artifacts of research at SIGgraph 2008

7/31/2019 artifacts of research at SIGgraph 2008

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Artifacts of Research: On SingularitiesJonathan D. Chertok (Universal Joint) 1

Figure 1: Computer images of recreated series of Rodenberg models with lines of ruling inscribed in their surface. Chertok 2008.

1. Introduction

The models presented at SIGgraph are 1:1 recreations of aclassical mathematical model collection originally made by

hand in plaster in the 1860s (fig. 2 and 3). Unlike theoriginal collection, these models were created using bothmesh-based and NURBS-based mathematical modeling,Computer Aided Design (CAD) software.

Fig 2. Original collection housed at the University ofGoettingen.

The physical models were then created utilizing plaster-based and resin-based Rapid Prototyping (RP) technologies,with each creating the models from the 3D files. Thesetechnologies use the CAD file to either drop a liquid binderonto a powder or to direct a laser as methods to solidify themodel material (fig 4).

The collection of 23 white plaster-based RP models arerecreations of a series of models designed to represent thetypes of singularities possible on a cubic surface. With asingularity effectively being an abrupt change in shape onecan look for the "double points" (in German "doppelpunkt")where a surfaces basically passes through to the other side.A close look at the series' nomenclature (describedelsewhere) will show that each model represents either anindividual or combined examples of the various types ofsingularities.

Figure 3: Double contact sheet of the author's photographs ofthe original plaster models c. 1860.

This series was created by Carl Rodenberg for his thesisunder the direction of Felix Klein (1849 1925) the founderof modern topology. Effectively this was an attempt to

catalog a part of the mathematical universe.

They exhibit an elusive beauty that I would characterize as"sublime". In a sense they appear so rational as to almost befacts of nature, while at the same time one has the distinctsense that this rationality justslightly eludes comprehension.

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Fig 4. Rapid Prototyping (RP) recreations being produced

Additionally presented in the Addendum are unique newmodels of the Clebsch Diagonal Cubic - a so-called "smooth"surface which is the "god-head" of the series from which theseries of singularities originate. Effectively the ClebschDiagonal Surface shows the 27 total possible lines on a

smooth cubic surface while the 23 models shown in theRodenberg series show "aesthetically pleasing" examples ofthe singularities that result when you remove some of theselines.

2. Exposition

Beyond their sublime beauty, the new models proved to be awonderful entry point for contemporary architecture andcontemporary construction issues related to digitaltechnology. In this vein, three observations regardingtechnique are germane.

First, the fact that the surfaces had to be modeled as mesh inmathematical modeling software (the only way to generatethese zero sum implicit equations of the surfaces) while thestraight lines were generated as NURBS lines in CADsoftware (from their parametric equations) resulted in the

lines which were inherently more malleable than thesurfaces. That is, their NURBS nature means that we could

parametrically control them which is a possible avenue forfuture work.2 Second, the relative precision of the

parametrically derived lines - when literally placed againstthe complicated curving mesh surfaces (generated by theMarching Cube Algorithm of the software) - allowed for avisual check of the algorithm, which would have beendifficult to discern otherwise.3 Third, the ability to "fly-through" using a 3D mouse provided unequalled power forquerying both the model and the relationships inherent in it.It also proved to be particularly useful for creating geometry

particularly when adding the sphere to show the varioustypes of intersections on the Clebsch Diagonal SurfacesConfiguration.

Thus, the use of CAD to work with these models providedobvious visualization and interactive benefits as compared toconventional and more static mathematical modelingsoftware.

Similarly, the ability to work with actual models providesobvious benefits (fig. 5 and 6). Utilizing newer two colortransparent RP model technologies can provide even moreunique feedback in these and other respects (fig. 9 and 10).

Fig 5. Cubic with 4 - A1 Type Singularities

Figure 6: Ruled Cubic Surface.



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3. Future Research

I'd like to explore the tendencies of the marching cubealgorithm in most mathematical modeling software to eitherfollow or not follow lines of principal curvature. In the caseof the cubics presented here there is a curious tendency toappear to follow lines of principal curvature but to thenthrow in facets which "dirty" the mesh.Similarly an understanding of when one can expect planar

quads using this algorithm has eluded me.; sometimesmathematical modeling software provides obviously planarquads and sometimes it does not.Also, there is a worthwhile kinematic exercise to explorewith respect to the 27 lines on the Clesbch Diagonal Cubic.15 of these lines may collapse to the plane (like those in ahyperbolic paraboloid) and I'd like to make someconstructive experiments for large scale supportingformwork for concrete supports for tranportationinfrastructure.On a computational note, it appears to me that the lines onthe original surface (see fig. 7) may have been numbered byhand when they arrived at the various Universities in the1800's or that the original numbering was destroyed over theyears and redone. As in some cases this numbering is

reported to be incorrect (more on this numbering in theAddendum). With help it should be possible to run a "script"that would generate all possible 36 combinations for thenumbering of the lines on this surface based upon the actualCAD configuration and in conjunction with the historicaldocumentation of the geometric relationships of these lines.This could then be used to check the numbering of theoriginal models.

4. Summary

The intellectual seed that started this work was aposthumously published work by Robin Evans called TheProjective Cast: Architecture and its Three Geometries(1998).

And since I first photographed these model in 1999, Ivebeen intrigued by their sublime beauty. In the end I have

concluded that this is partly a consequence of Evans'characterization in another context that by using moregeometry they appear to have less.

Acknowledgements

I want to thank Professor S.J. Patterson for allowing me tophotograph the original collection and for his continuousencouragement and assistance. A big thanks to RichardMorris for all his continuing help and the use of his software

program SingSurf. Thanks to Stuart Dickson for his expertiseand guidance. Finally I want to thank Giorgio Ferraresewhose great efforts allowed me to see this work to the finishline.

The models were produced with support and funding fromZCorporation, Objet Geometries and the ImagingTechnology Group at the Beckman Institute for AdvancedScience & Technology at the University of Illinois.

Figure 8: Jim Blinn studying the author's documentation andtaking a close look at the work at SIGgraph 2008.

Figure 7: New models on Flickr c. 2010.



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Figure 9: Clebsch Diagonal Surface c. 1860.

Addendum:The Clebsch Diagonal SurfaceConfiguration

1. Introduction

In the case of the Clebsch Diagonal Cubic we see a "smooth"cubic surface, as opposed to the singular cubic surfaces,which is in fact the "god-head" from which the other modelsof the singularities spring. Effectively, by removing linesfrom the Clebsch Diagonal Cubic, one can arrive at themodels for the other singularities.

The Clebsch Diagonal was originally discovered anddocumented by Alfred Clebsch (1833 1872). Both thetransparent models and white plaster-based models you seeare in some sense recreations of the original plaster modelfrom the model collection that was originally made by handin plaster in the 1860s.

This model in particular has historical significance and hasbeen touched upon by a number of early machine designersin addition to mathematicians.4

While no documentation of the original fabrication exists, wedo have the original recipe for the modeling clay. In additionto serving as an interesting analogue to contemporary RPmaterials, it is quite charming as it calls for white blotting

paper, 1 Liters of river water, 10 Marks of essence oflavender, 10 Marks of essence of clove, a tin container and a

porcelain bowl among other things.

Like the other models in this exhibition, you are looking atthe outside of the surface. If you can visualize the fact that

the surface continues upward, downward and beyond theextent of the cylinder that has been used to trim the exterior

portion, then you have a good jump on understanding thissurface. Just think of the model as hollow. Unlike the othermodels in this exhibition it has no singularity type associatedwith it, as it is non-singular; it is smooth.

In the following we will describe the geometricconfigurations of this surface. I trust you will find both the

models and the animation instructive.

Figure 10: Clebsch Diagonal Surface c.2008

2. 27 Lines

The surface has 27 lines on it. This is the maximum possiblenumber of straight lines on a cubic surface if, as a geometerwould say, "the lines are finite". If there are infinite lines onthe surface then it could be any number of ruled cubics - fourof which are shown in the Rodenberg Series. This is one ofthe reasons this model is so famous. Anyway, these lines arestraight lines but for mathematicians it is only necessary tosay lines. Oddly in NURBSNURBS-CAD we say"curves". The lines are either skew - they pass each otherwithout intersecting - or they are co-planar - and thus they

intersect. There is a special case of co-planar, which areparallel lines, but there are no parallel lines on the ClebschDiagonal Surface.

12 of the lines from the 27 were studied by Ludwig Schfli(1814 1895). These are called Schflis Double-Six inEnglish. This arrangement consists of six pairs of skew lines(remember this means they do not intersect). These arenumbered 1, 2, 3, 4, 5, 6, and 1, 2, 3, 4, 5, 6. A pairwould be 1 and 1 or 2 and 2 and so on.



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Most of the lines however cross each other. Lines numbered1 6 get crossed by lines numbered 1 6. The onlyexception here is that lines with the same number (e.g. 1 and1) do not touch each other. Again, these are the pairs ofskew lines. Thus each line from 1 6 is coplanar (theyintersect) with each line from 1 6 (excepting its prime asstated earlier). Similarly, each line from 1 6 iscoplanarwith each line from 1 6 (again excepting its non-prime). Inthe case of the Clebsch Diagonal Cubic, only four of these

five intersecting lines intersect within the model and oneintersects lower down beyond the base of the model.

Figure 11: Clebsch Diagonal Surface c.2008

There are 15 other lines called the Diagonal Lines. These are12, 13, 14, 15, 16, 23, 24, 25, 26, 34, 35, 36, 45, 46, and 56.These run horizontally and vertically. These lines are giventwo digit numbers. Line 12 intersects the plane described bylines 1 and 2 and the plane described by lines 1 and 2.Similarly line 23 intersects the plane described by lines 2 and3 and the plane described by lines 2 and 3. One does notneed a line 21 as it is coincident with line 12. Similarly for32 as it would be coincident with line 23.

3. Additional Geometric Objects

There are 30 points through which 2 pairs of lines intersect.24 of these exist within the extent of the model. 6 are beyondthe extents of the model. This makes for a total of 12 linesthat have 5 intersections along their full length. We discussedthese earlier when we talked about the numbering.

There are also 10 points at which 3 of the 27 lines meet.These are called Eckardt Points. 7of these intersections existwithin the extent of the model and 3 are outside the model.

The three horizontal planes are simply pointing out three setsof three lines that are part of the 15 Diagonal Lines. Thethree angled planes are formed by three pairs of lines, whichare pointing out the other remaining lines from the 15Diagonal Lines.

Figure 12: Clebsch Diagonal Surface with imprinted andnumbered lines c.2008

James Joseph Sylvester (1814 1897) discovered theSylvester Pentahedron, which is a five sided object described

by the bottom two horizontal planes and the three angledplanes. One triangle at the top and one at the bottom and fourisosceles trapeziums as the sides (just think of these astriangles with their tops chopped off). Apparently this isimportant because the planes of the sides of this pentahedroncan be described by equations which can also describe theequation for the actual surface of the Clebsch DiagonalSurface. For more on this we would need to consult ageometer.

4. Passages

Lets return briefly to this idea that this is a surface with twosides and look at the idea of Passages which was aclassical idea that was apparently used "intuitively".5 There

are seven passages in the model a fact that is related to theCoxeter-Dynkin Diagram which we will discuss in another

paper. Three are just obvious. They are the three holes in thesurface. The other three are a little higher, rotated by 60degrees and up at the ears. Actually these three passageshelp us find the seventh passage. Just jump into one of theears, continue toward the middle and then drop downthrough the waist this is the seventh passage.



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14. Coxeter-Dynkin Diagram

Figure 13: Clebsch Diagonal Surface c. 1860.

5. Numbering Script

In total there are 36 different ways to number the lines on theClebsch Diagonal Surface according to the rules above.Research indicates that the numbering on the original models

was done at the Universities that received the model, and notat the point of manufacture. As a final exercise I intend to runa computer based script in Rhinoceros 3D modeling softwaredefining all the possible combinations of numbering byquerying the CAD geometry and the documentedrelationships inherent in the geometry. This exercise would

provide an opportunity to verify the various numberingslabeled by hand on the various models in collections aroundthe world.

14. Still from Animation of the Clebsch Diagonal.

6. Conclusion

In summary, this portion of the research is model basedexercise, rooted in mathematics and CAD, and I trust that itwill provide interesting points of departure for future work.

Jonathan ChertokMay 2008Austin, Texas


1 [email protected]://www.universaljointdesign.com

2There is a range of mathematical software that will generatesingularities in a parametric way along with their lines and this isdefinitely an area which we would like to research.

3 As Professor Samuel Patterson pointed out to me, the lines on

the original plaster models are in fact highly curvedandmeandering due to the imprecision of the plaster model surfaces,which was related to practical issues in making the model.

4 Ferguson, Eugene S. 1962Kinematics of Mechanisms from theTime of Watt.

5 Fischer, Gerd (1986) Mathematical Models: Photograph Volumeand Commentary. Braunschweig / Wiesbaden: Friedr. Vieweg &Sohn.

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artifacts of research at SIGgraph 2008