I’ll end these posts about the configuration space metaphor where I began: in a big cube. I started the series in the Neapolitan room, a three-dimensional space where we could indicate our feelings about vanilla, chocolate, and strawberry ice cream with a single marker. From there we visited the Baskin-Robbins 31 Flavors space (which is tasty but beyond our ability to visualize).
Then I focused on spaces with only two-dimensions (which are easy to visualize). These are probably the best use of the metaphor; they turn a tug-of-war into a sensible place to stand. They also strongly differentiate “don’t care” from “care about both.”
Now let’s see what we can do with three dimensions…
One nice thing is that three can still be visualized. We’re trained now to “see” 3D when looking at an appropriately drawn 2D image. (It’s a part of some IQ tests.)
As three-dimensional beings we can easily think about three-dimensional space (up-down, left-right, forward-backward), so a 3D configuration space has real meaning to us.
(With a lot of aspirin, we might be able to sort of wrap our heads around four dimensions, but more than that seems too abstract to make sense.)
Before we start, a quick terminology refresh:
In a configuration space (aka parameter space), we use distances along perpendicular directions, such as east-west, north-south, and up-down, to mark feelings associated with that direction. The combination of distinct feelings form a single opinion point within the space.
The perpendicular directions are dimensions; the distance along one of them is coordinate. The value of all the coordinates gives us the opinion point.
A simple example of a 3D configuration space is the Color Cube.
This is a natural choice when dealing with color in terms of red, green, and blue — the three color primaries.
The resulting space contains all the colors we can form from those primaries. The space represents all possible “RGB” values.
The first diagram above shows the three axes, red, green, and blue, and also the three secondary color diagonal axes that combine pairs of primaries.
All six are shown in bright color, but they should really start off black at the center (the white sphere) and become brighter along their length. Only the tips are actually full brightness. (But showing them that way makes them hard to see.)
There is also a triple-diagonal axis that runs from the center to the opposite corner (the closest upper-right one). That axis runs from black to white with shades of gray along its length.
(Note the white sphere in that corner in the second diagram. You can also spot a nearly black one just in front of the large center (white) sphere.
Above is a rendering that shows of the coloration of the space by painting the walls to show the color combination at each point along the walls (and floor).
I also threw in some large spheres along the black-to-white diagonal axis. The spheres start off black, in the lower left far corner and progress through gray to white in the upper right near corner.
Note the distinct regions, of redness, greenness, and blueness, at the ends of their respective axes. Those primary color regions fill three corners of the cube.
There are also regions for yellow, cyan, and magenta, in three other corners, for a total of six. The far lower-left corner is the black (no color) region, and the near upper-right corner is the white (all colors) region.
The Task Cube is a fanciful 3D configuration space I doodled a while back thinking it might make a cute post.
In Task Space, the three dimensions are:
- Useful: How much the task results in something of value
- Difficulty: How hard the task is to perform.
- Fun: How enjoyable the task is to perform.
There are certainly other ways to set this up. My original version used: Investment, ROI, and Fun:
Cubes have eight corners, so this gives us eight combinations, or regions, of kinds of task:
- Not useful, not difficult, not fun. (Dusting areas no one sees.)
- Not useful, not difficult, fun. (Easy games. Watching TV.)
- Not useful, difficult, not fun. (??)
- Not useful, difficult, fun. (Many games.)
- Useful, not difficult, not fun. (Flossing.)
- Useful, not difficult, fun. (Going for a walk. Eating.)
- Useful, difficult, not fun. (Repetitious practice. Work.)
- Useful, difficult, fun. (Building, hunting, fishing.)
Your mileage may vary, and certainly many other examples exist. (It might make an idle party game, thinking of tasks that fall in the different corners.)
In general you can construct a cube for any three orthogonal axes that strike your fancy.
The cube is just an extension of the “four-square” that puts two orthogonal ideas in juxtaposition.
For example, on the one hand, whether the weather will be rainy (or clear) versus whether the weather will be warm (or cool).
We could add a third axis, say whether the weather will be windy (or calm), and that would give us a cube with eight basic regions or possibilities (giving us everything from clear, warm, and calm, to rainy, windy, and cool).
A “four-square” (or an “eight-square” in the case of a cube) reduces each axis to an either/or choice. If the axes have more choices, or are just linear scales where any value is allowed, the square or cube becomes a space.
School Scheduling Cube
Speaking of cubes with discrete choice scales, my ex-wife was a college registrar, and every fall she had the challenging project of scheduling teachers, the classes they taught, and available rooms.
I started thinking of a 3D software design with teachers, rooms, and times, as the axes.
In this case, the dimensions consist of discrete objects rather than numeric values. Teachers and rooms are obviously distinct objects, but the times are also “quantized” into 15- or 30-minute blocks.
As such, this is really a 3D “spreadsheet” structure with cells at each intersection. Each cell represents a given teacher, in a given room, at a given time.
Most cells would be empty, because the combination doesn’t result in a class being offered. A cell is filled with a class record to represent a given teacher teaching that class in a given room at a given time.
The main point here is that, just as thinking in 2D adds a (literal) new dimension that lets us escape the tug-of-war involved in 1D number line thinking, adding a third dimension doubles the possible “opinions” yet again.
Every dimension we add to a space doubles the number of basic regions!
Which means the Baskin-Robbins 31D space does have 2,147,483,648 basic regions. There are that many combinations of “thumbs-up or thumbs-down” on all the flavors.
This number is given that a region comes from dividing each axis into just a “high” and a “low” region along its length.
Conversely, the School Schedule Cube has only three axes, but likely many objects along each axis. If a schedule had 30 teachers, divided the day into 96 15-minute periods, and involved 200 rooms, the resulting cube has 576,000 cells.
Higher degree dimensional spaces, or more choices along each axis, both contribute to increasing the size of the space.
If all we care about is a coordinate, a point in the space, then the size doesn’t matter. But if we “quantize” the space into cells, or even divide it into general regions, then the bigger numbers become unwieldy.
Put simply, it’s hard to tell Baskin-Robbins Opinion #8,300,126 from the 2,000 million other opinions.
Even so, I find it, not only fun, but useful to think about higher-dimensional spaces.
For one thing, these are exactly what deep-learning neural nets are forming when they learn. When such a network determines a new input is a “cat” it’s because the input falls into the “cat” region of its configuration space. The distance from the center of that space determines the confidence of the identification.
It’s also just fun (if you’re the geeky type) to think about Beer Space!
Beer is a complex organic, with many paths to its making, and many variations in form, so it offers rich ground for multi-dimensional analysis! The same could be said for many other delights: chocolate, wine, even cigars.
Beer Space tries to capture some of the key characteristics of beer: How much malt, how much hops, and how much alcohol, being three major ones.
But we almost have to divide hops into early- and late-added. In the first case we’re talking about bittering; in the latter about fragrant hop oils. It’s possible we need dual axes for each style of hops. Mosaic hops, for example, is very distinctive (and yummy).
Malt is generally malt as far as how much there is. But its color makes a big impact on how the beer looks, so there should probably be a color axis. (And, in fact, one beer stat is SRF.)
And so on. This is a work in progress. Usually the progress of drinking beer. A great deal of further research is required.
Stay spaced, my friends!