1. EECS Computer Science Division University of California, Berkeley
ISAMA 2004
SMI / FASE, May 22, 2012
Prototyping Dissection Puzzles with Layered Manufacturing
Carlo H. Séquin
EECS Computer Science Division
University of California, Berkeley
Dissection puzzles are cool: They challenge your spatial perception; they often provide interesting sculptural forms; and in realizing them you have to address serious fabrication issue. So they are a good topic for this special track at SMI.

2. Dissection Puzzles A good educational component for UCB “CS 285”:
ISAMA 2004
Dissection Puzzles
A good educational component for UCB “CS 285”:
Graduate course: Solid Modeling and Rapid Prototyping
They train 3-D spatial thinking
“Hands-on” feedback on accuracy & tolerances
Fun artifacts to take home as souvenirs
 Good “motivators”
They also make for a good educational tool in a class on Solid Modeling and Rapid Prototyping. They not only train the student’s spatial thinking, they also give hands-on feedback . . .

3. Hamiltonian Dissections of Platonic Solids
ISAMA 2004
Hamiltonian Dissections of Platonic Solids
To get the students started, I show them some simple examples of dissection puzzles. One set of examples are these Ham. Dissections of Platonic Solids. In all cases the parting surface is created by a line anchored at the centroid and then swung around along a Hamiltonian edge-cycle until it meets up again with its staring position.
As far as puzzles go, these dissections may be trivial, but they still generate nice geometrical forms. The only slightly tricky puzzle is the split Icosahedron. It has 3 prongs on each part, so it is somewhat difficult to grasp, so that all fingers of one hand touch only one of the two parts. You may play with this model later…

4. Higher-Genus Dissection Parts
ISAMA 2004
Higher-Genus Dissection Parts
A more intriguing dissection of a Platonic solid
Here is a more intriguing dissections of a Platonic solid: First seems impossible to separate the blue and white parts into two rigid pieces.
--- But once you push on opposite corners, you realize that the parts are of genus ONE. The prong from one part goes through the other part.
--- Here they are completely separated!

5. Intriguing Dissection of a Tetrahedron
ISAMA 2004
Intriguing Dissection of a Tetrahedron
And you can make more complicated shapes based on the same principle. Here is an intriguing dissection of a tetrahedron…How is this possible?
--- Here is how … It consists of 2 genus-2 parts.
--- And here is the physical realization. But this is NOT the topic we pursued. CS285 is a course on procedural geometry generation, so I wanted to bring in some non-trivial curved surfaces.

6. Problem Statement Execution
ISAMA 2004
Problem Statement
Design a two- or three-piece geometrical puzzle in which a shape splits into all congruent parts via a helical screw motion.
Teams of 3-4 students
Conceptual discussions in class
A first design + Individual feedback
Initial design to be fabricated on FDM machine
2nd, “final” design, hopefully yielding a working puzzle
Execution
So I gave the students this formal assignment: “===“
--- I split the class into groups of 3-4 students and ask them to discuss this problem between them. Then we have some high-level conceptual discussion to clarify the task, and they have to do a first design as a homework due in the next class.
Then I give them some feedback on their proposals, and they create a first design to be built on an FDM machine.
With parts in hand, they then redesign it and get a 2nd chance to make a (hopefully) functional puzzle.

7. Simple Helicoidal Dissections
ISAMA 2004
Simple Helicoidal Dissections
Based on linear, twisting sweeps z
All teams quickly figured out that the parting surfaces would have to be one or more helicoids winding around a common straight line, say, the z-axis. It was then their choice to select a suitable overall shape that would lie symmetrically within the chosen system of helicoids, so that the resulting dissection (or tri-section) pieces would be congruent. Here I show a 3-way partition, so the cross section consists of three segments of 120 degrees. On the left there is no special tailoring of the outer shape, so you can just see the basic helicoidal parking structure ramp. On the right the overall volume has been tailored into a sphere.
The basic “ramp” Cross section Tailored envelope

8. Simple Helicoidal Dissections
ISAMA 2004
Simple Helicoidal Dissections
A sweep producing a tear-drop shape z
Here is an actual student project. Different teams picked quite different approaches. The students who were relying solely on the default modeling software offered with this course, our SLIDE system designed in the 1990s, were typically using rotationally symmetrical shapes, because SLIDE offers powerful and easy-to-use sweep constructs, but it has no Boolean CSG operations. So, using the cross section shown in the middle; they can just apply a linear sweep applying the scaling function on the right and an arbitrary amount of twist. This generates the tear-drop shape on the left.
This is the first phase of their design. . .
3-part CAD model Cross section Scaling function

9. ISAMA 2004
First FDM Parts
Then they get their first parts back. This is what they look like as they come off the FDM machine. Now reality sets in: These parts are not so nice and clean as the CAD model. Also there is this gray support structure that has to be removed; and the surface underneath that grey stuff is even more rugged!

10. Rapid Prototyping with FDM
Granada 2003
Rapid Prototyping with FDM
This is the FDM machine that we used for this assignment. On the right you see the concept of its operation.
There is a heated head with two nozzles – moving in X and Y. There are two spools of plastic beads that are forced through the nozzles and then drool out in semi-liquid form with the consistency of tooth past. The computer calculates the geometry of a slice through the part under construction at the current altitude, and the head moves back and forth to “paint that shape”. When a layer has been deposited and fused with the layer below, the purple stage moves down by 1/100 of an inch and the deposition of the next layer starts.
Yellow is the actual build material that forms the desired part. [Explain need for scaffolding … Stuff cannot be deposited in mid-air]

11. A Look Into the FDM Machine
Granada 2003
A Look Into the FDM Machine
2 NOZZLES
Here you see a build process in operation… On top is the heated head; you can see the two nozzles that dispense the plastic. White is the build material and the grey corrugated stuff is the support material.
A sculpture-build in progress; note grey support!

12. A Second Set of Parts There are still problems:
ISAMA 2004
A Second Set of Parts
There are still problems:
The parts may not slide together completely!
These parts were made on another FDM machine, one that has a soluble support material which can be washed away in an ultrasonic bath of a highly alkaline solution.
But these parts still did not fully intertwine! There is, of course, friction! But we discovered one more issue: This machine, unlike the older one, was placing beads not just INSIDE the desired geometry, but was running the center of a bead smack on the outline of the desired part – so everything was thickened by 5 or 6 mils all around!
LESSON: Always run some test parts on a new machine to find out how exactly it is behaving!

13. Simple Helicoidal Dissections
ISAMA 2004
Simple Helicoidal Dissections
A second approach – using a helical sweep path
A second design approach: These students used a helical sweep path.
Here, the cross section was varied only in width.
The thickness of the rectangular cross-section slab was defined by the pitch of the helical sweep path.
2-part FDM model Sweep path Scaling function

14. Clean-up and Sanding the Parts
ISAMA 2004
Clean-up and Sanding the Parts
On the left: you see how the pieces came off the FDM machine.
This puzzle had the same problem as the previous one: The 2 parts could not be screwed together further than what is shown top right!
After some tedious sanding they finally fit together as show in the lower picture.

15. More Helicoidal Dissections
ISAMA 2004
More Helicoidal Dissections
Bio-Hazard Symbol
This team had the opposite problem; their first design fell apart by itself.
They tried to model the Bio-Hazard Symbol. The ring is the basic helicoidal geometry. Each part in this version runs around that ring a full 360 degrees. This is just barely enough. In a first design each part covered just 240 degrees – and that was not enough.
Friction and extra thickening of the part surface helped in this case; and the helical rings are springy enough to adjust for any variations in surface thickness.

16. Advanced Helicoidal Dissections
ISAMA 2004
Advanced Helicoidal Dissections
This team created a more intriguing dissection puzzle: They started with a cube, then partitioned it with 3 helicoidal cuts.
--- And that is how the parts separate! This team was aware of the tolerance and friction problems; they counteracted it by leaving the central part of this object hollow, thus reducing the area of the surface areas sliding along one another.
The resulting cavity is also a good place to hide a surprise trinket.
This design started with the outer shape: a cube
then partitioned it in to 3 parts with helicoidal cuts

17. The Parts of the Cube Dissection
ISAMA 2004
The Parts of the Cube Dissection
Here is what the individual parts look like. They are rather complicated and cannot be modeled with a sweep.
This team had a student with expertise and access to SolidWorks. This CAD-tool package offers CSG.
So this team could start from an arbitrary shape and slice some part out of it bounded by two helicoid surfaces.
- - If you don’t have CSG operations then you have to calculate discrete steps of the helicoidal parting surface yourself.
Cannot be modeled with a sweep
Needs a CAD program with CSG operations

18. Helicoidal Dissection of Tetrahedron (done with SLIDE by C.H. Séquin)
ISAMA 2004
Helicoidal Dissection of Tetrahedron (done with SLIDE by C.H. Séquin)
Helicoidal axis
To show the students that this is possible, I designed this puzzle shape “by hand” within SLIDE.
Because this geometry has D2 symmetry, the path of the helicoidal cut on the tetrahedron had to be calculated through only two faces.
--- Six path points were calculated explicitly on the 2 surfaces (the pink dots) and connected with a piece-wise linear polyline.
--- In case it is not clear, this is the helicoidal axis.

19. Helicoidal Dissections of Rhombic Dodecahedron (George Hart)
ISAMA 2004
Helicoidal Dissections of Rhombic Dodecahedron (George Hart)
Here is a 4-way helicoidal dissection of a semi-regular solid – found on the fabulous website by George Hart. But this puzzle is too loose. It falls apart under the influence of gravity when placed on the table with its helicoid axis lying horizontal.
Too loose !

20. Generalization: Multi-Prong Dissections
ISAMA 2004
Generalization: Multi-Prong Dissections
Here is another conceptual approach to create helicoidal dissections: You start with a straight slide-apart partitioning and then you twist the hole geometry around the z-axis. And, of course, the part can have multiple interdigited prongs.
Design a straight configuration and then twist the whole thing

21. Two 3-Prong Parts … and it works … here are two such parts . . .
ISAMA 2004
Two 3-Prong Parts
… and it works … here are two such parts . . .

22. ISAMA 2004
. . . and they fit together!
And they do fit together … and it even makes some interesting sound! [ demo ]

23. Another Variant Four prongs of different width unevenly spaced
ISAMA 2004
Another Variant
Here is another example; in this case the prongs were all of different width.
Four prongs of different width unevenly spaced

24. . . . and they also fit together!
ISAMA 2004
. . . and they also fit together!
And they also fit together; and since I did bit more sanding, they produce a finer wistle.

25. Multi-Prong, High-Genus Parts
ISAMA 2004
Multi-Prong, High-Genus Parts
And this approach generalizes further. The principle is quite flexible: The prongs don’t have to be all the same. They can completely penetrate the other part to make part of higher genus.
And they can still result in two congruent parts!

26. PART II Fabrication issues
A New Problem Statement:
Realize a given puzzle geometry with a particular fabrication process.

27. Enlarging a Cubic Burr Puzzle
ISAMA 2004
Enlarging a Cubic Burr Puzzle
Here is a picture of a neat cube-dissection puzzle. I would like to have one like it, -- but at a bigger scale! Make a design that can be built at an eight time larger scale in an economical way with a layered fabrication process.
The problem is that for many layered manufacturing processes, the price goes up with the build volume.
A neat puzzle …
If only it were larger !
Price increases with 3rd power of scale !!

28. Burr Puzzle: Unit Elements
ISAMA 2004
Burr Puzzle: Unit Elements
45o 45
So the key goal is to keep the build volume small. Rather than Building solid cubes we could aim for hollow cubes or for Leonardo frames of cubes.
Now remember that arches and overhangs require support material underneath. So the details of the frame geometry and of the windows in them becomes really important.
--- Look at the 45-degree chamfer at edge-struts, which we introduced to minimize the support material required in the vertical windows.
Save building- and support-material:
Construct edge-frames of cubelets only
Shape all overhangs to minimize need of support

29. Details for Cubelet Frame
ISAMA 2004
Details for Cubelet Frame
Another problem is that our file for the FDM machine is not a clean manifold model. It has coinciding faces. Normally the FDM software does a pretty good job with such models. But here we had a problem that coinciding edges and faces were not properly eliminated, and thus prevented adjacent cubelets from being properly fused together.
Please see the paper for more details of problems we encountered and how we got around them.
Replicated, abutting geometry

30. A First Little Test Piece
ISAMA 2004
A First Little Test Piece
Finally we had figured out how to make clean cubelets properly attached to one another, and so then we could go for the real thing!
Just two cubelets

31. ISAMA 2004
One Puzzle Part
Here are several shots of one puzzle part. First the CAD model; --- the way it comes out of the machine;
--after some support has been removed; --- and with all support removed, and the sliding surfaces sanded.

32. ISAMA 2004
All 4 Puzzle Parts
Here now are all four parts of the original burr puzzle (with 13, 14, 16, 20 cubelets), realized with this grid frame approach.
After all the above design adjustments, the Stratasys 1650 FDM machine built the various composites of cubelets as we had intended. For the yellow part with 16 cubelets the build process took about 37 hours. Manually removing the support structure took about 30 minutes.

33. ISAMA 2004
Burr Puzzle Assembly
Progressive stages showing how the puzzle is put together.

34. Puzzle Part as a Sculpture
ISAMA 2004
Puzzle Part as a Sculpture
Since this conference track is also concerned with sculpting, it should be noted that these grid-frame puzzle pieces are rather intriguing and attractive shapes by themselves. I could readily see them enlarged to a 30-foot scale and then installed in some public plaza as a monumental constructivist sculpture. It depends how you look at it !
I give my students the advice: It is worthwhile to take good pictures of your work. Then you probably will get much re-use out of these pictures.!

35. ISAMA 2004
Conclusions
Designing and fabricating dissection puzzles has been a highly valuable experience for the students – as well as for me!
It is also a rewarding activity: Students can take home artifacts that appeal to and are readily understandable by lay persons.

36. ISAMA 2004
More . . .
Other inspirational work that combines puzzles, sculptures, and fabricational issues:
SIGGRAPH 2011

37. ISAMA 2004
More Inspiration
SIGGRAPH 2009

38. Interested in Burr Puzzles ?
ISAMA 2004
Interested in Burr Puzzles ?

39. Frederick Doering’s 3x3x3 Burr Puzzle
ISAMA 2004
Frederick Doering’s 3x3x3 Burr Puzzle
Inspired by the above design exercise, Frederick Doering, a student in the 2011 class, decided that for his final course project he would develop a program that helps in the analysis and design of such interlocking burr puzzles based on cubes. Relying heavily on the pioneering work by W. H. Cutler [2][3][4][7], he developed a program that determines the movability of individual parts and then uses an exhaustive search to find non-trivial combinations of moves that will take the puzzle apart. He found a 2-piece, 3×3×3 cube puzzle that takes three moves to come apart completely.

40. Burr Puzzle Theory and Software

41. Helicoidal Tiling of 3D Space
ISAMA 2004
Helicoidal Tiling of 3D Space
Matthias Goerner 2007
Matthias Goerner 2007

42. What is this ? Now for something completely different…
ISAMA 2004
What is this ?
Now for something completely different…
What is special about this puzzle?
All puzzles described so far allow the individual movement of one puzzle piece at a time. But there are other puzzles that “cannot be taken apart with two hands [11].” They require a simultaneous, coordinated motion of several pieces to get the puzzle apart.

43. A Puzzle That Cannot Be Taken Apart With Only Two Hands
ISAMA 2004
A Puzzle That Cannot Be Taken Apart With Only Two Hands
It needs a coordinated action of several (groups of) parts to be disassembled
Here is a simple example of such a puzzle: In this compound of 5 parts, each has an angle of 108° between the prong and the tunnel. One way to separate them is to move all 5 parts at the same rate outwards in a star-shaped manner. The parts have to be split into a minimum of 3 rigid groups to be separated.
The question arises whether a puzzle like this can be designed so that all the movements are helical screw motions…
SNOEYINK, J. and STOLFY, J Objects that cannot be taken apart with two hands. SCG '93 San Diego, pp