How do human beings produce knowledge? When we describe rational thought processes, we tend to think of them as essentially deterministic, deliberate, and algorithmic. After some self-examination, however, I've come to think that my process is closer to babbling many random strings and later filtering by a heuristic.
-alkjash

Introduction

I want to describe my first and so far only experience inventing a math problem. It's probably not an original or important problem. But I think there is value in describing the origins of ideas, especially when those ideas are actionable.

Here is the problem:

Draw a square bounding box of side length N. One at a time, drop a square of side length L, oriented identically to the bounding box, at a random location such that it is entirely enclosed within the bounding box, until two of the dropped squares overlap. What is the relationship between the average total area of the dropped squares (excluding the last one dropped), and L^2/N^2?

I did some analysis on this problem. As L increases, each square fills more area (increasing L^2/N^2), but the chance of an overlap increases as well (decreasing L^2/N^2). Until L^2/N^/2 = 0.25, the relationship closely follows a power law. After that point, the relationship is linear, as it becomes impossible to fit more than one square in the bounding box, meaning that the only remaining factor is the increasing size of the single dropped square.

It would be interesting to work out why this relationship holds, but my primary goal here is to describe how I formed this problem.

Circumstances

I am studying math, CS, and chemistry at the undergraduate level (as a post-bacc student). I was not passionate about mathematics until a couple years ago, and do not consider myself an unusually talented math student compared to my classmates.

I came up with this problem while smoking weed and listening to my girlfriend play the violin on the porch. I often notice that weed can make me ask more expansive or deep questions, follow a train of thought with more focus, and allow me to explore a vague idea that I might otherwise dismiss.

It also dramatically enhances my ability to visualize. I have a poor sensory imagination. But when I smoke weed, I can see and manipulate mental pictures and audio with much greater vividness and sustained attention.

Developing the Idea

Babble

In this case, I started imagining kaleidoscopic, geometric nonsense. Triangles fanning out from some nebulous origin point. Lines waving in symmetry.

Then I got the idea, I don't know from where, to imagine dropping a cube from the sky down onto the mental "ground." I imagined stacking them, or having them fall in adjacent columns. There was a pleasure in being able to visualize this. At some point, I had the idea that this could suggest some sort of math problem.

Then I imagined a lot more cubes falling, stacking up, at all different orientations, and realized they would have gaps in between them. And I wondered about how much space they would all fill up.

To make that make sense, I realized there would need to be some sort of container, like a crate or barrel. From that point, it became more obvious that the problem would be about the relationship of the size of the cubes to the size of the container.

Prune

But this problem seemed complex. So I simplified it to 2D shapes in a 2D bounding box. And to make it as simple as possible, I made the orientations all identical, and made the shapes and bounding box squares. This last step took place when I explained the problem to my girlfriend.

At this point, the problem had crystallized into a form I could solve. And of course, it could be extended by solving for other shapes, allowing the orientation to be random, and many other constraint changes.

What Was Important About The Babble?

One aspect of the babbling that I think was important is that it started with an atomic rule: dropping cubes from the sky. Such a rule is simple, but can have complex implications when additional constraints are added. Perhaps this is akin to introducing a piece of evidence to an argument, a new rule to game, or a character's decision in a story.

Another important aspect is the pleasure I took in free exploration of that first atomic rule. It was fun to visualize, and it felt pleasurable to just continue exploring the visualization, without having any particular goal in mind, or pressure to create any particular form of output. Rather than forcing myself toward the goal of defining a math problem, I was just allowing further ideas to bubble up, exploring the problem environment: this imaginary world of falling cubes. The weed helped. I don't think sober me would have found the dropped-cubes idea interesting to think about.

A third is the ability to mentally manipulate the materials in an easy and focused way. If I'd had to use some clunky computer software to play around with cubes, I don't think it would have been as easy to do, and it would have been more distracting. If I'd tried to do it in my head, I wouldn't have been able to visualize it or focus.

I think these three factors are fundamental to what alkjash calls "babble."

Can Deliberate Practice Increase The Manipulability And Pleasure Of Babble?

From Creativity in Science through Visualization (1965):

The hypothesis is advanced that the creative persons appear to have stumbled onto and then developed to a high degree of perfection the ability to visualize—almost hallucinate—in the area in which they are creative. And their visualizations seem to be of a sort that lend themselves to easy manipulation in the thinking process. This is illustrated by reports from many of the great inventors of the past and it is easy to demonstrate that individuals differ enormously in the kind and degree of their ability to think in such manipulatable visualizations.

What's happened since then?

The top-cited scholarly document referencing this paper is Visual Thinking by Rudolph Arnheim, published in 1969. He also published A Plea For Visual Thinking in 1980.

Perception and thinking are treated by textbooks of psychology in separate chapters. The senses are said to gather information about the outer world; thinking is said to process that information. Thinking emerges from this approach as the "higher," more respectable function, to which consequently education assigns most of the school hours and most of the credit. The exercise of the senses is a mere recreation, relegated to spare time. It is left to the playful practice of the arts and music and is readily dispensed with when a tight budget calls for economy.

How does visualization work? There are two neurologically distinct visual pathways: the object pathway, which processes color and shape; and the spatial pathway, which processes relationships and transformations in space. There seems to be a trade-off between them, as people strong in one tend to be weak in the other. but it's unclear to what degree this is due to nature or nurture.

Some approaches to improving visualization ability include:

  • Mental practice
  • Imagery rehearsal seems to involve simply setting aside time to practice visualizing, and can be combined with deep muscle relaxation.
  • Layered stimulus response training "helps individuals more easily generate and control their imagery experience by adding different elements of the image in progressive layers."

Although these techniques have their nuances, they all seem to boil down to setting aside some time to try and picture something in your mind. Beyond that, you have to figure it out yourself. But the advice here seems to be that it's relatively easy to develop this skill, if you try.

Five Minute Visualizing Meditation

So I set my timer for five minutes (sober), sat on the couch, and closed my eyes.

At first, I saw only blankness, with intrusive verbal thoughts crossing my mind.

I realized that there were two approaches I could take. One was to set a goal, like "visualize a white triangle." The other was to just allow my brain to start visualizing and let this develop freely.

I chose the latter, which immediately relieved some tension.

The images that arose in my mind where chaotic and not entirely visual. There was a sense of spatial distance that felt kinetic rather than visual, like a sensation of stretching. But visual images did arise. And along with them, a sense of being able to direct my attention, and to manipulate the visualization.

The way my attention and mental manipulations affected the visualization was often unpredictable. I saw a sketchy picture of a ceiling fan. Then I pulled the fan off the base, and replaced it with a white cube which continued to rotate. Then white cubes started to fall on the ground, which bent to form a sphere. The white cubes surrounded the sphere at regular intervals, then blasted off into space.

I saw a waving sheet-like surface. Although it was shadowy, I could consciously choose to change its color to blue, yellow, black, red. I could sort of "magnify" the edge of it, though I didn't have control over what that magnified image would look like, exactly, or how the waving motion would change in response to magnification.

I noticed the visualizations becoming more vivid, complex, and manipulable as time went on.

This seems promising.

A Ten Minute Visualization Meditation

After typing up my observations, I repeated the experiment for ten minutes.

The visualizations returned more quickly, and developed much greater vividness and complexity. There was a sense of being able to explore and control the visualization more reliably. Other senses were woven in.

I noticed some odd challenges. I pictured a house, and while I could visualize the interior, I couldn't make myself feel as though I was entirely inside the house. Instead, I felt like I was looking at the house from the outside, but had a drone-like secondary perspective that could go into the house. I tried to give this secondary perspective my full visual attention and my sense of being "the real me," but I wasn't able to do this completely or for very long.

My visualizations tend to move around, change rapidly, and appear and disappear quickly. I have to chase a picture down and make an effort to hold it in my mind, otherwise it will disappear over the "horizon."

I do seem to have to close my eyes to engage with my imagination. I wonder if I can improve my ability to visualize with my eyes open.

Ten Minute Visualization With Eyes Open

This is more difficult than the eyes-closed versions, but I am eventually able to sustain visualization. I notice a tendency to let my eyes slip closed.

There is a sense of some force, almost like a mental wind, that "blows the picture around." I visualize a leaf with a lady bug crawling on it, and this wind blows the leaf all around my mental space. It takes a strong effort to get it to stay in the center of my mental vision and to stay stable enough for the lady bug to crawl across it.

Many of the visualizations are nonsense: the moon gets caught in an accordion. Others are of eating or drinking, wandering around a city, abstract geometries.

I also get weird sequences of aimless curiosity. I'm standing on a rooftop. I look at the shingles. Then I imagine them popping off the roof. Can I see the individual nails? How many shingles are on the roof? Wait, who cares?!

What's missing here is both the atomic rule that gave the visualization some useful constraints, and the genuine pleasure in manipulating the rule.

Perhaps now that the visualizations are coming more easily, I can enforce some of these rules.

Final Ten Minute Visualization

Soon after closing my eyes, I picture an endless space containing regularly-spaced cubes. Quickly, I recognize this as a visualization of countable infinity. I picture this plane of countable cubes as resting on a much larger, uncountably infinite, column-like "foundation" of the numbers in between integers.

At first, I picture the number closest to the integer such as (6 and 6.000[...]1) as being on opposite ends of this "column," then realize this doesn't make sense. Instead, the uncountably infinite numbers would stretch from 6 to 7, with 6.000[...]1 as close to 6 as possible. And I recognize my error as a consequence of the way we write numbers, in which you need many more digits to represent 6.000[...]1 than you need in order to represent 6.1. Even though 6.1 is further away from 6 than the number 6.000[...]1 is from 6, 6.1 visually looks more similar to 6 than 6.000[...]1.

I didn't have a particular aim in starting this visualization, except that I wanted it to be mathematical in nature. It seems as though there is a genre of math-y visualizations that can be productive to explore.

Reflections

It occurs to me that my irritation at my inability to control my visualizations, and having no concept that it might be productive to allow myself to visualize nonetheless, has led in myself to decades of neglecting this faculty.

I am wondering what will come of continued visualization practice. I am also curious about visual memory. I notice that memories would sometimes pop up as I visualized, which is otherwise a fairly rare occurrence for me. People use memory palaces to remember better. It's often possible to fit much more information into a picture than into words or sounds, so perhaps if I can improve my visualization, a better visual memory will come along with it.

I am also curious about how my habits could be altered to make more room for visualization. Typing this blog post pulls attention away from my visual imagination. When I sit down to write or think, are my thoughts tending in certain directions because I'm not making an effort to engage my visual imagination?

I think I'll need more time to explore visualization in an open-ended way before I can say anything more definite.

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