In the past couple of weeks, I’ve been posting about seemingly simple mathematical problems that defy intuition, and where the answers we find on the internet turn out to be shallow or hard to parse. For a taste, you might enjoy the articles on Gödel’s beavers or on infinite decimals.
Today, let’s continue by asking a simple question: how many dimensions does a line have? A trained mathematician might blurt out an answer involving vector spaces or open set coverings, but there’s no fun in that. Instead, let’s take the scenic route.
The “container” dimension
What does it mean for a space to have a certain number of dimensions? Informally, we could say that a dimension is an independent axis along which we can position a geometric object. In one-dimensional space, a point can only be moved along a single path. In 2D, we typically talk of two orthogonal axes, x and y. In three dimensions, we have x, y, and z. There’s more nuance to certain exotic or stripped-down (topological) spaces, but we don’t need to go into any of that.
The definition lends itself to a simple, common-sense way to classify the dimensionality of geometric shapes: we can look at the minimum number of spatial dimensions required to contain the object in question. A pencil sketch fits on a piece of paper, so it’s two-dimensional; a rock in your hand is 3D.
The simplest way to define the dimensionality of a shape.
Yet, this common-sense definition is unsatisfying if we consider that a lower-dimensional object might end up straddling a higher-dimensional space. If a line segment is rotated or bent, does that make it 2D? Or is that object forever one-dimensional, somehow retaining the memory of its original orientation and curvature?
One dimension or two?
We could argue either way, but no matter which option we choose, the answer doesn’t feel particularly principled. This tells us to look for a more substantive approach somewhere else.
The “degrees-of-freedom” dimension
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