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June 22[edit]

Is the shape of the letter L single-dimensional or two-dimensional?[edit]

I don't have any significant knowledge in math besides some elementary knowledge.

Take for example the geometric shape of the letter L.

Is the shape of the letter L single-dimensional or two-dimensional?

I would argue that it is single-dimensional because, similar to a Disc (or Circle, if you will), it's a single line in a certain shape and not a polygon.

Thanks. 2A10:8012:11:A87A:8DB6:C45C:A180:55AB (talk) 13:11, 22 June 2023 (UTC)[reply]

It may depend on what kind of mathematician you ask. An L-shape without thickness is topologically "the same" as a straight line segment, since one can be deformed into the other without cutting or gluing. So a topologist might be inclined to answer that its dimension (its topological dimension) is 1. Its Hausdorff dimension is also 1. However, these answers, while not wrong, ignore an essential characteristic of the shape, namely that it has two legs that meet each other at a right angle. So the shape is one-dimensional only if one disrespects its identity. The geometric concept of an angle is not a topological concept. It requires that the meeting legs are embedded in a plane, which is 2-dimensional. Just like a fish needs water to survive and is therefore considered aquatic, an L-shape needs a plane to exist and is therefore a plane shape, which is a two-dimensional shape (in the sense that it requires a two-dimensional space to be defined).

Not just topologically. The intrinsic geometry of the L is exactly the same as of a straight line segment of the same total length. —Kusma (talk) 17:47, 22 June 2023 (UTC)[reply]

I'm not quite sure how to reconcile this with the statement in our article Differentiable curve that "a regular curve in an Euclidean space has no intrinsic geometry". Curves have no intrinsic length; they inherit it from a metric space in which they are embedded, so it is a bit strange to appeal to its length while denying a role to its embedding. --Lambiam 18:55, 22 June 2023 (UTC)[reply]

In any metric space (for example, in the L-shaped subspace of the plane) we can define the Intrinsic metric, where the distance between two points is the length of the shortest curve connecting these points. In this intrinsic metric, the L shape is isometric to any smooth or non-smooth simple curve with the same length. —Kusma (talk) 20:24, 22 June 2023 (UTC)[reply]

In this case, what is the metric space

(M,d)

(see Intrinsic metric § Definitions) on whose underlying set we are going to define this intrinsic metric? --Lambiam 22:47, 22 June 2023 (UTC)[reply]

The L with the Euclidean metric inherited from the plane. —Kusma (talk) 04:30, 23 June 2023 (UTC)[reply]

So then all simple plane curves of a given length have the same intrinsic metric, which makes the concept useless. So it is just a complicated way of saying, "the L-shape is a simple curve" (and so it has topological dimension 1). --Lambiam 09:54, 23 June 2023 (UTC)[reply]

Basically, one dimensional geometry is boring. —Kusma (talk) 10:11, 23 June 2023 (UTC)[reply]

Slightly more interesting than the L is the geometry of the X or the A, which do not embed isometrically into a one-dimensional space. —Kusma (talk) 15:16, 23 June 2023 (UTC)[reply]