In geometry, we want to work with different types of objects, and one of our goals will be to classify objects into various categories. One important category is the dimension of an object.

Dimension is a difficult idea in mathematics. One way that we can think about dimension is to think about how many coordinates we need to move around, but of course this would first require us to understand what we mean by coordinates. Perhaps you already have some idea what coordinates are, and in this case this definition is helpful.

However, there are some more informal ways to think about dimension, and they can also introduce us to one of our themes in this book: ideas at different academic levels. The discussion above can be helpful for high school students who are pretty used to graphing on a coordinate plane. But we would like to talk with kids about dimension starting as early as kindergarten, so we need an idea about dimension that will make sense to kids around middle school, and another idea that will make sense even to the youngest kids we talk to. These ideas about dimension at different levels don’t need to be a full definition, but they should help us help kids build their ideas about dimension towards the formal definition they will see in high school.

Let’s look at a middle school idea and then an elementary school idea.

Around middle school: the little bug

First, we imagine that we are a little bug – so tiny that when we live in or on a shape, that shape is so big relative to our own size that we can’t really tell what shape it is. (Think about this like being a human being living on the Earth: the Earth is a sphere, but you can’t really tell what shape it is just by looking at it from where you’re standing.)

Second, we want to understand how we can move in different directions. Think again about yourself on the Earth: you can move left or right, but left and right are kind of the same direction. One is a positive move, and the other is a negative move. This is a little bit like having East and West directions on a compass: we represent them with the same line, so they are two different versions of a “horizontal” direction. Or, you might connect any ideas about coordinates you had earlier to having a left-and-right number line. On the Earth, you could also move forward and back, and we would again say that forward and back are the same direction, but positive and negative again. This is like having a North/South direction in addition to your East/West direction. You can also move in all kinds of diagonal directions, but all of those diagonals could be written as combinations of the right-left direction and the forward-back direction. Continuing with the direction analogy, you could move north-west or south-east or any other diagonal direction. So, you have two different directions you could move, and all of the combinations of those directions. The Earth, in this case, is a two-dimensional object, because you have two different directions that you can move.

Another way of thinking about the number of directions the little bug could move is by thinking about how you might give instructions to the little bug. If the little bug lives on a line, you can give it directions like “move $2$ steps forward” or “move 8 steps back”. If the little bug lives on a piece of paper, you can give it directions like “move $3$ steps right then $6$ steps backwards” or “move $8$ steps left and $0$ steps forwards”. On the piece of paper, the little bug needs two pieces of information to be able to move everywhere on the object, whereas on the line the little bug only needed one piece of information to be able to move anywhere on the object. The line, then, would be a one-dimensional example, while the piece of paper would be a two-dimensional example.

Let’s take a look at another example to help us explore this idea.

Very young kids: examples

Another way to think about dimension is comparing to other examples we know. So, for instance, an object is $1$-dimensional if it’s like a line, or if we are using or measuring its length. An object is $2$-dimensional if it’s like a filled-in square, or if we are using or measuring its area. An object is $3$-dimensional if it’s like a cube, or if we are using or measuring its volume.

We can also use physical objects in these comparisons.

Another way that some teachers talk about dimension is to use rulers. If I need one ruler to measure the object, it should be one-dimensional. For instance, I can measure a straight line with one ruler. If I need two rulers to measure the object, it’s two-dimensional. For instance, if I want to measure the area of a square, I need two rulers: one to measure the width, and one to measure the length. If I need three rulers to measure the object, it’s three-dimensional. For instance, to measure the volume of a box, I would need one ruler for the length, one for the width, and one for the height. However, these ideas with rulers can also lead us astray: if I try to measure a wiggly line, how many rulers do I need? This is less clear. A wiggly line is still one-dimensional, even if the wiggly line has to live in three-dimensional space.

Some cautions

As we mentioned at the start, dimension is tricky. Consider the example of the piece of paper.

Finally, let’s consider the example of the circle.

One key takeaway: we want to be precise with our language. Always err on the side of over-describing what you are talking about so there’s no confusion.