30 December 2008

Tetrahedra with arbitrary numbers of faces

While reading a paper (citation omitted to protect the "guilty"), I came across a reference to an "n-dimensional tetrahedron", meaning the subset of Rn given by

x1, ..., xn ≥ 0 and x1 w1 + ... xn wn ≤ τ

for positive constants w1,..., wn and τ.

Of course this is an n-simplex. But calling it a "tetrahedron" is etymologically incorrect -- that means "four faces", while an n-simplex has n+1 faces. This probably occurs because most of us tend to visualize in three dimensions, not in arbitrary high-dimensional spaces.

I'm not saying that "tetrahedron" shouldn't be used here -- I'm just pointing out an interesting linguistic phenomenon.

10 comments:

Anonymous said...

What is the relation between k and n?

Michael Lugo said...

k and n are the same. I'm editing the post to reflect this.

Jacob Fugal said...

The obvious correction to me is to insert "analog of a" (or "analogue" depending on your spelling preferences). E.g. "n-dimensional analog of a tetrahedron".

I dealt with n-simplices a lot in some previous research, and this is the exact phrase I used when describing them to the layperson (those laypeople that understood "tetrahedron", that is).

Scott Carter said...

A similar example is given by the group of signed permutation matrices. Let e_i denote the standard jth basis column vector, and then consider matrices whose columns are obtained by \pm e_j and permutations of these (\pm e_{s1}, ... \pm e_{sn})
s is a permutation of 1 through n. The group of these is the "hyper-octahedral group." A hyperoctahedron is the convex hull of the union of the \pm e_j in n-space. Of course, a hypo-octahedron is a square (diamond). The n-dimensional figure has (n-1)-simplices as faces. Its 3-d faces are indeed octahedra, just as the 3-d faces of the n-simplex are terahedra.

The n-simplex has a nice projection onto the plane
as the complete graph on (n+1)-vertices that are the vertices of a regular (n+1)-gon.

Anonymous said...

Mathematicians think as they are as briliant as god,God gives the knowledge of numbers and they are so amused by the complexity and theorem they created.At last they tend to forget the existence of god and dare to made fun of god.

Remember gods never plays dice; mathematicians are created with a good intention and not by chance and ts the mathematician try to plays dice with god.

I have asked one mathematician of my country before who denied the existence of god but i am surpirised he admitted that the number "0" exist before the number "1" and zero is noting in life but meaningful in mathematic.

Anonymous said...

Looks like someone has come to the hockey game with a football.

K said...

Perhaps you meant "spitball" but spelled "football"?

Go away, anonymous. (the guy with
the spitball, that is).

:)

EastwoodDC said...

Clearly the author has never played Dungeons and Dragons, or they would know to use the word "polyhedron", or "polyhedral". This brings up another interesting question: Is it possible for a mathematician to not be geeky enough? ;-)

Michael Lugo said...

Eastwood,

I think some people take "polyhedron" to specifically mean a three-dimensional object.

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