Geodesic Dome

Published Tuesday, May 12, 2020 by Bryan

By a half-planned chain of events, I've spent the last six weeks of COVID-19 Shelter-In-Place over 2000 miles from my woodworking tools. Instead of diving right into a new construction after my dresser, I cleaned and then packed my shop, in preparation for a move. While our belongings have made their way across the country, we have stayed behind to "quaranteam" with a friend-couple, their young son, and their dogs.

We have entertained ourselves with other hobbies: walks to keep everyone moving, cooking delicious meals, reading books, and making music. A few ideas for construction projects have risen during that time, but with few tools and difficulty acquiring wood while maintaining social distance, none of them have been undertaken.

Then one of us saw a post about a geodesic dome made of cardboard. The shape alone immediately captured the attention of the four Xennial-age adults in the house. When we recognized that cardboard was the one material in abundance here, from six weeks of contactless deliveries, wheels set in motion.

Google found a calculator for ordering bits of PVC based on the size and complexity of dome desired. Reverse-engineering that math led to a very simple cut list for a "2V" geodesic dome of paper:

  • Ten equilateral triangles, with sides of length A
  • Thirty isosceles triangles, with one side of length A and two of length 7/8 * A

Seven-eighths isn't exactly what the calculator produced, but it's less than 1% off, it makes measurement simple, and it has worked in my experiments.

The size of the dome that is built is also related to A in a very simple way: the golden ratio. A compressible, bendable material, like paper and tape, worked with common tools like scissors or a box cutter introduced enough error that using many decimal places didn't make sense. Simplifying to estimating the height at 1.5 * A, and the width (diameter) at 3 * A proved close enough for toy structures.

I'll include some examples of how specific measurements work out later, but before I annoy people by showing how neatly these work out in Imperial units, I'd like to explain how no particular units are necessary at all. Grab a stick or a string, and I'll walk you through how to build your own geodesic dome without any arithmetic.

Step 1: Sizing your dome

Figure out where you want to put your dome. Is it a decoration for your desk, or a fort to play in? Find a piece of string, a stick, a strip of paper, etc. that you can cut to the desired width (diameter) of your dome. Before you cut it, find its halfway point, and hold it up in the approximate middle of where you will place your dome. This is about how tall your dome will be (the dome approximates a sphere, so you get one half diameter up from the ground). When you have found a size you like that fits your space, move on to step two.

Step 2: Making your tools

Cut your string, stick, strip of paper to length equal to the dome width that you chose in step one. Then cut that piece into three equal segments. I used a paper strip for my measuring device, so after cutting mine to the full length, I folded it into thirds, and then cut through the folds:

Label one of the cut pieces "8" (eight).

Cut off 1/8th of one of the other pieces. The easiest way to do this is to first find the middle point of that piece. Then find the point halfway between the middle point and one end. Finally, find the point halfway between that point and the end. Cut through that final halfway point. I folded my paper three times and cut through the third fold to do this:

Label the piece you just cut "7" (seven).

Step3: Equilateral Triangles

If the edge of your dome-building material isn't straight, draw a line on it using a straightedge. Using your "8" piece, mark divisions along your straight edge.

Now for the tricky part. Put one end of your "8" piece right on the left-most mark (or corner) of your straight edge, and angle it up so that the other end is somewhere near where you expect the point of an even triangle would be. Mark a dot on your building material at that end of your "8" piece. Do this a few more times, swinging both a little clockwise and a little counter-clockwise from that spot.

Connect these dots in the arc they form.

Next move the lower end of your "8" piece to the next division mark to the right on your straight edge. Swing the other end up until it crosses the arc you just drew. Mark the point at which it crosses the arc.

Draw a line from each of the division marks you just used to the arc-crossing point you just found. You have just marked your first equilateral triangle!

If you're building a large dome, and/or working with pieces of material that won't allow you to get multiple triangles out of one piece, you can skip the next few steps. Cut out this first triangle you have marked, and then use it as a template to trace out nine more identical triangles.

If you're working with a piece of material that will fit multiple triangles, repeat the arc-crossing process at the right-most division of your straight edge.

Draw a line connecting the points of the two triangles.

Using your "8" piece, divide the line between the triangle tips.

Connect the division markers on your straight edge to the division markers on the line between the triangle tips. You have now marked out many more equilateral triangles!

You will need ten of these triangles. If you've already marked ten, you're done. If you need to mark more, try extending your angled lines farther upward. When they cross, they will either make more triangles or diamonds. If they make diamonds, draw a horizontal line connecting the corners to make two triangles.

Cut out your equilateral triangles. Make sure you end up with ten!

Step 4: Isosceles Triangles

The process for the isosceles triangles is the same as it was for the equilateral triangles with one difference: use the "7" piece when finding the arc crossing. Use the "8" piece, as before, to mark divisions along your straight edge, but use "7" to find the crossing point from there.

You will need thirty of these isosceles triangles. Yes, 30.

Step 5: Pentagon Assembly

Time to start assembly. Looking at a finished geodesic dome, the eye is drawn to two (non-triangular) shapes: pentagons and hexagons. I've had success with assembling pentagons first, so that's what I'll show here.

Collect five isosceles triangles (the ones with two "7" sides and one "8" side). Arrange them in a circle so that all of the "8" sides are pointing out, and all the "7" sides are next to other "7" sides.

Connect four of the "7"-side seams together. A gap should develop in the fifth seam.

Draw the gap together. The pentagon will cup slightly. Seal the seam, and the pentagon will stay cupped.

Repeat this pentagon assembly five more times. You should end up with six pentagons, using all 30 of your isosceles triangles.

Step 6: Connect it All Together

This is where construction will really begin to get unwieldy. If you're building a large dome, I strongly suggest at least one person to help hold. Two if you can get them.

Collect one pentagon and two equilateral triangles (the ones with three "8" sides). Connect each triangle to two adjacent sides of the pentagon.

From here, connect a pentagon into the space between the two equilateral triangles. This will introduce more cupping, like when you sealed the fifth seam in the pentagon. Continue to alternate pentagons, and equilateral triangles, growing this strip until you have only one pentagon left unconnected (you should have five pentagons attached to five pairs of equilateral triangles). Connect the two equilateral triangles at the end of your strip to the pentagon at the start of your strip. You should have a ring that has a pentagonal hole in one side. Tape the remaining pentagon into this hole, and your geodesic dome will be complete!

Apologies for a lack of build pictures of these steps. The pieces pictured so far are being mailed in an envelope as a small birthday gift. But, here they are laid out ready for final taping.

And here is an annotated diagram of what gets taped where. Purple 1-5 are the pentagon seams. Yellow 1-10 are the remaining alternating-pentagon-triangle seams (9 and 10 appear twice to indicate where the wrap-around connects). Red 1-5 are the roof seams (and 2-5 are duplicated to show where the pieces connect.

And one more shot with a completed dome in the opposite color scheme, to aid in visualization.

What next?

If you followed along, I hope your first dome was successful. If you're wondering about the dimensions of the domes in my pictures, they are these:

Small dome, with blue pentagons: A = 2 inches. Isosceles sides = 1.75 inches. Height is just a bit over 3 inches. Width is just a bit over 6 inches.

Small dome, with green pentagons: A = no idea. I purposely didn't measure anything, to make sure I wasn't lying about being able to build this without numbers.

Large dome, made of cardboard: A = 24 inches. Isosceles sides = 21 inches. Height is just a bit over 3 feet. Width is just a bit over 6 feet. The additional ring around the bottom is ten inches tall. We have fit four adults and one child inside. It's close, but not cramped.

Good luck with your next build!

Categories: Miscellaneous