# Four dimension equivalent of a triangle and a tetrahedron, obtained *by translation*?

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.

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

Such objects are called $$n$$-simplices.

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

• 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. – Aleksandar M May 27 '19 at 16:13
• 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? – Lukas May 27 '19 at 16:21
• Please read my question and comment more carefully. Btw, you swapped 8 and 12 in your response. – Aleksandar M May 27 '19 at 16:28
• I don't suspect there is a name for such an object. – Lukas May 27 '19 at 16:30

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 $$n-3$$.

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