ADVANCED GEOMETRY LAB:
CONSTRUCTING TEMPLATES FOR POLYHEDRA
A polyhedron is a 3-dimensional solid with polygonal faces. In Part One of the lab, we will make
polyhedrons called Platonic solids. In Part Two of the lab, we will make polyhedrons called
Archimedean solids.
PART ONE: PLATONIC SOLIDS
A Platonic Solid is a polyhedron whose faces are all congruent, regular polygons meeting at each
vertex in the same way. An example of a Platonic solid is a cube. A cube is made of 6 congruent
regular quadrilaterals (squares) where each vertex “looks the same” when you rotate the 3D
figure. To see a spinning cube, go to mathisfun.com and type “spinning cube” in the search box.
Note how each vertex “looks the same”.
It would be nice if we could make a physical model of the cube. To do this efficiently, we can
use the custom square tool that you made in the most recent lab. If you open the file “Regular
Polygons” that contains the square tool, you will have access to your square tool in any new file
you open. So, let’s create a cube:
1.
Open the sketch “Regular Polygons” from the last lab.
2. Open a new file.
3. In the new file, click on the tool for Custom Tool and select the square tool.
4. Create 6 squares to form a 2-dimensional net for a cube. A “net” is a 2D
representation of a 3D solid which can be cut out and folded to make a physical
model of the 3D solid. Make sure to “attach” each square to the one beside it.
5. Once the cube net is complete, you can drag vertices on the first square that you
made to resize the net. Save this file as “Polyhedron Nets”.
6. If you would like to cut out your net and fold it to make the cube, feel free to print
the net. But, first, make sure to choose Print Preview and then Fit to Page.
There are only 5 Platonic Solids (proven by Euclid over 2000 years ago). (Recall, a platonic solid
is a polyhedron whose faces are all congruent, regular polygons meeting at each vertex in the
same way.) One is the tetrahedron (demonstrated in class). A second is the cube. The final
three are shown below:
Octahedron
Dodecahedron
(8 regular triangles) (12 regular pentagons)
Icosahedron
(20 regular triangles)
Choose one of the three Platonic Solids shown above and make a net using either the equilateral
triangle tool or the pentagon tool (made in the last lab). Follow these steps:
1.
In your Polyhedron Nets file, go to File and then Document Options and then Add Page
and Blank Page. You will now see a new page in your file. (You can go back to your first
page anytime you like using the page tabs on the bottom.)
2. Use a custom tool to “lock” triangles or pentagons together to create a net for one of
the shapes above. (If you are having difficulty with visualizing how to form your net,
feel free to go to mathisfun.com. In their search box at the top, you can type Platonic
Solids. (However, some of you may want to figure this out without the help of the
website.)
3. Once the net is complete, you can drag vertices on the first shape that you made to
resize the net. Click Save. You should still be in the Polyhedron Nets document.
4. You may want to cut out the shape and make sure it can fold up correctly to create the
solid. If so, make sure to use Print Preview and Fit to Page before printing.
PART TWO: ARCHIMEDEAN SOLIDS
An Archimedean Solid is also a 3D solid composed of regular polygons where each vertex “looks
the same”. But, an Archimedean solid allows for more than one type of regular polygon. (One
note: Archimedean solids are also “convex”, meaning that there are no “indented” vertices.)
There are just 13 Archimedean solids, shown below. (Note: One of them is the pattern for
making a soccer ball! The images and names are from uwgb.edu.)
1. In your Polyhedron Nets file, go to File and Document Options, then Add Page and then
Blank page. Choose one Archimedean solid that interests you and create a template for
it using Geometer’s Sketchpad software and your custom tools. (Nets are available
online. One site is mathisfun.com. Type Polyhedron Models in the search box at the
top. Or, just try to figure out how to construct the net on your own.)
2. Once you have completed your net, create a text box and type your name in it. Then, go
to File and then Print Preview and then Fit to Page. Click Print.
3. Cut out and fold up your Archimedean solid! To hold it together, try to keep the tape
hidden on the inside surfaces.
To score your lab, I will simply be going into your H: drive and opening your Polyhedron Net file.
I will need to see three nets: The cube on page 1, your selected Platonic solid on page 2, and
your selected Archimedean solid on page 3. And, I will drag vertices to check that all polygons
are “attached” to each other.
Bonus Opportunity:
Choose one Johnson solid (via internet search) with at least 7 faces and create a
template for it using Geometer’s Sketchpad software. Save the Johnson solid template
in the Polyhedron Nets file (perhaps as new page). Then, in a textbox, explain what is
different about a Johnson Solid as compared with the Platonic and Archimedean Solid
(in your own words!).