I’ve encountered assignments in this course that I’ve struggled to apply to my topic area (functions) but this assignment seems built for it! Dimensional thinking is thinking about a topic or concept in various dimensions and possibly coming up with models to represent that concept. One of the great things about modern graphing technology is that it allows us to push, pull, and explore functions. I want to explore functions both at the minuscule scale, the large scale, in three dimensions, and even explore higher dimensions.
We often look at functions in a standard viewing window. We generally like integers, especially numbers -10 to 10, when we first look at functions. Consider the image below that I’m sure most people remember from high school algebra, y=x^2 in the standard viewing window.
There is nothing special about this viewing window. In fact, as many math students find out, lot’s of other viewing windows are more useful. For instance, suppose this graph was modeling position (in meters) versus time (in seconds) and we wanted to know the velocity at exactly 2 seconds. This problem becomes difficult without calculus unless we change our window to make finding the exact velocity easier. See the image below.
This image shows that by significantly shrinking the viewing window we can easily find the average rate of change from 1.995 seconds to 2.005 seconds. Although this is the average rate of change (which I should point out is the same as the average velocity), the time interval (.01 seconds) is so small that the average velocity is essentially the exact velocity. It can be verified using calculus that the exact velocity is 4 m/s, which we got (see the green arrow).
In fact, this idea that we can zoom into any point on most functions and get something that looks like a line is incredibly important. If this can be done then we say the function is differentiable at that point and this is a pivotal idea in calculus. See the GIFs below to demonstrate this idea.
This may lead some students to believe that any function, at any point, has this property, but as you can see in the GIFs below, this is not true.
What I love about using Desmos is that it makes it easy to explore functions at the extremes. Let’s look at an example of functions at a large scale. For instance, in the image below we take a fairly ugly polynomial function and notice that it’s almost identical to a simpler function when we zoom way out. (Compare the two images. Click the images to enlarge them.)
Finally I wanted to take a look at functions in three dimensions. This is where things get more conceptually difficult as we don’t often spend a lot time thinking about functions in three dimensions (at least not in high school). Check out the videos I made below for a brief tutorial into functions in three dimensions.
I know what you’re thinking, what about four dimensions?
Any dimension higher than three is difficult for us to visualize as we live in 3 dimensions. However, mathematics doesn’t really care about the number of dimensions in which you live. In fact, if we think of two dimensional curves as slices of 3 dimensional functions, then 3 dimensional curves are slices of four dimensional functions. As I mentioned, this is hard to visualize but this video makes a good attempt. We can extend multidimensional mathematics even further. For instance some versions of string theory suggest that there are many very small, curled up dimensions. Check out this video for an explanation on how that works (it also has great visuals and metaphors to help you imagine multiple dimensions). This is what gets me excited about my content area. They aren’t visualizing these dimensions under a microscope. It all comes from the mathematics!
This activity made me realize how incredibly “flexible” functions are and how modern technology can help us see functions in different ways. By flexible I mean that you can look at them on the smallest scale and you (often) get a line and look at the largest scale and (often) get relatively simple behavior. You can also pull them in different dimensions. It just shows the plasticity of mathematics and I think that is very counter to how many people view mathematics. I know I’ve mentioned this in previous posts, but the idea that mathematics is rigid is simply wrong. The way we present it might be rigid but as I’ve spent a lot of time with functions over the last couple months I’ve seen that it’s even more flexible that I previously thought. I think it would be a good task to have students do what I did with the concept functions in this assignment. I would have them spend time exploring them at different levels and different ways. They need time playing with mathematics to get a full understanding of it and with modern graphing technology it’s never been easier.
Update: This article entitled “The XKCD guide to the universe’s most bizarre physics” came across my radar today and it has awesome visuals and insight into imagining higher dimensions.