Last time we considered a cube-shaped room where we could indicate our opinion about Neapolitan ice cream with a single marker. That worked well because we were dealing with three flavors and the room has three dimensions: east-west, north-south, up-down.
Later I’ll explore other examples of a 3D “room” but while we’re talking ice cream, I want to give you an idea where this goes, I want to jump ahead for a moment and consider good old Baskin-Robbins, who famously featured “31 flavors!”
So now the question is, can we set a marker for all 31 flavors?
Yes! We absolutely can.
But it takes a “room” with 31 dimensions.
There’s no way to visualize such a thing, but we can still think about it and use the idea as a tool.
In the Neapolitan room, we linked vanilla to east-west, chocolate to north-south, and strawberry to up-down. Then we ran out of dimensions. (Fortunately we also ran out of flavors.)
Since we can’t visualize this, we’ll forget about east or north or up. We can just label the dimensions with their flavor (or even just their number from 1 to 31).
The Neapolitan room has a Vanilla dimension, a Chocolate dimension, and a Strawberry dimension.
Our Baskin-Robbins “room” has a Vanilla dimension, a Chocolate dimension, and so on all the way to the Peppermint Fudge Ribbon dimension — 31 in all.
It doesn’t make much sense to call it a “room,” so from now on I’ll call it a “space.”
We have defined a 31-dimension Baskin-Robbins space.
Defining a point in that space takes 31 numbers, each indicating an opinion on a flavor.
In the Neapolitan space, we wrote our marker as:In the Baskin-Robbins space we write our marker like:
[Banana Nut Fudge: 2, Black Walnut: 7, Burgundy Cherry: 8, Butterscotch Ribbon: 10, Cherry Macaroon: 1, Chocolate: 9, Chocolate Almond: 3, Chocolate Chip: 8, Chocolate Fudge: 9, Chocolate Mint: 7, Chocolate Ribbon: 10, Coffee: 10, Coffee Candy: 10, Date Nut: 9, Egg Nog: 1, French Vanilla: 9, Green Mint Stick: 6, Lemon Crisp: 1, Lemon Custard: 2, Lemon Sherbet: 8, Maple Nut: 4, Orange Sherbet: 7, Peach: 2, Peppermint Fudge Ribbon: 4, Peppermint Stick: 8, Pineapple Sherbet: 2, Raspberry Sherbet: 7, Rocky Road: 3, Strawberry: 3, Vanilla: 5, Vanilla Burnt Almond: 2]
Note that, if we specify the order and use that order all the time, we can leave out the dimension names and just use the numbers, for instance [5, 8, 3], in the Neapolitan case.
(We really are just talking about coordinates in a space.)
It might seem that we’re not really doing much here, that our list of coordinates is just the same as a bunch of separate opinions.
There is some truth to that. If we want distinct opinions about, say, vanilla, chocolate, and strawberry, we do need three numbers. There’s no way around it.
But combining them as dimensions allows us to easily recognize similar opinions, because their markers are spatially close.
Creating such a space also creates regions.
Recall how the upper north-east corner of the Neapolitan space was the ‘loves all ice cream flavors’ region. (The whole north-east corner is a region of vanilla and chocolate lovers.)
There are broad regions, too. The lower part of the Neapolitan space was for those who weren’t into strawberry, whereas the upper part contains those who love the stuff.
The north is for chocolate lovers and east is for vanilla lovers.
In our 31-dimension B-R space, we can’t talk about “east” or other directions, but regions still exist.
Ice cream haters, for example, would still cluster around the center of the space. Ice cream lovers would still cluster in the far “corner” (although it’s a corner with 31 edges).
One way we might depict a single opinion in 31D B-R space will look familiar to those who watched Westworld on HBO.
On the show, interactive displays such as the above depicted (and allowed administrators to change) the personality properties of the “hosts” (intelligent robots).
On the show, each radial line (from center to edge) is a personality property. Here we can make each radial line an opinion on a given flavor of the 31.
Every distinct opinion on the 31 flavors produces a different figure:
Opinions that are similar produce similar figures, ones that differ produce different ones.
We can compare two to see how they overlap:
We can see that, in some cases, there is agreement about the flavor, but in other cases, one person likes it while the other really doesn’t.
Above I mentioned the center region of those who don’t like any ice cream and the “far end” region of those who generally love any ice cream:
Which is a pretty clear difference!
Unfortunately, these diagrams “flatten” the 31-dimensional space into two-dimensional diagrams, and that loses the sense of a space (or “room”).
It’s a bit harder to identify regions, in part because it’s hard to compare lots of shapes for common parts, but also because we’re no longer visualizing an entire space.
The thing to keep in mind is that each shape represents a specific location in the 31D Baskin-Robbins space.
Think of them as a visualization of a 31-dimensional coordinate.
Note that we could use a regular bar chart or even just a plotted line.
The effect is the same, but the polar chart makes shapes that are a bit easier to compare and work with than a regular horizontal chart.
Or maybe it’s just that I think they look cooler.
In the end, it really is just a Euclidean coordinate with 31 dimensions.
And frankly, when dimensions get that large it reduces the value of this metaphor because of the visualization problem. I just wanted to give you a feel for a more extreme use.
Ultimately, its greatest value may be when there are only two numbers. I’ll get to that in the future. (FWIW, one of the first posts I wrote here discusses the two-dimensional case.)
The key point is this:
Sometimes when we look at something that seems to be one thing, say Neapolitan ice cream, or Baskin-Robbins 31 Flavors, it feels like we can only have one opinion.
“One a scale of one-to-ten, what do you think of Neapolitan?”
Answering that question seems to collapse what we really think into far too simple a pigeonhole. It should be possible to think about it, and to talk about it, in a way that throw away important parts of our opinion.
Another important aspect, which I haven’t touched on in this post or the previous one, is that this way of looking at things allows for seemingly contradictory feelings to coexist without canceling each other out.
I haven’t touched on it yet because there isn’t anything contradictory about liking one ice cream flavor or another. There is no sense that liking chocolate mint has anything to do with how much you like butter pecan.
But imagine that you (A) think nuclear power is very dangerous, but also (B) think nuclear power is very useful and important.
It might seem those conflicting views oppose each other in some tug-of-war that leaves you in the middle. But if you view (A) and (B) as separate scales, there is no conflict. The winner is the one that is a stronger view.
This may seem obvious, the stronger feeling wins, but having a way to visualize it is (I have found) very helpful in myriad contexts (I explore this in that old post I mentioned above).
Essentially, configuration space is just a metaphor for how to think about a given situation.
In future posts I’ll apply this metaphor to a variety of real-world things.
Part of the fun, I think, is that it’s visual!
Stay dimensional, my friends!