Area, Volume, and the Fourth Dimension.

If you’re reading this post because you want to learn about the fourth dimension, skip down a bit. I’m going to start off talking about diameter, then area, then volume, or shapes in the first three dimensions. However, the conversation has some relevant issues which recur in the Fourth Dimension discussion so why not just read everything.

The First Dimension

As I move through the 3D printing world, an objective is to master all aspects of the art, including printing with different hardware, designing on different software, printing with unique materials, and printing at different sizes. A major limiting factor to the size challenge is the diameter of my printer’s nozzle (for fused deposition modeling machinery).

I recently acquired a secondary nozzle head. My original one had a diameter of 0.4mm, while the new is exactly half that at 0.2mm. Originally, this didn’t seem to me too impressive or a big deal: yay I can print things half as thin now. But I wanted to follow through and look at the math. What were the differences in print area and print volume? see chart 1 (below) for the results

The Second Dimension

Since the nozzle is shaped like a circle we can go straight to figuring the area of a circle, rather than any kind of polygon. Running through the math I find that slicing the diameter in half creates twice as much difference: the extrusion area of the 0.4mm nozzle is four times as large as the extrusion area of the 0.2mm nozzle. Again, not terribly significant of a difference, but we see a trend starting: the difference has doubled.

The Third Dimension

Ah yes, good old 3D. I’m interested in data from this dimension because, in reality, the printer is not extruding in the second dimension. It does not impress a two-dimensional stamp on the model, and it does not effect the color or quality of the material it’s being laid upon (like your office printer discolors paper with ink). Instead we deal with the volume of a sphere which is, of course, dependent on the diameter and area. This dependence means that we can draw conclusions of trend analysis.

Before we calculate it, let’s define volume of the sphere and say that this represents e, or one unit’s movement through the extruder. This is equivalent to the diameter of the nozzle (if with a 0.4mm nozzle then e = .4mm; if with a 0.2mm nozzle then e = 0.2mm).

By pushing the material through the extruder, we’ve entered the Third Dimension. Quick maths gives us the volumes:

Chart 1

note: the difference in volumes is actually 797.62% because we incorporate pi so much, but I’m just going to say 800% because it gets the point across.

The point being that the difference between the two nozzles gets more and more significant as we go up in dimensions. We also see that this difference is exponential by nature, obviously because we’re raising the units by a power as we progress through the dimensions. This knowledge helps set me up for some awesome 3D prints, but I wasn’t done with the math yet. This next section will deal with mostly theoretical maths.

The Fourth Dimension

Hopefully without causing too much panic amongst readers and purely for the sake of discussion, I’m going to assume for this post that the passage of time represents the fourth dimension. With this assumption, measuring the fourth dimension is essentially measuring how a given object interacts with the passage of time. Another way to put it is as a question: “what kind of ways can we expect this object to change as time interacts with it? What effects does the passage of time have on the object?”. Measurement is important because, in the words of entrepreneur Peter Diamandis, “If you can’t measure it, you can’t improve it”. Defining a system of measurement in the fourth dimension is important to improving how we operate in it. There’s two ways we can answer the question of the effects of time:

Analytical Perspective

From an analytical perspective, this can be analogized to observing an animal. Consider the scientist who travels to Antarctica to study the behaviors of a few species of penguin. For one species, we can expect a few things to be reported. First a good three-dimensional description of how the penguin looks, probably with lots of pictures of penguins in order for the reader to build a mental model. We could then expect a description of how the average penguin might spend their day, filling it with things like tending to offspring, engaging in social activities, and sustaining the body with food or sleep. The researcher has pinpointed some variables which we can expect as effects of the passage of time: the penguin will sleep, the penguin will be hungry, the penguin will age, and the penguin will eventually die. Through recorded analysis we can predict that these, as well as other calculable events (such as being eaten by a leopard seal), are the effects which time will have on the penguin. Thus is science.

However, our ability to accurately predict the effects of time are limited to what has been observed and analyzed. If an observer had never seen a penguin get eaten by a leopard seal then we’d have no data to base a prediction off of. Weird analogy? Good. Let’s look at a different perspective of measuring the fourth dimension, a perspective which has the possibility of 100% accuracy.

Design Perspective

Instead of analyzing an existing object, the designer creates one. In doing so they have the privilege of defining how the object interacts with time. Programmers might understand this interaction as a function, or a method of an object. The point here being that all variables are known. By starting from the beginning (in the design) rather than from an observation (the analytical perspective), there is a certainty that regardless of how many variables an object will have it is theoretically possible to perform a prediction with 100% accuracy how an object will react with time.

We need a formula, though. We know that to find the area of a circle we simply calculate πr^2. To calculate the volume of a sphere we calculate 4/3πr^3. In speculation of how to calculate the fourth-dimension, I believe that we can start to piece together a formula. We can assume that pi will be a factor as it has been critical in the two lower dimensions. We can also assume that our unit variable will be raised to the power of four (r^4).

That’s two factors already which can be reliably counted upon. For the rest we can only speculate, but I imagine that there will be a ratio which the other values can be multiplied against. Like the 4/3 which was used to calculate volume, we can count on a certain ratio which will be dependent on the space and time which our object will consume. Factors within this ratio will deal with the number of variables, a count of the changes possible through all those variables, and the amount of combinations in which these variables can exist with each other.

Conclusion

Though this by no means encapsulates all my thoughts or plans on working in the Fourth Dimension, it does well to outline the process of my thinking as the design evolves through the dimensions. Just like with the writing of this blog, I believe that the process of organizing thoughts and publishing them for criticism is the first step to discovering or creating something great.

Thoughts, opinions, or criticism? I’d love to hear from you!