# What precedes and follows: triangle, tetrahedron?

I have not studied geometry beyond high school myself. I’m looking for an answer that would satisfy (and be understood by) lower secondary school students.

Given that ...

Using straight lines, you need three lines to make the “simplest” closed object (polygon) in 2D space: a triangle.

Using flat surfaces, you need four surfaces to make the “simplest” closed object (polyhedron) in 3D space: a tetrahedron.

does it precede that ...

Using [?] points, you need two points to make the “simplest” closed object ([?]) in 1D space: a [?].

Using [?] volumes, you need five volumes to make the “simplest” closed object ([?]) in 4D space: a [?].

and what terms can fill in the [?] blanks?

• There is a nice pattern to consider; It takes 2 points to make a line segment, 3 line segments to make a triangle, 4 triangles to make a tetrahedron.... can you extend to say that it takes 4 tetrahedron to make a _______. Visit this page for more information Commented Aug 9, 2018 at 18:40
• You might be thinking of an "interval" for the 1-dimensional case, but that depends on what you consider to be an "object." Arguably a point is simpler. As for extending to higher dimensions, the more general term you are after is "polytope" (polygons are 2-dimensional polytopes, while polyhedrons are 3-dimensional polytopes). Tessaracts are easy examples of a 4-dimensional polytope, but the 5-cell is the example with fewest hyperplanes making it. Commented Aug 9, 2018 at 18:42
• I guess ‘flat’ and ‘straight’ don’t devolve any further into anything. I think you’re right on 1D: two points “close” an “interval” in 1D space. Reading Wikipedia on 5-cell ... seems like the candidate. Commented Aug 9, 2018 at 19:10

The general concept is called a simplex. The pattern goes:

• A point is a 0D figure.
• Two points can be the vertices of a line segment, which is a 1D figure. The boundary of a line segment consists of two points which are called its endpoints.
• Three points can be the vertices of a triangle, which is a 2D figure, and the simplest polygon. The boundary of a triangle consists of three line segments which are called its edges.
• Four points can be the vertices of a tetrahedron, which is a 3D figure, and the simplest polyhedron. The boundary of a tetrahedron consists of four triangles which are called its faces.
• Five points can be the vertices of a pentachoron, which is a 4D figure, and the simplest polychoron. The boundary of a pentachoron consists of five tetrahedra which are called its cells.
• ...
• 100 points can be the vertices of a 99-simplex, which is a 99D figure, and the simplest 99-polytope. The boundary of a 99-simplex consists of 100 98-simplices which are called its facets.
• ...

You can take the 99-simplex example and extend it backwards, too. A pentachoron is a 4-simplex, a tetrahedron is a 3-simplex, a triangle is a 2-simplex, a line segment is a 1-simplex, and a point is a 0-simplex.

• I’ll take this as the answer. You gave terms for the sequence, including names of the objects and their “endpoints”, which are all easy to follow. Commented Aug 9, 2018 at 19:15