# Part One: A Flatland Parable on Dimensionality

- Author: Kojin Glick
- Published: 2024-12-20
- Canonical: https://www.kojinglick.com/blog/motio-est-omne-divisa-in-dimensiones-tres

---

_crossing dimensions is less fantastical that you think_

```
This series is an attempt to capture the journey
to better understand weaponized linear algebra
as a thinker of words first, and numbers second.

These posts are dedicated to Maria,
who is subject to a far rougher version
of my thoughts than what appears here.
```

The most simple way humans experience dimensionality in information is via the fact that motion information exists in three-and-a-half dimensions. The main three components, describing spatial information, are the “x-dimension,” or the space computed, measured, and finally captured by the English word _width_, the “y-dimension,” captured by the English word, _height_, and the “z-dimension,” captured by the English word, _depth_. Like the other three spatial dimensions, we can measure it. But unlike the other three where we measure a value that involves computation, and we only get to observe a discrete slice of the “time-dimension” for our conceptual model of motion. We don’t actually traverse _time_ in a _moment_ like we traversed the x-, y-, or z-dimension to measure the earlier _width_, _height_, and _depth_.

For a moment, let’s consider a different situation. Rather than three computed values (x, y, z) and one observed value (time), we only have two (x, y) of the previous three computed values, and the same observed value (time). The following scenario may be more familiar to the Physics student as Flatland. In Flatland, Flatlanders experience motion in two and a half dimensions. Imagine drawing a Flatlander on a piece of paper, and then drawing a circle around them. It would be impossible for the Flatlander to move outside of that circle according to the laws of motion in Flatland. This is because the Flatlander can not use _depth_ to leap over what appears to us as a line on a surface. Now, feeling a little guilty for abusing higher-dimensional powers and imprisoning a Flatlander, how could we, three and a half dimensional beings, prove our existence to these Flatlanders, even just to apologize?

Even removing the question of the existence of intelligent beings in these dimensions, could we prove to Flatlanders even the existence of higher dimensions? What miracle could we perform, with our powers as higher dimensional beings, to the Flatlanders to convince them of our existence? Let us draw them other objects in their world with the pencil we used to ensnare the first Flatlander we encountered. We noticed that there was nothing on the Flatlander’s head. While the Flatlander is astonished to see a hat, a sideways P, appear in their reality and cover their nose with its brim, the conclusion that this has anything to do with the existence of higher dimensions is still a major leap. The pencil, never truly crossing the plane represented by the piece of paper, still remains an extra-dimensional agent, completely outside of the Flatlander’s conception of reality.

What we need to do is to fully utilize the dimension of which the Flatlander is blissfully unaware, _depth_, since moving the sideways P around and rotating it doesn’t do much to convince the Flatlander of the existence of other dimensions. We need to do something that is impossible according to the laws of motion in Flatland. We reach into Flatland, spinning the hat so that the brim rotates around the cap until the brim faces backwards leaving their nose uncovered by the brim. This miracle truly astonishes the Flatlander. According to other Flatlanders observing the newly hatted Flatlander, the brim started to shrink until the hat was a skull cap, and then the brim slowly started to grow on the other side of the Flatlander※.

Together we built a very small spatial model in our minds by imagining this scenario with the Flatlanders. Very similar mathematics are central to the architecture of most state-of-the-art Large Language Models. To summarize with a mathematical perspective, we projected an object with spatial information in two-dimensional space to an object with spatial information in three-dimensional space, spun it in a way only possible with a third controllable value, and then reduced it back to an object in two-dimensional space. In other words, we unfolded a new value of information inherent to the world that was hidden to Flatlanders, so that we could transform the hat, before folding that value back into the Flatlander’s native two-dimensionality. From the Flatlander’s perspective, the key to the solution of multiple universes lies in the transformations that occurred in a space that is already hard for them to conceive of using the rules of their environment.

In the rest of the series, we will examine the outputs from Large Language Models like Flatlanders. Just like the Flatlander who is aware of some spatial transformation happening in a space they can't see, we know that _something_ happens to words in a higher dimensional semantic space that allows Large Language Models to, well, _model language_ so effectively. Keep in mind this simple conceptual model we created involving Flatlanders, as we will return to this simple world repeatedly to understand language models. Stay tuned!

※ Some Flatlanders begin to worship the Flatlander with the hat, thinking that these powers were unique to the One with the Hat. Other Flatlanders argued that it was an external and exacting power that spoke through the Behatted One. This caused a great deal of disagreement and often resulted in violence. For a while, many Flatlanders adopted one set of beliefs, not out of ideological conviction, but in fear of being prosecuted by roving bands made up of those who had committed to one belief. Still, there were other Flatlanders still who disagreed with the whole premise, because if there truly were great and mysterious higher powers, why didn’t they free the recently Imprisoned One.