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The students in my physics class were playing with Rubik's Cubes this morning before class. This got us talking about solids. The traditional Rubik's Cube is a six-sided closed solid in a three dimensional space. Another student had a Rubik's Dodecahedron, which got us talking about the "simplest" possible Rubik's Solid. We decided that what we meant by simplest was "possessing the smallest number of sides". We settled on a tetrahedron (which apparently does exist in Rubik's form).

Then we got to talking about the simplest possible closed figure in spaces of varying dimension. It seems like the simplest possible closed figure composed of straight sides in a two dimensional space has three sides, a triangle. Similarly, the simplest possible closed figure composed of straight sides in a one dimensional space would be a line segment, with two sides (is this a stretch?).

Then class began.

Intuitively, it seems like the simplest possible closed figure whose sides are all straight in an $n$ dimensional space will have $n+1$ sides.

Is that intuition correct?

My formal training is in the Classics, so I'm not sure whether I've asked the question properly. What branch of mathematics thinks about questions like that, and is there a good introductory text for it?

Bill Carey
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    An entertaining introduction to thinking about shapes in different dimensions is Edwin A. Abbott's Flatland: A Romance of Many Dimensions (1884). Your intuition is correct; in $n$ dimensions, the $n$-simplex has $n+1$ faces, each being itself an $(n-1)$-simplex. – mjqxxxx Jan 09 '13 at 15:41
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    @mjqxxxx: Shouldn't that be an answer? Bill: The Wikipedia pages on regular polytopes and the list thereof are a little advanced, but your students might find them interesting still. They are full of fascinating pictures. –  Jan 09 '13 at 16:27
  • Both of your comments are really helpful. I'd love to be able to vote them up as answers. "Simplex" is, indeed, the word I was looking for. – Bill Carey Jan 09 '13 at 16:39

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An entertaining introduction to thinking about shapes in different dimensions is Edwin A. Abbott's Flatland: A Romance of Many Dimensions (1884).

The generalization to higher dimensions of the triangle and the tetrahedron is the simplex. Your intuition is correct; in $n$ dimensions, the $n$-simplex has $n+1$ "faces" that are $(n-1)$-simplexes themselves. More properly, these are its $n$-faces. Each $n$-face is the convex hull of $n$ of the $n+1$ vertices. There are also $m$-faces for $2 \le m < n$, which are defined similarly, and which generalize the edges of the tetrahedron (i.e., they are faces of the faces, or faces of the faces of the faces, etc.)

mjqxxxx
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