Product ID: Student Research Corner

Supplementary Print

High School

Which 3-dimensional convex polyhedra,
all of whose faces are triangles,
can be made with isosceles
triangle faces? In an earlier research column (this column is essentially
self-contained but you might want to
look at the earlier column in Consortium
116), I raised the question of whether it
might be true that all possible (finite)
triangulations in the plane with at least
four vertices (see Figure 1 for examples)
might be "realizable" as 3-dimensional
bounded convex polyhedra with faces
that were all isosceles triangles, an isosceles
triangle being one with two sides
of one length and one side of another
length. Here I don't allow a triangle described
as isosceles to be equilateral, a
triangle where all three of its edges
have the same length.

In the earlier column I mentioned the issue of trying to better understand which triangulations could be realized as polyhedra with all faces congruent isosceles triangles. I knew of several infinite classes of such congruent isosceles triangle polyhedra, and I knew examples of triangulations where it was not possible to realize the triangulation with a convex polyhedron having isosceles triangle faces that were congruent to one another. Below, when I use the term polyhedron I am referring to a convex bounded polyhedron in 3-dimensional space where all of the faces of the polyhedron are convex polygons.

I am not thinking about a polyhedron gotten, say, by pasting two congruent cubes along a face and counting the vertices of the faces where the pasting occurs as "vertices". Thus, two 1 • 1 • 1 cubes pasted in such a way would be the polyhedron of the same combinatorial type as the cube, having 8 vertices, 6 faces and 12 edges, with edges of length 1 or 2, and is the "solid" we would think of as a 1 • 1 • 2 box. Below, the polyhedra I have in mind are the convex hull of their "extreme points."

In the earlier column I mentioned the issue of trying to better understand which triangulations could be realized as polyhedra with all faces congruent isosceles triangles. I knew of several infinite classes of such congruent isosceles triangle polyhedra, and I knew examples of triangulations where it was not possible to realize the triangulation with a convex polyhedron having isosceles triangle faces that were congruent to one another. Below, when I use the term polyhedron I am referring to a convex bounded polyhedron in 3-dimensional space where all of the faces of the polyhedron are convex polygons.

I am not thinking about a polyhedron gotten, say, by pasting two congruent cubes along a face and counting the vertices of the faces where the pasting occurs as "vertices". Thus, two 1 • 1 • 1 cubes pasted in such a way would be the polyhedron of the same combinatorial type as the cube, having 8 vertices, 6 faces and 12 edges, with edges of length 1 or 2, and is the "solid" we would think of as a 1 • 1 • 2 box. Below, the polyhedra I have in mind are the convex hull of their "extreme points."

©2021 by COMAP, Inc.

Consortium 121

4 pages

- Polyhedral

You must have a **Full Membership** to download this resource.

If you're already a member, **login here**.

Browse More Resources

Search

COMAP develops curriculum resources, professional development programs, and contest opportunities that are multidisciplinary, academically rigorous, and fun for educators and students. COMAP's educational philosophy is centered around mathematical modeling: using mathematical tools to explore real-world problems.

Policies