How many dimensions do we live in? Many would say three, some would say four — as Einstein taught us with the concept of *spacetime* that time is a dimension existing along with the familiar three spatial dimensions. In some theories of physics, there are thought to be $10, 11, 24,$ or $26$ dimensions! What are all these other dimensions?

But this isn’t a science problem, this is a math problem. For a look at how we define dimension geometrically, keep reading. Or jump straight to today’s challenge.

Let’s unpack the “familiar three spatial dimensions.” The **dimension** of an object is the number of perpendicular directions that we can use to measure distances on that object. (Confusingly, these directions are also called dimensions.)

$3$ dimensions:

Here’s how some examples from geometry fit into those- A point is $0$-dimensional; it can mark a position but doesn’t take up any space.
- A line is $1$-dimensional; it only has length.
- A flat shape is $2$-dimensional; it has length and width.
- A solid figure is $3$-dimensional; it has length, width, and height.

Dimension also relates to how an object can be made up of smaller copies of itself. Consider a line segment. It’s a portion of a line, so it’s $1$-dimensional.

If we scale its length down by a factor of $2,$ we can put $2$ copies of the line segment together to recreate the original line segment:

If we scale its length down by a factor of $3,$ we can put $3$ copies of the line segment together to recreate the original line segment:

Now, instead, let's look at a right triangle. It’s a flat shape, so it's $2$-dimensional.

If we scale both its length and width down by a factor of $2,$ we can put $4$ copies of the triangle together to recreate the original triangle:

If we scale both its length and width down by a factor of $3,$ we can put $9$ copies of the triangle together to recreate the original triangle:

What is the relationship between dimension, scale, and the number of copies here? It’s your job to find it in today’s challenge.