Consider the operations of a sequence of simple affine transformations being carried out on an ordinary equilateral triangle. It should be obvious, in that case, that the angle bisectors and the midpoint bisectors of each angle and side respectively, all intersect at the same point, a point which is commonly referred to as the centroid.  This property is invariant under simple shifting and rotation operations.  However, if the triangle is stretched along any axis, or pair of axes, to transform it into another triangle, such as one that is isosceles, then even though this can be accomplished through ordinary linear operations, the nature of the angle bisectors is such that the apparent symmetry is broken, even though they remain convergent amongst themselves.

Now as far as the midpoint bisectors are concerned, they still result in the creation of a set of points that also can be used to subdivide any arbitrary triangle into four similar triangles, each one-fourth of the size of the original – allowing for recursive tessellation, of course.  Yet if we introduce the angle bisectors into the process, and include them in our tessellation plan, this might provide a way to so also algorithmically generate a seemingly pseudo-random distribution of points, as it were, which might be useful for doing such things as defining the location of trees, blades of grass, hair follicles, or for defining other geometric forms.

The fact that Alan Turing himself was fascinated by the idea that oscillators regulated by chemical gradients might play a role in the determination of the distribution of stripes, or spots on such animals should not go un-noticed, either.  Even if this is all regulated by DNA of course, a cell still has to ask as if it were, not just “What am I”, or “Who am I”, but “Where am I?

Here is an interesting question:  What do you think about the idea of “intersectional mosaicism?”  Not so much as a social construct, but rather, from the point of view of epistemology. Yet this is something that I should clarify since I am more likely to be influenced by such modern as Chomsky, Penrose, or even Escher.  Yet another name comes to mind, and it is not Nietzsche. Sarte or Hagel, by the way.   I will get to that later.

Let’s get back to the idea of “intersectional mosaicism” therefore, based upon an epistemological framework rooted in the approach to physics known as “deep reductionism”.  Now some will theorize that everything is either male or female, or else is matter vs. energy, or it is math vs. geometry, or something like that.  I am not saying that those things aren’t relevant, rather I should say that the simple cherry-picking of taxonomies just because one can use PowerPoint or draw Venn diagrams is not the answer either.

Yet, if we embrace “intersectional mosaicism” as a social construct, a question might arise whether there is some formal operational scheme, like Maslow’s hierarchy of needs, or Ericson’s stages of maturation that must somehow describe everything.  Most of the time, such outlines are just as often as not, rooted in otherwise ad-hoc hypotheses at best, or else they are mostly pseudo-scientific constructions.

Whether to say, “to be or not to be”, or whether it is better to discuss the “is-ness of is”, or else “the being-ness of be” and what that means, let’s say, from the point of view of pure hedonism, is, another matter altogether.  Or is it?  In physics, spontaneous symmetry breaking, either is or is not, whatever Wikipedia says that is or is not, and that is all that I am going to say about that, for now. 

Yet what if from the point of view of pedagogy, something else altogether has been overlooked?  Something, that from the point of view of not only the mathematical framework of physics but so also from the point of view of the philosophical notion of the “construction of reality”, which might therefore have profound implications.

I started with some observations about triangles, which hopefully imply whatever that might imply.  Maybe this is a good time to mention Minsky, and Winograd, among others, as promised earlier, and then move on to the next topic – a perennial favorite:  wave matching in quantum systems, as well as applications, like turning audio into sheet music, and doing spectrum analysis using polyphase filter trees, when can then encapsulate multiple FFTs, each running with different bandwidths, sample rates, frequency ranges, sampling intervals, and so on.  With everything being re-split, re-overlapped, and of course re-combined, as if by magic, as it were, so that one model will dominate, insofar as providing a higher temporal resolution concerning note onsets, than is normally possible, that is according to the Rayleigh limit as it is normally perceived.  Then other methods of analysis provide enhanced frequency resolution, which is especially important at low frequencies.

An “intersectional mosaic” therefore, is not merely a “collage” as it were – as in some random art project, but something that has the potential to simultaneously blur traditional boundaries in regions of overlap, whether in a well-defined way or not.  Yet this should hint, therefore, as to the notion that so-called principles of “deep reductionism” might apply, so also in some social constructs even – if I dare to say it – and I just did.  For just as it should be obvious that the roots of logic are largely shared with geometry, and therefore algebraic topology.

Some will say that such ideas might invite some into the framework such things as theories about global economics, political upheavals, or those paths elsewhere that lead to the various wars, or opportunities for peace.  But that gets way ahead of things. In the meantime, LLMs I think are deficient in their implementation details, insofar as the quality of the framework is concerned, that is to say – to the foundations of their world model.  So maybe it is time to return to Euclid.  Or at least it is for me at least, at least that is for a while.