Making a Frabjous

George W. Hart is a professor at Stony Brook and is one of our favorite artists, making a wide variety of stunning geometric sculptures. On his of his many works that has particularly captivated us for some time is a sculpture called Frabjous.
When we realized that George had posted a template for this sculpture we dropped everything, grabbed the cardboard and hot glue, and raced to build our own.

You’ll need papercraft type building materials: Paper, cardstock, or cardboard, and tape or glue. Also good scissors and/or a hobby knife with sharp blades. You can also build this with wood, plastic, or other materials, of course, but cardstock and cardboard are inexpensive and effective. Hot glue also proved to be excellent, providing sufficient strength and flexibility, and good working time.
You can download the PDF template for Frabjous on its web page.
   
The sculpture is made out of 30 identical pieces, with this S-curve shape. (We’ll look at the reason for the shape a little later.) The design is scaleable: print out the template at the size that you would like to make it. Larger means more time cutting but it might be easier to assemble.
   

Cut out your template at your scale of choice. You can directly make 30 of those as a paper model or trace and cut out the model, like we did in cardboard.


You need 30 pieces so this can *ahem* take a while. The good news is that the sculpture will still work even if your pieces are banged up, have lost some stiffness, or have occasional mistakes.
Big hint for cardboard: replace your blade often. (Here is where we get ours.)
Once you have all the pieces, then comes the challenging part, putting it all together.

The first hint about construction is to look at the corners. Every intersection between the pieces is attached the same way: the flat ends are glued together, in a symmetrical arrangement. Unfortunately, this only gets you so far, and constructing the rest from that principle is a true puzzle indeed. To understand which pieces should go where, it’s helpful to look at the design from a geometric standpoint.

The parent shape donating its symmetry is a simple dodecahedron.

Within that shape we inscribe a line segment between two vertices: this is where our “puzzle pieces” will go– each will go where a line segment is and connect two vertices of the dodecahedron.
Each vertex of the dodecahedron is also a vertex of three bordering pentagons. To draw a line segment, pick one of those three pentagons, and connect the vertex to the corresponding opposite vertex of the neighboring pentagon as shown. This is probably the single most important thing for understanding how the sculpture goes together– that each piece fits in that relation to the overall shape.
Now, we’ve only drawn segment coming out from that top vertex, but obviously there are three such possible segments because there are three pentagons that touch there. This intersection of three “puzzle pieces” at the corners is exactly what we saw in the completed sculpture, where three pieces get glued together at each intersection.
Now what happens when we start drawing the segments coming out of more than one vertex?

Here we’ve drawn one segment originating at each vertex of the top face. Unfortunately, this shows that our model– connecting vertex pairs as we have, with simple segments– leads to a problem. All five of these puzzle pieces would intersect.
The solution is to “bend” the segments to avoid the intersection:

Here’s a corrected curvy “segment” that will hopefully avoid bumping into others as we start to add more to our sculpture.

Yes– looks like we’re now avoiding intersection.

Taking a look at the model from the top, we can also now begin to see the “vortex” shapes from the original sculpture start to appear.

Adding the rest of the curves, between every relevant vertex pair, the sculpture emerges. In particular, note that we can now see the three pieces that come together at each vertex of the dodecahedron.

Without the framing dodecahedron, the shape looks more mysterious once again.
We have the geometry now, but we still need to replace our curves with flat shapes to begin to approximate the appearance of Frabjous.


It’s a slightly crude drawing, but it might be helpful in seeing how the real-world version fits together.

You can see more pictures and renderings of this sculpture in this flickr photo set.