## Introduction: Geodesic Paper Disco Ball

I normally design pop-up cards, but lately I've been captivated by a 3D form which will not collapse: the geodesic dome. From lampshades, greenhouses, to massive event spaces, triangles carefully assembled together will create a deceptively simple sculptural form. In this instructable I am using a geodesic dome to make a paper disco ball with 320 facets -- but no worries, you'll only need to cut, fold and assemble 14 pieces to make your own disco ball!

This template can be made with 6 sheets of standard paper, and will create a sphere with a diameter of 20 cm (almost 8").

## Supplies

Template (download for free from www.makepopuocards.com)

High tech (recommended): digital cutter -- low tech: scalpel knife (aka Xacto knife) and cutting mat.

(Please note: I am an Amazon affiliate, which means if you purchase an item using these links I may get a small portion of the proceeds -- but the price you pay will be the same).

## Step 1: The Geometry

Skip to step 3 if you just want to make the damn thing.

For me, the purpose of the exercise was to understand the geometry (one of the reasons I didn't use a dome calculator), so I'm including this step in case other non-mathematicians are interested in the process.

There are an infinite number of geodesic domes... One of the simplest, the icosahedron, is a volume composed of 20 equilateral triangles, is a bit angular for a dome, but it still qualifies (it's the globe you see in the background of the first photo). The more triangles which form the dome, the more spherical it becomes, and (important if you are making something big) the stronger its structure.

Domes are generally classified as frequencies 1V, 2V, 3V, etc -- names which refer to how many times the triangles which form it are divided (more details here) -- but I find it easier to think of the domes in terms of their underlying geometry.

Take the icosahedron (1V). Divide each of its 20 triangles in 4 (by halving each side). That's a 2V dome made of 80 triangles. Still a bit lumpy, but definitely a sphere. Do you want to do better?

Take that same icosahedron. Chop off all the vertices (pointy corners). You get a "truncated icosahedron", also known as the classic soccer ball, a sphere formed of a combination of 12 pentagons and 20 hexagons. Turn the pentagons into 5 triangles, the hexagons into 6 triangles, and you've got yourself a 3V sphere, with two different sized triangles, coming to a grand total of 180 triangles. I used this structure to make a lampshade, but for the disco ball I wanted even more surfaces/facets/triangles.

So I started out with a different volume, the icosidodecahedron -- a sphere composed of 12 pentagons linked together with 20 equilateral triangles. At the hemisphere it is a decagon (polygon with 10 equal sides) which is helpful, because it makes it easy to set the diameter of your sphere. Also I'm very fond of this volume, and I've been wearing various versions of it on my face since the Spring of 2020... but I digress. Take those pentagons, divide them into 5 triangles, and then divide all the triangles into 4 (by halving each side)... and now you have your 4V sphere, made of 320 triangles of 4 different sizes! Now at the hemisphere you'll have an icosagon (polygon with 20 equal sides).

There is space here for some esthetic choices: when you split the 5 triangles forming your pentagons, you don't have to half them evenly -- for example if you want to emphasize the star pattern this form creates -- but for the disco ball I wanted my triangles to be as even and regular as possible so I split them exactly in half.

Then I needed to dust off my very rusty high school trigonometry skills to figure out the correct angles and dimensions of each triangle so they curve on the radius of the dome... the trigonometry itself is simple enough (thanks to online calculators, I will admit), the complicated part was figuring out which plane I was thinking about, what angle I was looking for. And there is no room for error. I thought I'd figured it out many times only to be crushed by a 3D model which didn't behave as I'd expected (picture #3). If you're off by 1/2 degree or a fraction of an inch it will look the same on your drawing, but multiply the error by 320 triangles and you won't have a proper closed volume.

## Step 2: The Net

A "net" is the name of a pattern for a 3D volume, and even with a simple volume like a cube there are many options (11, to be precise) so you can imagine that with a volume with 320 facets there are quite a few possibilities! The colorful image above shows one such possibility -- not the one used here, but it's pretty. Plus it shows the 4 different types of triangles which form the sphere, which is harder to see with just the cut and fold lines.

Believe it or not, you actually can draw a net for this dome from one single piece... but it would be a pain to assemble, it would require a huge sheet of paper (with a lot of wasted space/paper) and some of the edges would not be glued together because the slits between the facets would be too small to allow a proper glue tab.

So I made a compromise between ease of printing and fabrication and assembly and structural integrity. I drew a net which can be printed on standard paper and is decently easy to assemble. You only have to cut and fold 14 pieces to make your dome... not bad considering you're making 320 triangles! It is reasonably intuitive to assemble, though gluing the last piece is a bit tricky.

## Step 3: Cut, Fold, Glue

Download the template (for free) from www.makepopupcards.com. It is formatted for letter sized paper (8.5" by 11") but it can be printed on A4 or even scaled up to make a bigger sphere. You will need 6 sheets of mirror paper to make a whole sphere.

You can print the template multiple times to cut and make it by hand, with an Xacto knife and a cutting mat, or you can cut the pieces with a digital cutter. The download includes a printable template with instructions, and a second file with just the die lines so you can import the pieces into your digital cutter software.

If you are making this with a digital cutter, all fold lines should be made with a "kiss cut" -- a very light cut which goes through the mirror layer but not through the thickness of the paper, This will give nice sharp edges to the folds, and it will allow you to peel off the mirror metallic layer on the tabs, so you can glue the form together with regular white glue.

If you are making this by hand, it is more efficient to score the fold lines the regular way, by using a ball point pen to trace the printed lines on the reverse (and pressing down to make a dent in the paper). After you've scored all the fold lines you can flip your sheet over and carefully kiss cut the mirror layer only for the tabs.

Specific instructions on where to glue what pieces are included in the template.

Inevitably, you will smudge the mirror surface as you glue the pieces together. Don't worry about that, if you are using white glue it will wipe clean easily.

Enjoy making your very own geodesic dome paper disco ball!