By a regular tetrahedron, I’m referring to the tetrahedron formed by connecting the vertices of the $3$-simplex with the “far corner” $(1, 1, 1)$. That is, the four vertices of this regular tetrahedron are $(1, 0, 0) ,\, (0, 1, 0) ,\, (0, 0, 1)$, and $(1, 1, 1)$, shown as blue edges one on the right.

My question is this:

What do you call the counterpart (generalization) of such tetrahedron in higher dimensions?

That is, connect the vertices of the $n$-simplex with the "far corner" $(1, 1, 1,\ldots 1)$ so that one gets a shape which vertices are the unit vectors $e_i$ and $\sum e_i$, and the sides have the same length $\sqrt{n-1}$.

If there's not a name for such construction, how about alternatively connect the origin $(0, 0, 0)$ with $(0, 1, 1) ,\, (1, 0, 1) ,\, (1, 1, 0)$?

This alternative tetrahedron is shown below with orange edges. One can compare it with the previous one involving the simplex with blue edges.

enter image description here

The higher dimension counterpart of the orange tetrahedron has vertices being the complement (binary flip) of that from the blue one, e.g. $(0,1,0,\ldots, 0) \to (1,0,1,\ldots, 1)$. It is congruent to the previous one, also with side length $\sqrt{n-1}$.

  • 3
    $\begingroup$ You are probably looking for a simplex. $\endgroup$
    – robjohn
    Dec 9 '16 at 7:59
  • $\begingroup$ @robjohn ha, thanks, I just wasn't sure about the terminology. $\endgroup$ Dec 9 '16 at 8:16
  • 1
    $\begingroup$ They can be called regular simplices (plural of simplex). The orientation is irrelevant. $\endgroup$
    – user65203
    Dec 9 '16 at 8:25
  • $\begingroup$ @YvesDaoust Thank you. Just to make sure: previously I thought a regular simplex is one that is sort of orthogonal, like sitting in the corner of a Cartesian coordinates. In fact, regular simplex refers to the fact that the side lengths are all the same, right? $\endgroup$ Dec 9 '16 at 8:36

I'm posting to remove this question from the unanswered list, and I'll check it as soon as the system restriction is lifted.

I previously thought $n$-simplex (as I used it in the question) refers to only the "orthogonal" ones.

Yeah the very first few sentences in e.g. wiki for simplex states clearly that a $n$-simplex refers to generally a convex hull formed by its $n+1$ vertices.

I guess there's no special names for the special cases I talked about in the question.


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