Shaking the Rattle

Thanks to Ryan of Polydactica, whom I first met through Andrius of Anaheim and Lithuania (I’m near Anaheim right now), I became more aware of two trending topics: Category Theory and Liminal Spaces. 

Category Theory (CT) is a shoptalk embedding the idea of a “topos” which my spellchecker doesn’t like. Lets give that word some context, with a quote from the Stanford Encyclopedia of Philosophy (SEP):

All this work culminated in another notion, thanks to Grothendieck and his school: that of a topos. Even though toposes appeared in the 1960s, in the context of algebraic geometry, again from the mind of Grothendieck, it was certainly Lawvere and Tierney’s (1972) elementary axiomatization of a topos which gave impetus to its attaining foundational status. Very roughly, an elementary topos is a category possessing a logical structure sufficiently rich to develop most of “ordinary mathematics”, that is, most of what is taught to mathematics undergraduates. As such, an elementary topos can be thought of as a categorical theory of sets. But it is also a generalized topological space, thus providing a direct connection between logic and geometry. (For more on the history of categorical logic, see Marquis & Reyes 2012, Bell 2005.) 

CT has a “Bourbakian” flavor, which to me means it revels in abstraction, but not without some diagrams involving ovals and arrows (should we credit American pragmatist C.S. Peirce as an influence here?).

SEP again:

Categories, functors, natural transformations, limits and colimits appeared almost out of nowhere in a paper by Eilenberg & Mac Lane (1945) entitled “General Theory of Natural Equivalences.” We say “almost,” because their earlier paper (1942) contains specific functors and natural transformations at work, limited to groups. A desire to clarify and abstract their 1942 results led Eilenberg & Mac Lane to devise category theory. The central notion at the time, as their title indicates, was that of natural transformation. In order to give a general definition of the latter, they defined functor, borrowing the term from Carnap, and in order to define functor, they borrowed the word ‘category’ from the philosophy of Aristotle, Kant, and C. S. Peirce, but redefining it mathematically.

I thought Perplexity did a good job answering my question about whether the idea of a functor was analogous to an analogy, making “natural transformation” mean “analogous analogies”. 

I liked that as a specific example it featured the hydraulics-to-electronics analogy, as being of the functor type. 

Functors are akin to translations between natural languages, giving rise to different ways of expressing the “same thing”. 

Natural language processing (NLP) makes analogous analogies a central focus, seeking to encode the generic essence of relationships in a generalized semantics. I sound like a chatbot now. I’m shaking my rattle. 

A natural transformation takes us from one analogy to another (a meta-functor).

A phrase which stuck with me, an analogy of sorts: “toposes could prove to be for the 21st century what Lie groups were to the 20th century.” 

Both provide overarching schemas, conceptual frameworks, inside of which more specific systems (of concepts) do their thing. Conventionally such overarching schemas are called “foundations” but if we zoom out to just see “in-falling to form a spherical network” then our “foundation” is more like a “spaceship” (freely floating).

When used to characterize a specific mathematical domain, category theory reveals the frame upon which that area is built, the overall structure presiding to its stability, strength and coherence. The structure of this specific area, in a sense, might not need to rest on anything, that is, on some solid soil, for it might very well be just one part of a larger network that is without any Archimedean point, as if floating in space. To use a well-known metaphor: from a categorical point of view, Neurath’s ship has become a spaceship.

I’ll be pestering Perplexity about Neurath’s ship.

Friendly Skies

I picked up a book of Quaker writings in Quilcene, from a little outdoor booth where you’re trusted to simply leave the asking price in a jar. 

I’m impressed by the level of community living skills expressed among so-called anarchists, a word with too many connotations to be that useful, and oxymoronic to boot, as structured living requires discipline, self regulation, organization. 

Food Not Bombs, my neighborhood chapter, was likewise a pretty tight ship. Anarchists see what needs doing and do it. How Bucky Fuller of them. 

That African Heritage restaurant in Everett, Washington lived up to our expectations. I was with some dance school students local to the area. I enjoy touring in the company of indigenous people vs trying to figure it all out on my own.

Tuning In a Namespace

Didactic Manga

from Summer Term, 2026

Asylum District Dining

Here’s one of those classic food pix. Cold sesame noodles. I’d been craving em ever since Fujin went away years ago. Hard to find. Could I make em myself? No doubt. I’ve held various prep kits at HongPhat, wondering if this was my least action path (a physics allusion).

Anyway (a) Sapphire Hotel had em on the menu and (b) I’d never been to Sapphire Hotel, which is in my neighborhood, where I’ve lived for over 30 years (the longest I’ve lived in any neighborhood). Sapphire Hotel is a restaurant, not a hotel.

I was reminded of going to The Fork long ago with Kiyoshi Kuromiya. Kiyoshi was a true foodie who wrote about Philadelphia eateries and in so doing helped elevate the tastes and cuisine of that entire city (per my narrative account anyway). When we showed up at The Fork that time, the staff was well aware this was Kiyoshi Kuromiya, the one and only. Great food by the way.

I’ll link to my Dinner with Kiyoshi in the comments, a play off the movie My Dinner with Andre. He was indeed a super interesting dude, a deep intellectual not only a gourmand. He’d started studying architecture with a focus on Louis Khan but then found Buckminster Fuller and felt he’d finally hit the jackpot. They became co-conspirators. Bucky called him “adjuvant” which is closer to “catalyst” than “lieutenant”.

In those days, I represented Friends (Quakers) to American Friends Service Committee (AFSC). To earn its reputation as working on behalf of Friends, AFSC would host these annual corporation meetups where they’d impress us with all their good works, and we’d spread the word. North Pacific Yearly Meeting insisted on paying my way, not accepting AFSC money for our visits, cuz we were in a sense auditing them and so should not be like unscrupulous members of congress who gladly allow lobbyists to wine and dine them (and so so much more) in exchange for protecting their own interests.

So I’d be in Philly anyway, for the corporation meeting, and I’d pad the trip with visits to old and new haunts, including meetups with CJ and Kiyoshi.

Closing Time

Per the yin-yang way of thinking, the ending of a chapter creates the space for a next chapter to begin. Novels jump around a lot more than real life, but then real life includes reading novels. 

In this ending, or closing, in my role as a Wanderer (capital W, cuz it's a practice), I found myself up and out the door early, taking bus 75 to the Hollywood District (Portland's Hollywood, which is likewise into movies), then walking the length of East Broadway to Lloyd Center, a Portland landmark, and soon (within hours) to open to crowds for a last day as an indoor shopping mall.

However I didn’t stick around for those crowds. People I know were there for the festivities, which included lining up for candy treats and ice skating (that rink will be missed, but few were using it, and the mall had become a ghost of its former self). 

They sent me pictures, which I added to my album. But I was back in my “outstairs office” long before then, content with my almost-no-people vacant streets look. That’s a slice too, early morning hours on a Saturday when few people have any reason to be out and about.

I’m accepting of the fact that systems have more than just boundaries in the sense of inside and outside. They have boundaries, or call them lifespans, of before and after. However based on my latest observations of all the new construction going on, this will be more of a transformation than a complete replacement. 

The plan is to turn the place inside-out so it becomes less of an enclosed space and more a shared outdoor space where people also want to live. I’d say look at the Pearl District and NW Portland for a preview. That being said, I’m have no privileged access to these plans, or maybe they’re public and I simply need to look them up at the city website.

I left the bicycle in the repair shop today in preparation for a next loop.

Quadrays History

Johnny Tri Goode sent me this set of quadray coordinates for an icosahedron, which I ran through my framework to compare with my default icosahedron (cyan) from the concentric hierarchy (a matryoshka of polyhedrons).

Johnny is researching the history of Quadrays. As I was saying to fellow faculty in the faculty lounge (virtual):  

The way I see it is: Quadrays are a shelf of books in a math library or bookstore, but only a few of them bother with Synergetics. That’s a metaphor as “shelf of books” implies wood pulp, but we’re actually doing all this in the cloud without much paper.

You might be wondering "what are quadrays?". Spellcheckers complain about it (squiggly red underline) much as they complain about Synergetics. English is hostile to our curriculum in a lotta ways (we’re mostly immune to toxic aspects of English, having escaped the Anglosphere). 

In my version of the narrative, Quadrays are an escape from the linear algebra “hegemon” (spellchecker complains again), but not in such a way as to undercut LA in any way. Machine learning is all about LA and I’m one of who teach (some might say preach) ML (my company bosses were near Istanbul — not sure where exactly given I was in a cloud-based company using Google glue).

Some of us (picture a subculture) like exploring in other namespaces with an apparatus that has a lotta family resemblance to the principal Hilbert Space exploration device: the extended Euclidean version of XYZ. Quadrays operate well within the IVM, which ML (NLP) will call a “substrate” for same, per this output from Charlie (Charlie = generic chatbot).

### 3.4 Fuller's isotropic vector matrix (IVM) and the coining of "quadray" **R. Buckminster Fuller (with E. J. Applewhite), *Synergetics: Explorations in the Geometry of Thinking*, Macmillan, Vol. 1, 1975; Vol. 2, 1979.** Full text: [monoskop.org](https://monoskop.org/images/4/46/Fuller_R_Buckminster_Synergetics_1997.pdf); scan at [archive.org](https://archive.org/details/synergeticsexplo0000full_t1i6). Sections 420.01–420.02 describe the isotropic vector matrix (IVM) as "four-dimensional and 60-degree coordinated" — Fuller's own coordinate framework, based on the FCC/CCP (cubic close-packed) lattice, i.e. geometrically the same lattice underlying quadrays and (via the \(A_3^*\) permutohedral identification) GBT in three dimensions. **Fuller's IVM predates the specific "quadray" coordinate formalism by roughly six years** and is the acknowledged geometric substrate on which quadrays were later built, but Fuller's own texts describe vector *directions* and a lattice, not an explicit \((a,b,c,d)\) 4-tuple coordinate algebra with defined arithmetic.

Social Engineering for Dummies

Warm Springs Circuit

Drone Zone?