Stephan Bridger

δ(x)

[hexagr]

Hofstadter’s INT

It’s a little known fact that before Benoît B. Mandelbrot first pioneered his work on fractal geometry in the 1970s, Douglas Hofstadter stumbled over the phenomenon while working on problems in number theory.

During the 1960s, Hofstadter discovered a family of graphs that exhibited a specific kind of discontinuity with the integers of a particular function. Hofstadter dubbed the main graph INT. The graph of INT(X) contains infinitely many distorted copies of itself. INT(X) is discontinuous at all rational values of X but continuous at all irrational values of X. The function replaces the partial quotient, forming a nearest integer continued fraction, like so [1].

Note, this LaTeX will only render in the browser view of this post and not in Tumblr’s dashboard.

\[ \displaystyle{\displaylines{a_i+~\leftrightarrow~a_i+1-.}} \] \[0+\cfrac{1}{3-\cfrac{1}{4+\cfrac{1}{5+0}}}\leftrightarrow 1-\cfrac{1}{2+\cfrac{1}{5-\cfrac{1}{6-0}}}. \] \begin{align*} \frac{1}{2}=\frac{1}{2+0}&\leftrightarrow1-\frac{1}{3-0}=\frac{2}{3},&\frac{1}{3}=\frac{1}{3+0}&\leftrightarrow1-\frac{1}{4-0}=\frac{3}{4},\\ \end{align*} \begin{align*} \frac{2}{5}=\frac{1}{2+\frac{1}{2+0}}&\leftrightarrow1-\frac{1}{3-\frac{1}{3-0}}=\frac{5}{8},&\frac{3}{5}=1-\frac{1}{2+\frac{1}{2+0}}&\leftrightarrow 0+\frac{1}{3-\frac{1}{3-0}}=\frac{3}{8}. \end{align*}

[hexagr]

Objects decay. Information decays. Let's periodically reiterate obvious concepts as a safety measure.

[mostlymarginalia]

Humans are complex multicellular diploid organisms, differentiated into two sexes often called “sperm” and “eggs.” While human reproduction involves a multicellular diploid phase, the haploid phase consists solely of gametes. The evolution of gametes into two extremes evolved independently across nearly all lineages of multicellular organisms.

[hexagr]

Objects decay. Information decays. Let's periodically reiterate obvious concepts as a safety measure.

[hexagr]

I’ve noticed that a lot of people are suddenly reblogging this post again. For anyone wondering what it means or what I was referring to, this post is about the concept of spaced repetition.

[hexagr]

Thinking about how one time, a friend's lover died and their family needed (and wanted) access to his laptop because it contained important records and family photographs, etc. And so I was tasked with decrypting and gaining access to his machine. That was a strange experience.

[hexagr]

Mirror life presents potential dangers. For example, a chiral-mirror version of cyanobacteria, which only needs achiral nutrients and light for photosynthesis, could take over Earth's ecosystem due to lack of natural enemies, disturbing the bottom of the food chain by producing mirror versions of the required sugars.

[hexagr]

"Scientists rush to stop mirror microbes that could threaten life on earth" - June 13, 2025

[hexagr]

"Your description of [thing or process] is actually just an oversimplification of something much more complex." Hold up there, partner.

Complexity is real. But descriptions aren't merely simplifications. Nothing is mere. Descriptions are data--they're encodings of real (and sometimes causal) information.

In fact, fine-grained analyses sometimes miss this. In the desperate search for perfect granularity, context and emergent properties can get lost.

But sure, feel free to continue chanting, "correlation isn't causation" at all times if it makes you feel superior.

[prokopetz]

Wait a second: cuckoos are nest parasites. Do "cuckoo" and "cuckold" share a common root?

Huh. Apparently the etymology of "cuckold" is literally something along the lines of "cuckoo-lord".

[hexagr]

what does your handle mean?

Oh, "hexagram" is a nod at the study of cognitive science. Six fields birthed cognitive science -- philosophy, linguistics, neuroscience, artificial intelligence, anthropology, and the study of human and nonhuman mental phenomena. As for the shortened 'hexagr' handle -- I think I chose it after seeing the way urbit did usernames back in 2016, which reminded me of six-digit character encoding, and the MIT research "HAKMEM" memo, a compendium of technical trivia. Hence, my unfinished website (and domain hack), hexagr.am.

[todaysdocument]

Letter from Thomas Jefferson to Benjamin Franklin Regarding the Eastern Boundary

Record Group 59: General Records of the Department of StateSeries: Domestic Letters

123

To Dr. Benjamin Franklin and Henry Laurens Esq.

New York March 31st. 1790.

Sir

Encroachments being made on the Eastern limits of the United States by settlers under the british government, pretending that it is the western and not the Eastern River of the Bay of Passamaquoddy, which was designated by the name of St. Croix in the Treaty of peace with that nation. I have to beg the favour of you to communicate any facts which your memory or papers may enable you to recollect, and which may indicate the true River the commissioners on both sides had in their view to establish as he boundary between the nations. It will be of some consequence to be informed by what map they traced the boundary.

I have the honor to be

(signed)Thomas Jefferson

[hexagr]

Fascinating. Also, Benjamin Franklin's autobiography is a great book—an incredible bit of history. And it's free on Project Gutenberg. Highly recommended.

[hexagr]

C.S. Lewis on temperance and virtue, from Mere Christianity.

[hexagr]

While I was typing up some documents this afternoon, I needed a throwaway email address to receive some information.

So I logged into an older account I have, then clicked the drafts folder. I found a draft of an email from 2017.

The body read, "This email will have to wait." I guess I must have been hyperliteral at the time, since it's now been waiting for eight years.

[hexagr]

Today I learned about the British military conquest of Sindh and famed British General, Sir Charles James Napier.

General Napier was asked to acknowledge and respect the sati practice of burning women alive on funeral pyres, to which he responded:

"Be it so. This burning of widows is your custom; prepare the funeral pile. But my nation has also a custom. When men burn women alive we hang them, and confiscate all their property. My carpenters shall therefore erect gibbets on which to hang all concerned when the widow is consumed. Let us all act according to national customs."

[bobdylanyaoi]

you used to be able to put a dvd in your laptop and play it. you used to be able to burn cds.

[hexagr]

Maybe I'm old-fashioned? But I still have five vintage(?) computers from the early 2000s that double as servers. And I still burn CDs as a way to keep safe, offline, redundant backups.

[hexagr]

Every relationship is a parasocial relationship. It's a matter of degree, not kind. Just like a map-territory problem.

Your relationship to anyone is based on assumptions and mental models of the other person. We don’t interact with people directly. We interact with our internal representation of them.

Every relationship is a hallucination shaped by feedback. The more accurate the feedback loop, the healthier the illusion.

[hexagr]

It appears some users have found my blog and are trying to perform an internet maneuver known as the 'ackshually' on this blog post. But my post is both joking and serious. And interpreting it in a charitable, reasonable manner requires being able to think about it for more than five seconds. For the perplexed, I offer a helpful analogy: in atomic physics, we infer the existence and properties of particles through indirect measurements. That is, we don't actually see individual charges. We measure them by how they scatter or interact with other fields through spectroscopy. Initially, we are taught the atomic model, and we learn that electrons "orbit" the nucleus of the atom like a planet orbits a star. But as we learn more about physics, we learn that this model is actually just a useful lie. And it is eventually replaced by quantum mechanical models, where electrons exist in probability clouds — and we can only predict the likelihood of finding them in a certain region. Much like the more accurate-yet-probabilistic models of atoms, the nature of our relationships with others varies in the same probabilistic fashion. Some relationships produce a feeling of great trust and alignment; others not so much. A great deal of social decision-making relies on guesswork from an external point of view by proxy measurements. But the point I wanted to most highlight is that even in the extreme case of sensing great trust and alignment, those measurements are still based on proxy measures—our very fallible senses—and our approximate mental models of the other person. Trust is a spectrum. And yes, some nodes of trust are greater than others. For example, when we make sacrifices for or have patience with each other, we are essentially minting a proof of work—a proof of our convictions, commitment, or alliances with one another. But even commitment, taken to an absurd extreme, can become self-destructive — where you get in deep shit, and justify it post hoc with, ‘I am here because I am committed.’ That, however, is a different dilemma altogether. Even those constructive, meaning-making rituals do not guarantee absolute trust or understanding. If it did, there would be no literature about betrayals—a category of story that exists precisely because it is such a common theme in nature and our collective anxiety.

[hexagr]

Every relationship is a parasocial relationship. It's a matter of degree, not kind. Just like a map-territory problem.

Your relationship to anyone is based on assumptions and mental models of the other person. We don’t interact with people directly. We interact with our internal representation of them.

Every relationship is a hallucination shaped by feedback. The more accurate the feedback loop, the healthier the illusion.

[hexagr]

People often assume that states like Texas, North, or South Carolina are conservative. But they're not at all like people assume.

For example, I live in the Carolinas. And there's a lot of progressive culture. I mean, a dose of healthy liberalism isn't such a bad thing.

Yet sometimes, it's surprising. Like, there's a lot of sex work culture here. That's maybe the most ancient profession of all time, right?

But almost every time I hear from people who are involved in it, it's genuinely horrifying.

It seems like a lot of stories go something like, "I got into sex work. Then I started using drugs. Then one day I got pregnant. Then I kept doing drugs while I was pregnant. Then I learned my child and I both have [HIV/hepatitis/preventable disease] and my child now has [lifelong condition related to drugs or disease]."

And this is just a super unfortunate and heart breaking set of affairs. Unfortunately it's a super common pattern in the city I currently reside in.

Maybe it's not like this everywhere. But I sort of can't help but view it as an indictment on certain overlapping cultures.

[hexagr]

Emotions and rationality both arise from the same place, i.e., the mind. This nearness can make emotions harder to see or check, since we are almost always swimming in them. But it's worth considering taking up the habit of disambiguating and checking our mental activity so as to not let it take us off the rails.

We tend to myopically assume that our emotions are always real or an obvious guide. But upon closer examination, we find that sometimes our emotions are really signifiers without referents—floating, disconnected ideas, perhaps springing from doxa (δόξα) or unexamined beliefs, rather than episteme (ἐπιστήμη) or knowledge gained through reason and evidence.

These empty or floating signifiers—which are ambiguous, open to interpretation, or seem to resist being defined—are analogous to dangling pointers in software. They can pull us around, lock us into situations we’d otherwise avoid, or lead us in circles as if we were without a compass.