Burning Man 2013 Honorarium Project / Impossible Triangle Now Made Possible

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.   

Kickstarter is Live

The Burning Man organization’s honorarium is designed to partially fund the project. They also expect the rest of the community to rally around the project and take it the rest of the way there. Therefore, we’ve put together a Kickstarter to get us the rest of the way.

Be sure to watch the destructive testing of the cube in the second half of the video.


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.

Everybody likes making cubes

How does one assembly the twelve edges of the cube, along with four diagonal pieces?  First, get yourself a three-way right-angle fixture and weld up four corners:

Now, each of three cubes has four faces with diagonals, and two adjacent ones that are open.  The seven tubes surrounding the open faces get 0.12″ wall, and the others get 0.08″.  That means that each cube gets one vertex with all thin, one will all thick, and two with two thick and one thin.  Blake has to put all that together, along with the diagonals.

Blake doesn’t much like to talk about what happens next.  I just inspect the results and make sure that everything is within tolerance.  Everything important on the cube is within 1/8th.  (That’s a 40″ cube with 1.5″ OD.)

What is important is that we have a nice square corner and well-placed flanges, which we’ll talk about next.


More Beam Making

Thought I’d show you the setup for notching beams.  Went to check the automatic uploads from my phone to find that somehow something had automatically made an animated gif of Surly working the Bridgeport:

A few observations:

  • The rotary table and right angle unit belong Surly’s shop-mate and patriarch to the Boston burners, Doug Ruuska.
  • We had some crazy ideas for how to cut these before, such as using the type of heavy-duty hole saw used by some tube notchers, but they did not cooperate.  This one relatively cheap end mill, however, is delivering the goods.
  • Some of you might have noticed that piece of square tube securing the beam in the vice with an hex key.  That’s a jig to orient the beams for the cube corners.  Once clamped into place, he can rotate the beam to get new notches at right angles.
  • The most observant of you might have noticed that the beam Surly is notching is way too short to be part of the structure.  You are correct.  What it’s for, however, will have to wait for another post…

Plasma Cutting the Flanges

Dewb, Jacob, and Max all have space at Artisan’s Asylum, the makerspace of all makerspaces, right here in Somerville MA, less than a mile from Surly’s shop.  Surly also seems to have access to  Artisans’ as I think he has placed some of his gear there.  Something about a forklift, too.  In fact, Surly seems to have something on loan to every cool outfit in the greater Boston area, somehow entitling him to VIP status wherever he goes.  And he might have had something to do with the fact that Artisans’ now has a large CNC plasma cutter.

And Dewb just got checked out on this thing.  Here’s Dewb, astonished that it actually worked:

You can’t look at it with your naked eyes, but there’s nothing wrong with a little video:

And fine, a rare photo of your intrepid narrator.  That expression is my realizing that there is an outside chance I’ll still be a free man at the end of the summer.


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…