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If a straight line is given, and its copy is created and translated (for conveniently choosen amount), we get a square. If the analogous thing is done with a square, we get a cube. Further, we get a hypercube.

But what about triangle instead of square? What happens if we do same process starting from triangle? What are properties and names for such geometric bodies?

Clarification note: I am not talking about adding a vertex in higher dimension, but translating the whole body. In such scheme, a tetrahedron is not 3-dimensional extension of a triangle, for example.

The same question for a tetrahedron as a starting point.

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The name for the $n$-dimensional analogue of triangles and tetrahedrons is a simplex.

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Such objects are called $n$-simplices.

The so-called standard $n$-simplex $\Delta_n$ can be described as $\Delta_n = \{ x \in \Bbb R^{n+1}: x_0 + ... + x_n = 1, x_i \geq 0 \}$

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  • $\begingroup$ Are you sure? The simplices in k dimensions have k+1 vertices, but thet are not what I had in mind with my question Or I am wrong in understanding simplices? My 4 dimensional tetrahedron has 2*4 vertices, and for 4 dimensional triangle 2*2*3 vertices. $\endgroup$ – Aleksandar M May 27 '19 at 16:13
  • $\begingroup$ I'm not sure what scheme you'd like to apply to get $8$ vertices for a triangle and $12$ for a tetrahedron in $4$ dimensions. Could you please elaborate? $\endgroup$ – Lukas May 27 '19 at 16:21
  • $\begingroup$ Please read my question and comment more carefully. Btw, you swapped 8 and 12 in your response. $\endgroup$ – Aleksandar M May 27 '19 at 16:28
  • $\begingroup$ I don't suspect there is a name for such an object. $\endgroup$ – Lukas May 27 '19 at 16:30
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In dimension $3$ your object is called a triangular prism.

Extended to $n$ dimensions, there is no special shorthand name. Instead, your object is best described as the Cartesian product of a triangle with a cube of dimension $n-2$.

And if you want to start with a tetrahedron in $3$ space and carry out your process extended to $n$ dimensions, then the resulting object is the Cartesian product of a tetrahedron with a cube of dimension $n-3$.

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  • $\begingroup$ Cool, can I name them by my name? :) $\endgroup$ – Aleksandar M May 27 '19 at 16:48

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