Strap in folks. We’re getting theoretical today!

*Note: I'll be updating this later today or tomorrow with images to help show what we're talking about.

I often tell people that I tend to think of fitness as an N-dimensional space. And if the person I’m talking to doesn’t have a strong statistical or math background, their eyes often glaze over at this point because they have no idea what I’m talking about. Hang with me though, and I’ll explain it to you.

Basically, it’s a visual way of thinking about anything with a lot of variables that go into it. It’s easy to imagine a 2-dimensional graph (X and Y axes), where the variable we’re changing is the x-axis (the horizontal line) and we’re interested in how high or low the line goes when measured on the vertical axis (the y-axis). i.e. y = (some multiplier)x + b (look familiar?) where b is where the line starts on the y-axis. If we use this to think about “fitness” then we could use any metric we could think of that we associate with being physically fit as our x variable. For instance, if x is how strong we are, then we could measure how strong we are and calculate our “fitness”.

If we take it a step further and add a second variable we’re interested in, all we have to do is add another horizontal axis (the z-axis) that goes away from us as we’re looking at the graph. So now the axes look like three lines that are all perpendicular to each other. This allows us to have two variables (i.e. strength and speed) we’re putting in to determine where on the y-axis our line (now a plane) our answer is. So this would look like y = (some multiplier)x + (some other multiplier)z + b.

Now if we want to get more variables into the equation, we need to start using our imagination a bit. I remember when I saw a graph on a poster in grad school that showed how to display 4 axes on the same graph that I was able to wrap my head around how to do this. I don’t remember what the exact question was they were looking at, but it was a forestry poster that had 3 variables they had measured to determine how trees would respond to different kinds of forestry practices. What I do remember was that they had the typical 3-D graph you’d use if you had two input variables (the x- and z-axes), but then they color coded the plane on the graph, with each color being a different value on that 4th axis. And it occurred to me that you could probably take it another step further if you could picture that color coded plane moving (allowing a 5th axis). I can’t honestly say that I’ve seen anyone use this technique, but you couldn’t print it on paper anyway, you’d have to use a video. The important part is that it allowed me to figure out how to think about complex systems in a simple way. Each thing that could influence the end product gets its own axis.

Once we get to this point, we can flip this around a bit and think of the end product (i.e. fitness) as a three dimensional object. Let’s say we’re measuring speed, strength, and cardiovascular capacity (or VO2-max) to determine how “fit” we are. If we treat each one of these as a measurement for our 3-D object, the higher any value is, the more volume our shape is going to have, right? And that volume is how “fit” we actually are. And the beautiful part of this is that once we understand it, it’s a lot easier to understand if we put it into math: x(strength) + y(speed) + z(VO2-max) = fitness.

But we all know there’s more that goes into being “fit as an endurance athlete than just speed, strength, and cardiovascular capacity, right? What about flexibility? Metabolic endurance? Muscular endurance? Neuromuscular fitness and coordination? The ability to buffer our blood pH and not suffer acidosis? The potential list of things we could measure goes on and on.

But if we can wrap our heads around those 5-dimensional forestry management graphs, then we can use that same concept with our “fitness” volume. We just have to add another axis to the shape we’re drawing. It has more dimensions than just height, width, and depth. It has “n” dimensions:

x(strength) * y(speed) * z(VO2-max) * a(flexibility) * b(metabolic endurance) * (etc.) = “fitness”

So what does this mean for training and athletic performance in endurance sports? Typically it means we’re measuring “fitness” (or the volume of our n-dimensional space) by our finish times. The more fit we are, the faster we’re going to run, bike, and swim. The point of training is to maximize the volume of that n-dimensional space. We want to get as “fit” as possible so we can finish as fast as possible. At this point, the big question is “what variables have the biggest impact on the volume of our n-dimensional space?”

If you stuck this one out to the end, hopefully we were able to stretch your brain a bit today. Stay tuned for next week when we dive into how to actually apply this to get the most out of training

## Comments