Welcome to Math Craft World! (Bonus: How to Make Your Own Paper Polyhedra)

Welcome to Math Craft World! This community is dedicated to the exploration of mathematically inspired art and architecture through projects, community submissions, and inspirational posts related to the topic at hand. Every week, there will be approximately four posts according to the following schedule:

• Monday: Highlights from member submissions to the community corkboard.
• Tuesday: Introduction to the new project of the week.
• Thursday: Extensions, inspiration and more mathematical details for the current project of the week.
• Friday: Inspirational posts about artists and artwork in the field, including historical projects and works.

My goal is to host a public forum in which people can learn, participate and contribute. With that said, please post anything of relevance in the comments section of posts, the community corkboard, or start a thread in the forum. I'm hoping the community will learn even more from each other than from my posts.

Since this is the first post, and future Mondays will be dedicated to presenting community submissions, I'm going to go off schedule and share a simple DIY project for exploring the basics of geometric art.

Paper Polyhedra

Polyhedra are the three-dimensional extension of two-dimensional polygons. They are composed entirely of flat faces and straight edges. Since they are made entirely of flat faces with straight edges, you can often unfold them to a two-dimensional shape, as you would with a cardboard box. This unfolding of the polyhedron is called a net. One of the easiest ways to make a three-dimensional shape is by making the net out of paper and folding it.

To show what amazing forms can be made from paper—using techniques similar to folding nets—I present some images of work by Father Magnus Wenninger.

These are truly amazing geometric shapes. The objects in the last group are actually three-dimensional projections, or shadows of objects that can only exist in four dimensions! Some day, perhaps, we'll take a look at these polychorons in detail.

For now, let's look at nets for folding up simpler paper geometric objects. Gijs Korthals Altes has a great site for finding these nets. All you have to do is download the object, and then use your printer to print it out on regular paper or card stock. Next, cut out the shape of the object and fold as directed, and then glue or tape the object closed.

I suggest starting out with the Platonic Solids, which are the simplest of polyhedrons. They are composed entirely of regular polygons of the same size and shape, and are convex so that all angles bend towards the shape's center. The templates (nets) can be found here. Once you've exhausted the platonic solids, I suggest the Archimedean Solids, which can have more than one type of polygon. Those nets can be found here.

Finally, to make really cool Christmas ornaments, you should try some convex polyhedra like the Kepler-Poinsot polyhedra (download here). Please note, these are significantly more difficult and time consuming. Some involve hundreds of folds. The great icosahedron, while beautiful, took me close to 3 hours to cut and fold out of 1 piece of paper.

Let's go through the process of making one of these in a bit more detail. We'll start with the dodecahedron.

Materials

• Something to cut with (scissors, or Exacto knife).
• Cutting mat or board, if using an Exacto knife.
• Paper or card stock. I use the thickest my printer can handle, so I can make stronger objects, but this does make it more difficult to fold. You can buy 110 lb card stock at any office supply store. Usually mass retailers such as Walmart or Target carry thinner cardstock, which might be preferable for some of you.
• A scoring tool like a blank pen or the back of a table knife. You only really need this if you use card stock. I actually just lightly use the blade of an Exacto knife, but this takes precision, so be careful if you use this technique.
• Tacky craft glue, super glue gel or tape.

Download, Print and Cut

Go to the download site and find the polyhedron you wish to build. To follow along with me, go to the section on platonic solids and download the template for the dodecahedron. I recommend the .pdf files, and I usually print the color page rather than coloring my own. Print out the net. Cut it out using scissors or knife.

Make sure you cut the space between the tabs and the other polygon it starts off touching. Only leave the tab connected to the main polygon.

Score

If you used card stock, apply pressure with your scoring tool across the edges and tabs that are to be folded. This makes the paper weaker in these spots, allowing the folds to be near perfect. If you used paper, the paper will fold fairly easy without this step. It can help, but it can also destroy the paper if you're not careful.

Fold and Adhere

Fold and glue, or tape, the object together. I usually partially fold all of the edges and tabs so that I can see how the object is going to be formed.  

Put glue on the tab you wish to glue.

Press the tab together to the polygon it is supposed to attach to. For superglue, you only need to hold for a couple of seconds. For other glues, you might need to hold for close to a minute.

Repeat the placing of the glue on the tabs and pressing the pieces together until you're done. The final object should look like this:

You're Done!

Now go play with your growing polyhedral collection. Come up with ways to use them in other projects. You could use them as Christmas ornaments perhaps. Have fun and post the results to the corkboard!

Here's mine: This weekend I decided to use some polyhedra to make a mobile for my newborn son. Here's a quick picture of what I made using paper polyhedra, some 1/4 inch copper tubing, and some copper speaker wire.

Lastly, here's the my most helpful tip: Don't get too frustrated. Some of you will be able to do this easily right away. Others might take quite a long time to produce something that looks like an elementary school child made it. Your hands may be covered in glue. This is normal. Like everything else, this might take some practice. It did for me. Have fun. And know that it is addicting. You might end up with a house, apartment, or dorm room completely full of these.

On Tuesday, we'll learn how to make polyhedra out of ordinary playing cards like this: