Wiring Euclid for manufacturing
In this blog post we consider the practical tradition of Euclid’s elements, in contrast to its more familiar role as a model of axioms and proof. Inspired by that tradition and combining it with the notion of wiring diagrams we propose an abstract model to think about and problem solve issues of manufacturing, especially using the power of modern digital manufacturing tools.
1 Introduction
Euclid’s elements has an almost mythical role in the history of modern mathematics. In particular it had significant influence on the notion of axiom and proof systems. Establishing a fixed set of rules and axioms and then seeing what can be built from those using proofs. Yet it is also a practical text, the propositions are often not just proofs, but operations that use straight edge and compass to create geometric figures, from the equilateral triangle to the regular polyhedra1.
In modern mathematics the notion of proof has been well developed to be independent of the notion of construction. What happens if we instead explore ideas of construction and abstract models of tools independently? In doing so we will show how wiring diagrams provide a model to structure manufacturing methods and add detail to David Spivak’s notion of plausible fiction as a model for manufacturing.
2 Euclidean construction
Euclidean construction begins with two tools, the straight edge and compass. The axioms establish a mapping between physical actions with these tools and abstract notions. For example a straight edge can be placed next to two points on a piece of paper and the line between them drawn. This is abstracted as the axiom that any two points lie on a straight line. Similarly the point of a compass can be placed at one point and the circle through the second traced. Which gives the abstract definition of a circle.
At this stage it is worth noting that the rules allow our actions to have abstract meaning as well as physical content. The points we begin with will not be perfect points, but dots drawn with area. The line between them will thus not really be unique, even if there is sufficient skill to draw the line so it does actually pass through them. Yet, abstractly we can consider the small dots to represent pure points, and then the line we draw represents the line between them. Similarly the surface we are working on, for example a sheet of paper, is clearly never the infinite Euclidean plane. It has boundary, and if we zoom in far enough it will no longer be flat. Yet in many cases the assumption it is just Euclidean is incredibly useful.
The beautiful aspect of this abstraction is that these issues can often be ignored, and thus the physical drawings can help understand the world created by the axioms and in contrast more advanced propositions can create useful physical processes. In the case of geometry we are quite comfortable with this relationship and often do not even reflect on it. It is therefore worth considering a case where the abstract and physical traditions of geometry diverge, in the construction of regular polygons. We therefore have a conversation between the abstract system and the physical where each can inform and help understand the other.
3 The challenge between theory and practice
The relationship between the two can be quite subtle and not always what is expected. For example, take the powerful results of Gauss and Wantzel showing exactly which regular polygons can be constructed with straight edge and compass. In particular the heptagon can’t be and the 17-gon can. Yet in Islamic art, that used straight edge and compass extensively, we find heptagonal tilings. These were constructed not by bringing in an additional tool but by a process of approximation:
- Take the circle you want to construct a heptagon, and guess the right length for the edge.
- Run the compass round the circle making points at that distance.
- Adjust based on the final error.
- Repeat 2 and 3 until the error is small enough.
In practice, therefore, the lack of an exact construction for the heptagon is not a significant barrier. In addition the slight blurring between the physical and the abstract, mentioned above, ensures that the resulting heptagon is no less accurate than a pentagon constructed perfectly by Euclidean methods. When considering the 17-gon this becomes even clearer, as the “perfect” construction is technically challenging and complex. When I have tried it, my initial guess of the correct edge length is often more accurate than the length I obtain following the supposedly exact method. The approximation method is also significantly simpler. Thus, even when a perfect abstract method exists, it might not be the best option in practice.
This illustrates a key challenge in considering abstract models of manufacturing. Especially as they are developed these might seem to be ill-defined and non-rigorous to mathematicians, on one hand and a lot of work for little practical benefit to engineers and practical makers. Yet it is already happening informally, especially with digital manufacturing, which turns computer designed models into physical objects with CNC and 3d printing. Creating good abstractions to look at the systems as a whole has the potential to make the processes more visible and thus help with communication of ideas from prototype to manufacturing, reveal significant challenges and enable more direct application of research into practice.
A good candidate for this abstraction is the wiring diagram. One of the ways these can be implemented is in data flow programming, for example Grasshopper (part of the CAD software Rhinoceros 3d). The following examples will be implemented in Grasshopper to keep a consistent visual language, even when there is no computational content.
4 Example: Euclid Book 1 Proposition 1
Let us begin by going back to Euclid for inspiration. Here is the construction from Proposition 1 of book 1 of Euclid’s elements rendered in Grasshopper:
Note that this is a functioning operation that gives a triangle for a given base. This implementation immediately reveals two limitations in Euclid’s version, for example the lack of any axiom to say when two circles intersect (that box is labelled with a question mark). There are also two intersections and both are drawn here, giving two triangles.
After the construction, however, the proof of Proposition 1, continues, showing that the resulting triangle is indeed equilateral. So the large green region can be seen as a component in its own right, taking in a line and returning an equilateral triangle with that line as base. In Euclid this proposition is used as a component in many later proofs starting with Proposition 2.
5 Proposal: Augmented Wiring Diagrams
The key proposal here is to augment each component in the wiring diagram with assertions about the nature of the result. For a full wiring diagram these assertions could be used to prove properties of the output, perhaps with additional assertions on the input. In the case of Proposition 1, above, the base level components are the axioms and definitions of the elements.
At the lowest level these assertions would be just that, clear assumptions about the nature of the inputs and outputs. They can then be combined into standard workflows (just like propositions) where the behavior of a collection of components has an argument (or even proof) that it will work a certain way, given the assumptions of the sub-components. The behavior of more complex components thus rests on the hopefully simpler assertions of its components, rather than just an assertion that the whole thing works.
The goal might be best described as creating a more local mathematics. Instead of proof being needed for powerful universal statements within a broadly accepted axiom system or field of study, the proofs would just be of interest to a specific case, the axioms required just being assumptions about how certain tools work (or should work). Below I give an example of how the process of milling wood can be considered in this frame. One distinct aspect of this local mathematics is that the axioms are far more lightweight. The solution to a problem might be including a new tool, and thus axioms about its use, rather than clever working with what is already on hand.
In developing a techniques for unfamiliar or new problem, this process connects directly to David Spivak’s notion of plausible fictions. An overall claim is made that can be broken down into a series of steps. At each stage the steps can be asked if they are plausible. Any step that is perhaps plausible but certainly not straightforward—maybe on the verge of “magical”—must be broken down further. In this case instead of building up the wiring diagram from already accepted components we start with a component we want to create. To achieve this much of the work might take place through standard processes, so the magic unknown component can then be pushed down into a simpler subcomponent. Over several rounds everything is perhaps reduced to known work and the whole process can be tested. Alternatively a key step might still be missing but new software, machine or tool can be developed to solve that issue.
The multi-level nature of wiring diagrams, combined with the notions of plausibility and proof also allows for details to be blurred out an revealed as needed. For example in the planning stage a task that has not been done can be deemed plausible to achieve and left until it needs to be done. An example might be work holding of material in a machine. This always requires a little bit of work, with standard techniques like vises, but can generally be left until the time comes to machine the part.
In this picture there are obviously multiple levels of plausibility. In mathematics for example, everything must be reduced to an accepted axiom set as the only possible plausible set. For manufacturing plausibility is allowed to rest on concrete practical claims about the abilities of certain tools, but the list of tools available is flexible, so new axioms can be created. This is still more concrete than the examples of social change in Spivak’s work that allow plausibility to include predictions about reasonable future acts.
6 Example: Milling Lumber
We now consider a more practical example, turning rough cut lumber into a dimensioned board2. Here is the diagram:
This process begins by using a jointer which is able to create a flat surface on one side of the board, and a second flat surface at ninety degrees to the first. This creates a reference edge. Then a planer is used to make the other side of the board parallel to the first. Finally a table saw makes the far edge parallel to the jointed edge and cuts the board to length. These operations also give the height, width and length of the board.
Each tool here behaves like an axiom, we asserted its abilities. Much like the Euclidean axioms assert the abilities of straight edge and compass. Combining these assertions we are able to prove that the resulting board can be considered abstractly as a cuboid with the given dimensions. A rather complex assertion is thus reduced to assertions about the abilities of individual machines. At each stage a different machine might substitute as long as it is capable of the equivalent operation. A CNC router for example can be used to flatten a face, and give a straight edge, replacing the jointer. Similarly a mitre saw can replace the table saw in crosscut to cut the board to length.
It is worth noting that the assertions made here can also be augmented, for example to add claims about the time spent, or the quality of the finish that a tool gives. These extra assertions might mean that a previously possible replacement tool cannot be used.
7 Example: Curved Metal Sculptures
The examples given so far are more illustrative of the idea, placing classic ideas of mathematical proof or practical process into the language discussed here. To conclude I have a final example where versions of these ideas were used in a research setting to develop a new technique; the zipform system I developed with Emily Baker, initially to create the Gearhart Courtyard Curvahedra sculpture (above) at the University of Arkansas.
The nature of practical examples means that the process has a larger technical load that might be of general interest, but is not all directly relevant to the specific discussion here. I give a more detailed discussion in an appendix. Here is the wiring diagram:
In order to make this sculpture the challenge was to create curved metal beams. In order to achieve this we already had a powerful tool, a plasma cutter that can cut sheet metal into nearly any shape we ask of it. Sheet metal can also be bent in many ways, as long as its intrinsic geometry is not changed. Mathematically these bent sheets can be considered as developable surfaces. In the physical space we can therefore assume we can make arbitrary shapes and then find ways to bend them precisely. This created two smaller challenges, firstly could we create developable surfaces with the properties we wanted, and secondly could we bend the cut sheets into the right form.
The first problem was solved with differential geometry, and the Bishop or zero-torsion frame which creates developable surfaces, at right angles containing a given curve. The physical solution also used this frame, building a jig to keep the two flat sheets at right angles as they were bent together to recreate the curve in physical space.
This theoretical process, with its assertions, was also being tested in practice. Our first tests were not even in steel, but in a different material (paper) cut in a different process (laser cutting). Like sheet metal this can be assumed to be a developable surface. We then made scale models in metal. For both these test models we used augmented reality to compare the designed beam on the computer with the physical model. Abstractly this gives a different mapping from the computer model to the physical world. We therefore achieve a form of commuting diagram to validate that the beam built is the one we think it is. The accuracy was rather necessary for the large sculpture as the beams formed loops and needed to be rigid, so the tolerances needed to be tight enough that the loops could be connected without bending.
The use of paper implies a different abstract relationship. As both paper and sheet metal can be considered as developable surfaces a paper model can be scaled up to one from sheet metal. Here, though, one does have to be careful. Not all properties are shared, for example paper is very elastic when bent, and so in combining the two pieces hand holding and glue were sufficient for the integration phase. This is why, while the paper model was encouraging, the scale model in metal was necessary. This is an example of how making assertions clear is useful even when the assertion is incorrect or not complete. The argument here that gives a path from a paper model to metal just uses the developable surface assertions. These do not say anything about the elastic properties of the materials. Understanding this reveals where work might need to be done. The use of paper models as a design space for metal ones can then take into account the precise differences as well as the relationships.
After validation, and just like the propositions of Euclid, this technique can now be used in many different situations. For example we have used it to create a completely different shape a hyperboloid, rather than a sphere (shown below). Part of the mathemalchemy exhibit. The zipform technique was also used to create precise developable surfaces to cast optimised concrete beams. In each of these cases additional geometry was required to create the curves involved, but the method described then smoothly recreated this geometry in physical form.
8 Conclusion
The ideas described here are still very much in development. To me, however, they reveal some powerful aspects of flexibly mixing abstraction into physical manufacturing. In particular I feel there are several potential benefits, such as:
- Breaking problems down into component pieces. (like the milling of wooden blocks, or the propositions in Euclid’s elements).
- Combining tools and processes to create new reliable technologies (like zipform).
- Clearly describing the power and limitations of moving between different physical modeling spaces (like the difference between constructing polygons in theory and practice, or moving between paper and metal models).
The Author
Edmund Harriss is an assistant professor at the University of Arkansas holding a joint appointment between mathematics and art. Much of his work considers the ways mathematical ideas interact with the physical world other than in the classic notion of scientific models. This includes methods of illustrating mathematics to reveal the power and beauty of the ideas, but also open up questions for mathematical research. To do this he often works to create illustrations and artworks using digital manufacturing, leading to an interest in the right ways to conceptualize those processes, and thus the work discussed here. He can be reached at eharriss@uark.edu and as Gelada on Mastodon and Bluesky. Other examples of his work to bring mathematics into the physical world include his two coloring books and the construction toy curvahedra.
9 Appendix: Zipform details
The heart of zipform uses the rigidity of a T-beam, but creates one that is curved, rather than straight. In fact where the joint between the vertical and horizontal followed an arbitrary smooth curve in 3d. In the resulting wiring diagram (below) information about the bending of the curve is split into two curvature functions (using the zero torsion or bishop frame). These functions can then be used to created two curves with that curvature. Thickened versions of these curves are then cut out of sheet steel. The actual curve is the centre line of one piece that will form the crossbar of the T and the edge of the other (forming the vertical). The two flat sheets are then placed together in a T-shaped jig (shown above) and welded together. They are then pulled a little way through, in each case the thin direction of the metal allows it to bend to match the curve on the other sheet. As they are pulled, bent and welded together the curve along their intersection recreates the input curve. Here is the diagram:
We therefore start with an assertion about this process as a whole. The curve that is actually made is the input curve. In development this was broken down into smaller assertions that could be combined with proof to create a theoretical process.
We began with a commonly used assertion, that sheet metal cut on a plasma cutter, and then carefully bent is a realisation of a developable surface (a surface with constant Gaussian curvature 0). This assertion for a sheet material (thinking of it as having no thickness) is quite close to the assertion that a straight edge creates lines, or a compass draws circles.
With this assertion, on the abstract side we can therefore create developable surfaces knowing they can be turned into physical objects. The zero torsion frame of the curve is perfect for this. Taking any line at right angles to a point on the curve the zero torsion frame sweeps it out along the curve in a unique way so that it never twists around the curve’s tangent (this twisting is torsion). The animation above shows curves and the resulting surfaces for different initial lines. Note that they twist along the whole length of the curve at once. The resulting surfaces are always developable. These developable surfaces can then be flattened onto the plane. When this happens the intersection with the original curve becomes a 2d curve. The curvature of the two curves produced are called the geodesic and normal curvatures. These two curvature functions determine the original curve up to rigid motion.
The resulting flat shapes can now be cut out. They then need to be bent into shape, to reverse the flattening process and recover the initial curve. In this case we began not with a process we had used before but the assertion we wanted. Essentially we needed a physical process that would work like the integration of geodesic and normal curvature. The jig described above was the result of this development.
Footnotes
For a recent discussion of the messy history, geometry and cultural impact of Euclid a good resource is Benjamin Wardhaugh’s Encounters with Euclid.↩︎
If you are not familiar with the process of milling lumber for woodworking you can find out about the whole process here, as well as how you need to be careful with exactly what the machines are doing here.↩︎