Google SketchUp
Math Project:
Grades 10 - 12

It's amazing what you can make when you start with the famous rectangle known as the “golden section.” Here are three such objects.

• View this project as a printable PDF.

• View the entire Teacher Guide as a printable PDF.

If you need some basic information on downloading Google SketchUp, or about its user interface and basic tools, please see our printable “Intro to SketchUp” PDF.

The PC version is at www.3dvinci.net/SketchUp_Intro_PC.pdf.
The Mac version is at www.3dvinci.net/SketchUp_Intro_Mac.pdf.

1. Use the Rectangle tool to draw a golden section on the ground (red-green plane).

2. It's easiest to complete this project if the rectangle is centered at the origin. So use Move to place it there. (If you need, you can make a dummy line to help while moving, then erase it.)

3. Rotate-copy the rectangle once, then rotate the copy, to get this. (Having the center at the origin makes rotating much easier.)

4. Repeat the last step in the other direction. Now you should have three golden rectangles that are all centered at the origin.

5. Use Line to connect a corner point of one rectangle to the two corners of the edge closest to it. This makes your first triangle, which is equilateral (all sides equal).

6. Create the rest of the triangles the same way. There should be 20 triangles in all. You've just made an icosahedron.

7. Here's a challenge: color the icosahedron using only five colors, so that no color is repeated at any corner.

8. Make a copy of the icosahedron, and on this copy erase each of the 20 triangles. You will have this object, where each “main” face is a pentagon with a star sticking out of it. Another challenge: color these faces using only six colors, so that each pentagon / star face has no repeating colors.

This object is called a “great dodecahedron” (http://mathworld.wolfram.com/GreatDodecahedron.html). Another great exercise is to “unfold” it, using the methods described in the Math Project for Grades 6-9.

9. We’ll do one last thing with the icosahedron (make a copy or use the original one). On one of the triangles faces, make a hexagon. To do this, you can divide each triangle edge into three segments, by right-clicking on each edge and choose Divide. Then you can connect the endpoints of these new segments.

10. Copy this hexagon to the other 19 faces. Use the Rotate tool for this, using the click-and-drag method to set the rotation axis. Remember to use the Ctrl / Option key to make copies.

11. You now have five sets of triangles between each ring of five hexagons. Erase the edges of these triangles, and use Line to replace the face where you're left with a hole. This new face is a pentagon.

12. Do this at every corner, until you have filled in all 12 pentagons. Behold: a soccer ball! (Or buckey ball, or football if you're in Europe.)

There is no limit to what you can create with a set of three golden sections. And for more advanced users, if you understand about components and scaling, you can complete these projects a bit faster.

If you like this project, please check out 3DVinci's GeomeTricks Series. The books shown below are appropriate for this age group, and teach how to create Platonic and Archimedean solids, as well as some amazing derivative compounds and other objects. Click each book graphic for more information.  

       

And if you want to receive three ready-to-use projects like these each month, sign up for our
SketchUp Projects of the Month subscription. Each month, you'll get one project related to geometry, and two projects that teach a 3D design concept.

Also, be sure to check out our free geometry projects at www.mathforum.org/sketchup!

All of the models in the Teacher Guide can be downloaded from Google’s 3D Warehouse: http://sketchup.google.com/3dwarehouse.

• Click this link to see the entire Warehouse collection of models used in the Teacher Guide.

• Click this link to locate this exact golden section model in the Warehouse.

If you’re reading this as a printout and can’t click on the links above, here’s how to find the Teacher Guide models:

1. Open the 3D Warehouse.

2. In the Search field, type “3dvinci teacher guide” and click Collections. Then press Enter or click the Search button.

3. Open this link:

 

4. In the collection, find the model you want, and open or download it.