am interested to study how mathematics unfolds, and in particular, constructions in triangle geometry. I think this can also relate to:
- Daniel's interest in modeling motion.
- Jere's pictorial thinking.
- My study of the 24 ways of figuring things out in mathematics.
- My insights from teaching and tutoring mathematics.
- Projective geometry over {$\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O},\mathbb{F}_2,\mathbb{F}_1$} which all relates to Bott periodicity and modeling Wondrous Wisdom.
Triangle geometry can be related to how meaning arises in
- synergetics
- biochemistry
- irony
- humor
- prayer
Triangles can also naturally encode triples of concepts.
We can look at how a pictorial language of triangles can be used to encode meaningful experiences in life.
Constructions will triangles will also relate to constructions with circles. It is interesting how mathematics can unfold in various conceptual directions.
What I'd like to think about together is choosing data to study to make
sense of language. It would be awesome to study the same data. In
particular, I am considering studying how meaning arises in triangle
geometry. There is a lot that one can do with a triangle, but then again,
there is only so much you can do with a 
triangle.
Or so it seems. But there are about 200 Wikipedia articles.
Furthermore, there is an Encyclopedia of Triangle Centers with 65,000 triangle centers (or notable points)
starting with incenter, centroid, circumcenter, orthocenter, nine-point
center, and so on...
Thus is it possible to study how this conceptual language evolves from the
most basic concepts.
As a precalculus teacher in San Diego, and then a tutor in Chicago, I taught my students fundamental principles such as "Every right triangle is half of a rectangle" or "Every triangle is the sum of two right triangles." These can be used to prove, for example, that the area of a triangle is 1/2 base times height. They also formulate pictorial thinking (such as drawing an altitude) which is very much in the spirit of Jere's Relational Symmetry Paradigm.
Cognitively, I think there are four basic ways to think of triangles - in terms of three paths, three intersecting lines, three angles or sweeping out an oriented area. Pairs of these ways make for six Mobius transformations which arise in the Wondrous Wisdom theory of emotion, see A Geometry of Moods.
Triangles are also key in Buckminster Fuller's Synergetics.
Note that Douglas Hofstadter, author of "Goedel, Escher, Bach", studied triangle geometry and related visual cognition. At the University of Michigan and Indiana University, Hofstadter and Melanie Mitchell coauthored a computational model of "high-level perception"—Copycat—and several other models of analogy-making and cognition, including the Tabletop project, co-developed with Robert M. French.[22] The Letter Spirit project, implemented by Gary McGraw and John Rehling, aims to model artistic creativity by designing stylistically uniform "gridfonts" (typefaces limited to a grid). Other more recent models include Phaeaco (implemented by Harry Foundalis) and SeqSee (Abhijit Mahabal), which model high-level perception and analogy-making in the microdomains of Bongard problems and number sequences, respectively, as well as George (Francisco Lara-Dammer), which models the processes of perception and discovery in triangle geometry. Wikipedia: Douglas Hofstadter
Constructions in Triangle Geometry
’’Active Inference is a normative framework to characterize Bayes-optimal2 behavior and cognition in living organisms. Its normative character is evinced in the idea that all facets of behavior and cognition in living organ- isms follow a unique imperative: minimizing the surprise of their sensory obser- vations. Surprise has to be interpreted in a technical sense: it measures how much an agent’s current sensory observations differ from its preferred sen- sory observations—that is, those that preserve its integrity (e.g., for a fish, being in the water). Importantly, minimizing surprise is not something that can be done by passively observing the environment: rather, agents must adaptively control their action-perception loops to solicit desired sensory observations. This is the active bit of Active Inference.’’
Generative model
The prior {$P(H)$} and the likelihood-what (of evidence-whether {$E$} given hypothesis-why {$H$}) {$P(E|H)$} together are the generative model. Whereas the posterior probability {$P(H|E)$} expresses how.
The generative model supposed could be that the triangle has sides whose lengths can be described in terms of integers. The agent acts to discover those integers, which may or may not exist, as the lengths may be irrational.