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Is there a name for the tetrahedron/pyramid (four vertices, four triangular faces, six edges) where three edges meet orthogonally at a point? Three of the faces are right triangles.

Another description: taking $\vec e_1,\vec e_2,\vec e_3$ as the standard basis vectors of $\mathbb{R}^3$, the vertices are $\vec 0,c_1\vec e_1,c_2\vec e_2,c_3\vec e_3$. The faces are composed of the coordinate planes with the plane $x/c_1 + y/c_2 + z/c_3 = 1$.

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In Coxeter's book it is called an orthoscheme. –  Will Jagy Jun 23 '11 at 15:37
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I've also seen this figure just called a 'right tetrahedron', by analogy with 'right triangle', and that's probably the first name that I would use for it in informal math writing. –  Steven Stadnicki Jun 23 '11 at 16:30

4 Answers 4

up vote 3 down vote accepted

According to Wolfram's Mathworld, it's called a trirectangular tetrahedron.

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A right-cornered tetrahedron.

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A common name is rectangular tetrahedron.

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Any solid polytope of the form $\sum_{i = 1}^{n} \frac{x_i}{a_i} \leq 1$, where $a_i > 0$ is called a solid, orthotopal, simplicial $n$-polytope and can also be written as the convex hull $\mathbf{conv} \{ \mathbf{0}, a_1 \mathbf{e}_1, \dots, a_n \mathbf{e}_n \}$. For $n = 3$, it is called a solid trirectangular tetrahedron. If you mean the boundary complex, then simply trirectangular tetrahedron is fine.

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