Representing Figures in Two & Three Dimensions

Instructor: Garrett Taft

Garrett has taught college level mathematics and has a master's degree in Applied and Computational Mathematics.

How can you made a 2D representation of something in 3D? What about the other way around? In the lesson, well use a polygon, polyhedron and a net to see just how this is done.

Exactly Wrapped Gifts

It's Susan's birthday. Before cake and unwrapping presents, she looks at her pile of gifts all neatly wrapped. Most were simple boxes, rectangles on all sides, but there were two that caught her eye.

A soccer ball is made up of hexagons and pentagons
soccer ball

One was exactly wrapped to shape and was clearly a soccer ball. Who could mistake the familiar six sided hexagons and five sided pentagons? The other one, probably a basketball, was not so nicely wrapped. She started pondering why some objects could be more neatly wrapped than others.

To answer this question we will need to talk about nets.

Boxes and Nets

Before directly diving into the problem at hand, let's start with the humble box. A polyhedron (pl. polyhedra) is a 3 dimensional figure that has many faces made of polygons (2 dimensional shapes) like triangles, rectangles, pentagons, etc.

The general shape of a box is a polyhedron with 6 rectangular faces

So if Susan's dad, Steve, wanted to wrap a sweater that was in a box using only one connected piece of wrapping paper, without overlapping, he could lay out the wrapping paper and draw the box as a net.

In geometry, a net is a way to represent a polyhedron on a 2D surface, like the wrapping paper, by drawing all the faces as one connected polygon. You could then cut and fold it to make the 3D object that it represents. Nets are a useful way to study polyhedra, as it allows you to easily build a model.

Since a box has six sides, the resulting net will need to be made up of six rectangles. In order to draw the net of the box, Steve can place it on the wrapping paper and trace the outline of each face and side by rotating the box along the paper. In the end, Steve would have something like this:

box net

All Steve would have to do is cut out this net and fold along the lines, and the box would be neatly wrapped with no overlapping wrapping paper.

Alternate Nets

Could Susan's father have drawn a different net and still have wrapped the sweater? The answer is yes. Nets are not unique, meaning that there is more than one way to represent a polyhedron with a net.

Other net options for the sweater box
other box nets

Although there are multiple correct ways to draw nets for polyhedra, there are also incorrect ways. Some of the common ways to incorrectly draw nets would be to either not have the correct number of faces, or to draw one or more of the faces in such a way that if you were to try and fold the net, you would end up overlapping faces.

Two incorrect nets for the sweater box: one five-faced (left) and one that would overlap upon folding (right)
box nets wrong

Recognition From a Net

Suppose that before Susan's birthday she were to see just the outline of a net for one of the boxes. Could Susan guess the shape of gift? Could this possibly help her guess what she is getting? Let's find out!

Susan sees this shape traced on the back of the wrapping paper:

What 3D shape does this 2D shape indicate?
toblerone net color

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