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Math Craft

How to Make Icosahedral Planet Ornaments

Dec 13, 2011 06:36 PM

In honor of the new Astronomy World, I thought we should look at a few planetary icosahedrons. The icosahedron is the most round of the Platonic solids with twenty faces, thus has the smallest dihedral angles. This allows it to unfold into a flat map with a reasonably acceptable amount of distortion. In fact, Buckminster Fuller tried to popularize the polyhedral globe/map concept with his Dymaxion Map.

The other advantage of an icosahedral globe is that it's quick and easy to make, only requiring a paper net. Here's a picture of the planetary icosahedra I made last night—in less than a hour.

Geometric models representing planets in the solar system.

Tools and Materials

  • Paper (1 sheet per planet)
  • Scissors or X-Acto knife
  • Templates
  • Glue (I prefer Loctite brand superglue gel for its speed and ease of use)
  • Ornament hooks (if you want to hang them)

How to Make Icosahedral Planet Ornaments

Pick out your celestial template. On the template site, click on one of the planetary net images on the left side. This will take you to a new page where you can download either a .jpg or .pdf. Print out whichever one you prefer. I used glossy photo paper, which looks great and is stiff enough so that the ornament should survive a fair amount of handling.

World map displayed in a zigzag pattern.

Carefully cut out the net.

Illustration of a folded paper model depicting a world map.

Glue the tabs, connecting the net together. You can pre-fold at the lines if you want, but I opted not to in this case. I wanted to let the paper curve, which makes the globe appear more round.

Folded paper representation of a world map highlighting continents and oceans.

Keep connecting the tabs with glue.

Folded paper representation of a world map.

Here you see there's only one more top tab to be connected.

A colorful paper model of the Earth with a torn section.

And now the last tab on top has been connected.

3D model of the Earth with blue and green colors representing land and water.

Now it's time to connect the side tab.

Origami figure made from colorful paper on a grid background.

Connect the tabs on the bottom.

A blue and white geometric paper model on a grid surface.

Almost finished.

Geometric paper model with a blue base and white top.

And... finished.

3D globe model with detailed continents and oceans

Now I'll make the moon, only to show you how to make the folds crisp, instead of round.

Moon topography illustration.

After cutting out the net use a straight edge as a guide to folding. You could probably fold directly across the straight edge. I instead used a light touch from a knife to slightly weaken where the folds are going to be before folding. You could also use a dead ball point pen to do this.

Start from one angled line on the left, then continue along all of the parallel folds.

Protractor measuring angles on textured surface tiles.

Then do the same thing the other way, making sure to hit all of the folds.

Protractor and paper templates on a drafting surface.

And finish with the top and bottom lines.

Protractor placed next to triangular paper cutouts on a cutting mat.

As you can see below, the folds are now crisp.

Once you have pre-folded the icosahedron, you can start gluing the tabs together.

Crumpled piece of paper on a gray cutting mat.

Here's the finished product.

Pyramid-shaped geometric sculpture on a grid background.

Notice the difference between the two globes below. They both look great. Folding results in the more mathematically pure form, but takes more work. Letting the paper bend results in a rounder globe and is faster to make.

Geometric shapes depicting Earth and a textured surface.

Show Off Your Work

If you make the icosahedral planets or any of the other previous Math Craft projects, please share with us by posting to the corkboard. I'd love to see these as Christmas ornaments hanging from a tree, or maybe you can figure out how to make them into usable boxes by adapting them with a lid. Perhaps you have an original project or something you've seen on the web that you'd like to share.

If you like these types of projects, let me know in the comments. If you have any other ideas you would like to pursue, let me know in the forum.

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