Burning Man 2013 Honorarium Project / Impossible Triangle Now Made Possible
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Credit where Credit is Due

I work for an incredibly cool software start-up, GrabCAD, which makes engineering collaboration tool called Workbench.  Workbench is the center of our design infrastructure, and is the primary way that we collaborate with each other outside of email and shouting over heavy machinery at the shop.

The Penrose Triangle in Workbench

When we started the project, Workbench was in early development and was fairly limited.  Earlier this month, we released is as a commercial product.  As a product manager, using it in the real world and learning from our experiences has been incredibly valuable.  Additionally, the GrabCAD as a company has been incredibly supportive, and company resources such as our 3D printers have helped tremendously.

For those who are interested, I shared my experience using GrabCAD Workbench on this project on the GrabCAD corporate blog.  For anyone who gets excited about design collaboration, I think you will find it an interesting read.

Also, I would like to thank SpaceClaim Corporation, a company I helped found in 2003 and for whom I worked for ten years, who graciously provided me with a license of their incredibly powerful direct modeler after I departed in January.  SpaceClaim is the primary 3D CAD tool used on this project.  Almost all of the CAD shots you see of the project were generated in SpaceClaim.

One of many screen shots from SpaceClaim

For simulation, we relied on the Autodesk stack.  I’ve had early access to Autodesk Fusion 360 since January, and I used its Simulation 360 tools to perform all of the structural analysis.  In addition, I used Autodesk’s free Labs product Falcon to estimate wind loading on the structure.

Simulation results from Autodesk Simulation 360

Although Burning Man fosters a non-commercial atmosphere, the reality is almost everything at Burning Man has an original commercial source.  These companies have let us use their incredible technology at no cost to the project, and this project could not have happened without them.

Designs that design themselves

In the last post, I mentioned that the design was determined from the level of subdivision.  Let’s walk through how that happens, as there are a few fun twists and turns.

Last time, I showed the first three items of the sequence of subdivided Penrose triangles.

We’ll choose the second for this exercise.  The actual sculpture uses the third, so we’ll end up making our triangle’s little brother.

Corner Cubes

The first step is to create properly-oriented corner cubes from their hexagonal silhouettes.  Let’s draw it’s cross section, which is the hexagon inscribed on the midpoints of the edges of the silhouette.  

All CAD work performed in SpaceClaim.

We need to reconstruct the faces of the cube from those six cyan lines, each of which represents a face.  Let’s start by getting those faces in 3D.

Three of those side faces on the hexagonal prism will tilt one way, and the other three will tilt the other, all rotating along their cross section lines, to form the cube.  What’s the correct angle?  Here’s the cube in a delightful cross with a right triangle with the edge length (l), the length of the face diagonal, and the length of the cube diagonal.

We then give that angle to the CAD tool’s draft command, which pivots face geometry about a neutral plane, for the heavy lifting.  First the top set:

It’s almost hard to see the top three edges of the cube beginning to form.  Do the exact same trick to the bottom set, and like magic:

And there’s our cube in exactly the right position.  Taking advantage of symmetry and viewing from the front:

Inserting the arms

We now need to make a design decision: What is shape of the arms?  What is the obvious thing to do?

We know the start and end points of the arm beams, so we really just need to agree on an interpolation technique.  Linear and spline interpolation are fast and easy, but observe that introducing arcs leads to an elegant solution.  There is exactly one pair of tangent arcs that are tangent to both cubes for the bottom-most beam.  This pair of arcs is symmetric, which matches the needed symmetry.

If we continue from there, we can copy curves to the middle position and the top.  Observe that the middle pair of arcs’ ends are both touching and  perpendicular to the faces of the cube.  Therefore offset arcs will also touch the cube.  We then generate the actual curves by offsetting those arcs by half of a face diagonal in each direction.

Lofting between these curves provides the arm surfaces.   Note that each face is defined by concentric, offset arcs, so they are all cones and are therefore are developable (able to rolled from flat patterns).  

Back to the symmetric view:

Locating the beams

We should be sufficiently pleased that we have constructed the triangle out of cones, but now we need to turn it into a truss structure.  Let’s extrude our 2D beam pattern to see where it intersects the arms.

The intersection curves aren’t straight, but they are close enough that our straight beams will be sufficient approximations.  We can draw them in place:

Notice that the arm has some symmetries we can use.  The beams on the bottom are a reflection of the ones we just drew.  We can also rotate them 180 around the center of the arm to get the other side.  Adding all the symmetry gives us:

Or in our symmetric view:

The rest is just a sizing exercise and a lot of detailed design.

 

Counting the Truss Triangles

Designs that have relatively few degrees of freedom have always been attractive to me. Perhaps one of the most rewarding aspects of designing the truss model of this illusion is that it almost completely designs itself. There are very few arbitrary choices for the engineer to decide. In fact, there are exactly as many choices for the truss model of the Penrose triangle as there are for geodesic domes, which makes this triangle some sort of kindred object to the domes of the playa.

When building a geodesic dome, there are thee variables you need to decide:

  • Dome diameter
  • Beam type and size
  • Level of subdivision
The dome diameter is continuously variable; you can make it any size you want, provided you can find beams long enough.  The beam type and size is limited to your supply resources.  The level of subdivision is an integer.  Here is an example of the subdivision incrementing [1]:
Images derived from the dome calculator at Desert Domes.
Similarly, for the Penrose triangle realized by a truss structure, there are three variables
  • Size of triangle (continuously variable)
  • Beam type and size (as supply allows)
  • Level of subdivision (integer)
And here is how the subdivision increments for the triangle:
From here, the location of the vertices is fully determined if you decide that you want the arms to be made out of arcs (up to handedness).  Maybe I’ll walk through how to build it in 3D in a future post.
[1] Note that for the purpose of illustration I’m skipping odd geodesic subdivisions, for which the typical hemisphere domes don’t get divided down the middle.   
 

What you can do with an impossible triangle

Was reading some of Melinda’s copy about the project, and it reminded me that there was this one other time I flirted with the genre.  Way back in January, 2008, I made this fun video showing off SpaceClaim, CAD software I was working on at the time.  

It takes like half the video for me to get through the CAD explanation, but that’s when it gets fun.

Tubes.

Surly and Andrew aka “Shop Monkey” have started banging out some tubes.

Cube edges with 0.083″ wall are in the foreground, 0.12″ wall is in the background, only one of which has been prepped.

The triangle requires four types of beams on the cubes, fourteen types of straight beam on the arm and three types of bent beam.   All of the straight beams come in mirrored pairs except the thin and thick cube edges.  Here’s how the arm breaks down:

The beams need to get  a 1.5″ diameter notch where they meet the bent sections.  The other cuts are miters.  The result is a lot of fussy little cuts:

Each beam has 2-4 miters and notches per end, creating 28 different unique types of ends for the beams in the arm alone.  Each cut had three degrees of freedom, so that’s a lot of information to communicate.  The solution?  Stickers!

Now if we can only figure out how to feed the sticker paper the right way into the printer…

View the Penrose Triangle in 3D

Want to explore the Penrose Triangle in 3D?  I’ve shared the full Penrose Triangle model on the GrabCAD Workbench, the CAD collaboration that also happens to by my employer. By day, I work on the product side—this 3D viewer is one of my current projects.

I will never get salesy on this blog, but I will say it is incredibly cool to be working on a collaboration platform by day that I then put our team’s own work on by night.