Nets for Polyhedra

net In this challenge, we ask you to find nets for shapes other than a cube.

The Boxes, Cubes, and Nets [1] puzzle featured finding nets for cubes. A net is a two-dimensional shape that can be folded up to form a three-dimensional shape. For example, the hexomino (shape made from six squares) shown in the figure can fold up nicely to form a cube. The red squares fold up to form the base and sides of the cube; the green square folds over to form the top of the cube.

A cube is a special kind of three-dimensional figure called a regular polyhedron. A cube is a regular polyhedron because all of its faces are congruent regular polygons.

In this puzzle we will try to find nets for three related polyhedra: a regular tetrahedron (four equilateral triangular faces), a regular octahedron (eight equilateral triangular faces), and another common polyhedron, the square pyramid (four equilateral triangles and a square).

Challenge 1

How many nets can you find for a tetrahedron? A tetrahedron has four faces, so each net consists of four equilateral triangles attached along their edges.

Challenge 2

How many nets can you find for a square pyramid? A square pyramid has one square face and four triangular faces, so each net consists of one square and four equilateral triangles, attached along their edges. Try to find all the possible nets.

Challenge 3

Find at least one net for a regular octahedron. See if you can find more than one. Each net will consist of eight equilateral triangles attached along their edges.

Background

First, a couple of definitions. A polygon is a closed two-dimensional shape whose edges are straight lines that do not cross each other. A polyhedron (plural: polyhedra) is a three-dimensional figure, solid or hollow, all of whose sides are polygons The word comes from the Greek, poly, "many," + hedron, "side" or "base." A regular polyhedron is one whose faces are all congruent regular polygons. Actually there are only five regular polyhedra, sometimes called "Pythagorean solids," the cube and four others: a tetrahedron (4 faces), octahedron (8 faces), dodecahedron (12 faces) and icosahedron (20 faces).

net

There are many, many Web sites that deal with the mathematics of polyhedra. Here are just a few:

Unfolding Polyhedra [2]

Two-dimensional and three-dimensional geometry are often studied as if they were different subjects. In this puzzle and "Boxes, Cubes, and Nets," we study one way they are related—through the construction of nets, two-dimensional shapes made of polygons that fold up into polyhedrons.

This puzzle makes use of three other important mathematical ideas.

The first of these is the concept of congruence, which is needed when we try to decide whether two shapes that look different are actually identical. Mathematically, two shapes are congruent if they have the same size and shape, regardless of their orientation. If you can pick up one of the shapes and rotate it, flip it over, and/or slide it so that it fits exactly on top of the other shapes, then the two shapes are congruent. This is one simple test for congruence. There are many others. As an example, consider these three shapes, each made up of four equilateral triangles and one square. Two of them are congruent? Can you tell which ones?


1	2	3

The second is the idea that you sometimes have to test to be sure whether two mathematical objects are the same or different. Just because two shapes look different, you can’t be certain they are different until you’ve picked one up and made sure it’s not congruent to the other. For nets, congruence is the test—but mathematicians have many other tests they can use for different types of problems.

The third is the idea of finding all possible ways to solve a particular problem. This is a common type of problem in combinatorics. For example, find all the possible ways to arrange the numbers from 1 to N. Or for a more everyday example, a restaurant serves pizza with four different toppings: onions, mushrooms, sausages, and extra cheese. How many different types of pizzas can they make using 1, 2, 3, or 4 of the toppings? (Of course, a real restaurant would probably provide many more than four toppings.)

Combinatorics problems involving geometry can be much more complex. For example, if N is the number of faces in a polyhedron, there is no known rule or formula that can tell you how many different nets can be constructed for a polyhedron with N faces. (Mathematicians have worked out the exact number of nets for many different polyhedrons, but no general rule has ever been found.)

There is an important problem solving strategy that can be used in this puzzle. Once you’ve solved one problem, you may be able to use the solution to the first problem to solve a harder one.

Course:

Math [3]

Result/Solution(s)

Challenge 1

This challenge asked you to find as many nets as possible for a regular tetrahedron?

This one’s fairly easy. There are only three ways to arrange four equilateral triangles in patterns with their edges aligned.


1A	1B	1C

It’s pretty easy to visualize that the first pattern, 1A, works as a net: just fold up the three outside triangles. Their vertices meet at a point, and the center triangle forms a base.

Now look at the second pattern, 1B. It has four outside edges and two “inside edges." The inside edges are the ones that form a “V.” Visualize folding the four triangles so that the inside edges meet at the central point forming a vertex. What you get is an open “tent” with four triangular sides and no bottom. So pattern 1B is not a net for a tetrahedron.

The pattern 1C is harder to visualize in a folded position. You may just have to make this pattern, cut it out, and fold it to learn that pattern 1C does fold into a tetrahedron.

Challenge 2

In this challenge you needed to find all the possible nets for a square pyramid.

This is bit harder because there are a lot more possibilities of arranging four triangles and one square into a pattern. However, if we start from what we learned from the tetrahedron, we can simplify the problem pretty quickly. First of all, we can eliminate any patterns that include a net for a tetrahedron, such as patterns 1A and 1C. This is because the four triangles in such a pattern will fold up on themselves, leaving no room for a square base.

On the other hand, we already learned that pattern 1B folds up into a four-sided "tent." So if we add a square to one of the outside edges, we get a net for a pyramid. Patterns 2A and 2B are nets for a pyramid. However, pattern 2C does not work, because putting the square on the inside of pattern 1B makes it so that the "tent" can’t be formed.


2A	2B	2C

Now we’ve used up all the patterns based on the "tent" of four triangles. But there’s one more easy net we can make by putting the square in the middle of four triangles.

Now that we’ve used up all the easy ways to make a net, we have to consider possibilities that include a square and two pairs of triangles. There are four patterns we can make with this kind of arrangement.


2E	2F	2G	2H

These are harder to visualize, so it may be easiest to make them and cut them out. If you do, you’ll find that patterns 2E and 2F form nets for a pyramid, while patterns 2G and 2H do not. If you look carefully at patterns 2G and 2H, you may be able to visualize that for 2G there is no triangle that can meet the top edge of the square and for 2H there is no triangle that can meet the right edge of the square.

We’ve got five nets so far: 2A, 2B, 2D, 2E, and 2F. Can we find any more? To answer this, we need to consider patterns that have one pair of triangles together and two triangles separated. There are three of these.


2I	2J	2K

If we look at these, we see that only 2J can work. For 2I and 2K, there is no triangle that can meet the far edge of the square.

Finally we need to consider all the patterns that have three triangles together and one triangle separated. There are six patterns like this:


2L	2M	2N

2O	2P	2Q

Imagine the three triangles together, folding into a little three-sided tent around the middle triangle, leaving a space open for a square or triangle to meet the open sides. Consider patterns 2L and 2M. When the three triangles are folded, there is no way for them to meet one side of the square. The same is true for 2O. For 2N, on the other hand, the three-triangle tent meets both sides of the square and the opposite side folds up to make the pyramid. So pattern 2N works. This leaves patterns 2P and 2Q to investigate. If you cut out both of them, you’ll find that 2P does not work and 2Q does work. So from this last set, we have two more nets, for eight nets in all:


2A	2B	2D	2E


2F	2J	2N	2Q

Challenge 3

Lastly, we asked you to find at least one net for a regular octahedron.

The octahedron consists of eight equilateral triangles. There are 40 or more different patterns possible using eight triangles. It’s possible—but tedious—to generate all of them and eliminate ones that don’t work. But if we’re looking for just one or two, we can use what we just did for tetrahedron pyramids and apply it to octahedrons.

For instance, an octahedron can be thought of as two square pyramids put together.

So, if we can make two tents of four triangles, we can put them together to make an octahedron. A group of four triangles that makes such a tent is pattern 1B.

If we make two of these and attach them—outside edge to outside edge—we’ll have a net for an octahedron. There are six different ways to do this:


3A	3B	3C

3D	3E	3F

So, there may be several more nets for the octahedron, but using the “tent” we were able to find six different ones very quickly.