What precedes and follows: triangle, tetrahedron?

Question

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.

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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 [?].

and follow that ...

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

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

Answer 1

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.