Nets in a Geometrical Sense

Date: 03/07/99 at 19:15:11
From: Abby Shields
Subject: Nets in a "geometrical" sense

In our geometry project, we are supposed to draw the "net" of various 
shapes. What is the net of a shape? 

Thank you.

Date: 03/08/99 at 12:00:14
From: Doctor Rick
Subject: Re: Nets in a "geometrical" sense

The net of a polyhedron (a 3-dimensional shape made up of flat faces) 
is a plane diagram that shows how the edges of the polyhedron are 
connected. The edges in the net should not intersect.

You can picture making a net by making a hole in one face of the 
polyhedron, then stretching the hole out as if the polyhedron were made 
of very stretchable rubber, and flattening the whole shape onto your 
paper. It does not matter if the edges change their length; sometimes 
they even have to curve, and that is okay too. You just want to show 
which other edges each edge meets.

For example, here is a cube:

            D_____________ C
            /            /|
           / :          / |
          /  :         /  |
         /___:________/   |
        A|   :        |B  |
         |   :        |   |
         |   .........|...|
         |  . H       |  / G
         | .          | /
         |.___________|/
        E              F

Here is a net of the cube, with the vertices labeled. I put the "hole" 
in the bottom face, EFGH.

    H_________________________ G
    |\                       /|
    | \                     / |
    |  \                   /  |
    |   \                 /   |
    |    \_______________/    |
    |   D|               |C   |
    |    |               |    |
    |    |               |    |
    |    |               |    |
    |    |               |    |
    |    |               |    |
    |    |               |    |
    |   A|_______________|B   |
    |    /               \    |
    |   /                 \   |
    |  /                   \  |
    | /                     \ |
    |/_______________________\|
   E                           F

You can see that these lines divide the plane into 6 regions, matching 
the 6 faces of the cube. Five of these are quadrilaterals like EADH. 
The sixth is the region outside the figure (that is, the whole plane 
except what is inside EFGH). This matches the face that I put the hole 
in.

I hope this helps you complete your project.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/   

Date: 03/09/99 at 09:12:58
From: Doctor Peterson
Subject: Re: Nets in a "geometrical" sense

Hi, Abby. I saw Doctor Rick's answer to your question, and want to add 
something to it.

I think the kind of net your teacher wants may be slightly different 
from what Dr. Rick told you about, which is the "topological" version 
of a net (that is, it allows you to stretch things and ignores the 
actual shape and size of the faces of the polyhedron). Here is a page 
that shows you several nets in a "geometrical" sense: the shape is 
flattened out not by stretching, but by cutting apart along the edges. 
You can cut these nets out and actually build the shape. Doctor Rick 
and I (twin brothers!) used to make a lot of these when we were kids:

    http://mathforum.org/alejandre/workshops/net.html   

There is a lot on the Web about making polyhedron models, but it is 
hard to search because the word "net" is a little too common! But this 
is probably enough to get you going.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/