In this short, chatty paper, I describe how my attempt to use mathematics to create a 3D print of a school portrait led me a group of early 20th century French artists known as the Fauves

Aboufadel, Edward

English

arXiv

Scholarworks@GVSU

Grand Valley State UniversityScholarWorks@GVSUOther Faculty Publications Mathematics Department7-20163D Printing & the Artists Known As the FauvesEdward AboufadelGrand Valley State University, aboufade@gvsu.eduFollow this and additional works at: http://scholarworks.gvsu.edu/math_otherpubsPart of the Physical Sciences and Mathematics CommonsThis Article is brought to you for free and open access by the Mathematics Department at ScholarWorks@GVSU. It has been accepted for inclusion inOther Faculty Publications by an authorized administrator of ScholarWorks@GVSU. For more information, please contact scholarworks@gvsu.edu.Recommended CitationAboufadel, Edward, "3D Printing & the Artists Known As the Fauves" (2016). Other Faculty Publications. 3.http://scholarworks.gvsu.edu/math_otherpubs/33D Printing & the Artists Known As the FauvesJuly 2016Prof. Edward AboufadelGrand Valley State University,Allendale, MI, USAaboufade@gvsu.eduAbstract—In this short, chatty paper, I describe how myattempt to use mathematics to create a 3D print of a schoolportrait led me a group of early 20th century French artistsknown as the Fauves.I. 3D PRINTING: SOMETHING FROM NOTHINGAt the beginning of the decade, I started to hear about low-cost 3D printers, and I decided that I wanted one. As the endof the 2012-2013 fiscal year approached, I asked my Deanto purchase a Makerbot Replicator 2 for my department, andhe agreed. Soon, we had a new 3D printer set up in myoffice, and I had no idea what to do with it. With two REUstudents that summer, we started asking the question, ”Whatcan a mathematician do with a 3D printer?”The first thing to do was to “print” something. TheMakerbot came with several stereolithography (STL) files,which are large files that describe the object as a set oforiented triangles. STL files are processed by software knownas a “slicer”, for reasons that will be obvious in a moment,and the results of the slicer direct the printer’s actions.Following these directions, a 3D printer creates an object bymelting filament (a spaghetti-like spool of plastic, usuallya type known as Polylactic acid, or PLA) at temperaturesover 200 degrees Celsius, and then extruding the hot plasticthrough a nozzle onto a build plate, layer-by-layer, or slice-by-slice.Fig. 1. 3D print of a cube.Mathematically, working in three dimensions, the printerstarts by laying down plastic in the plane z = 0, at variouspoints (x; y; 0), following paths created by the slicer [7].Then the build plate moves down vertically a very smallamount z so that more plastic can be extruded in the planez = z. (The nozzle moves in the x and y directions, but notthe z direction.) The material for this second slice usually isplaced on top of the material in the z = 0 plane. The printercontinues to z = 2z, z = 3z, etc., and there are a varietyof rules to take into account curved surfaces, overhangs, andother aspects of a three-dimensional object, or else all wecould ever print would be rectangular prisms and pyramids.One of our first prints can be found in Figure 1.There are a variety of ways to create STL files. For verysimple designs, STL files can be written in a text editor,where the vertices and outer normal vector of each triangleare listed, such as:facet normal 0.0 1.0 0.0outer loopvertex 0.0 40.0 0.0vertex 40.0 40.0 0.0vertex 0.0 40.0 40.0endloopendfacetHowever, nearly all STL files are created with a softwareprogram of some sort, and saved in relatively efficient binaryfiles. These files can be created using 3D analogues ofMicrosoft Paint available for free online such as Tinkercad(https://www.tinkercad.com/), open source pro-grams like OpenScad (www.openscad.org), and propri-etary programs like Mathematica. Influenced by Segerman’spaper [15], as well as my affinity for mathematical analysis(as opposed to geometry), my students and I have primarilyused Mathematica to design objects for 3D printing. Some ofour creations from the past two years can be seen in Figures2 and 3.II. CAPTAIN, I’M AN ANALYST, NOT A GEOMETERThere are other mathematicians who havedesigned and printed fantastic objects with 3Dprinters in the past four years. Laura Taalmanspent a year doing daily mathematically-inspired3D-printing projects that she documented on herblog (http://makerhome.blogspot.com/).George Hart has created 3D printed fractals (https://www.simonsfoundation.org/multimedia/3-d-printing-of-mathematical-models/).Links to many other projects can be found on myweb site (https://sites.google.com/site/aboufadelreu/Profile/3d-printing). Most ofthese projects are based on mathematical topics like knotFig. 2. 3D printing examples: Che; Star; Data capture from Kinect; Pendantcelebrating the Mathematical Association of America’s 100th anniversary.theory, polyhedra, and minimal surfaces that have geometryas a foundation.However, I have always been more of an analyst, thinkingin terms of functions defined on a domain, so my 3D printingprojects have had more of a flavor of computing a functionf(x; y) that represents a height above points on a compactset in the xy-plane. The MAA pendant in Figure 2 is a goodexample of this approach, with the domain being a circle ofthe form (x h)2+(y k)2 = r2, and the function having avalue of 0 where the hole for the clip is, 2 (millimeters, whichis the standard unit in 3D printing design) for the parts of theouter parts of the pendant, and then something significantlymore elaborate for the combination of the initials and theicosohedron.The first major project with my REU students was to usetwo photographs of the same object (specifically, a hand) tocreate a 3D print of the object, as seen in Figure 4. The ideawas that the stereo effect of two photographs, like the wayour pair of eyes works, allows us to deduce the distance ofvarious points on the object to the cameras, and from therea depth function can be defined [4]. After we completed thisproject, I started to wonder to what extent this could be donewith just one photograph. This is not a new question (forFig. 3. Victor’s cross Christmas ornament.example, there is the new “123D Catch” app from Autodesk,see http://www.123dapp.com/catch). However, Iwanted to figure out how to do this with my methods:create and generate a 3D rendering of a function f(x; y)in Mathematica that represents height or depth, export therendering as an STL file, then slice and print. Of course, thehard part is the creation of the function. And my naive use ofgrayscale, which I will describe in the next section, wasn’tgoing to get me very far.Fig. 4. Based on two photographs of a hand.III. FIFTY SHADES OF GRAYSCALEOne of the tricks to create f(x; y) from an image like alogo [5] is to base the values of the function on the shades ina grayscale image. Grayscale describes intensity (or, as wewill discover below, luminance), with brighter shades havinglarger grayscale values. There are two equivalent versions ofgrayscale: a scale of 0 (black) to 1 (white), and a scale of 0to 255 (which comes from using bit strings).A JPG image can be imported into Mathematica andconverted to 0-1 grayscale, represented in a large matrix, andthen this matrix, or a scalar multiple, can be used as a heightfunction defined discretely in a table. The ListPlot3Dcommand in Mathematica can then be used to nicely renderthe function. We can do more by transforming the matrixvalues with some sort of filter, such as re-assigning all valuesgreater than 0.5 with a height of 10, and all values lessthan 0.5 with a height of 5, using if-then-else commands.An example of this results of this procedure can be seen inFigure 5.Fig. 5. Grayscale becomes height.Without thinking things through, I thought I would use thisapproach on my son’s high school portrait. I will spare youthe picture of the rendering of this function – let’s just saythat the nose is not necessarily the lightest part of a portrait,the ears are not necessarily the darkest, and hair color playsextra havoc on an attempted design. This led to the question:how does the conversion from color to grayscale work? Andthat led to the idea of luminance.Photographers, painters, and other artists understand theintensity of a color is measured by its luminance. Theluminance of various colors can be seen in the color wheel inFigure 6, and when these colors are projected into grayscalespace, luminance becomes grayscale. There is an excellentvideo by neuroscientist Margaret Livingstone on this topic[12]. Now, it would be hard to find a portrait with a limegreen nose and dark blue ears. But in the video, Livingstonementioned a group of early 20th century artists that createdthe type of portraits that I could work with.IV. THE WILD BEASTS OF FAUVISMLes Fauves, or the “Wild Beasts” was the name givento a set of young painters, primarily in Paris, in the earlytwentieth century [17]. They were known for their wildand unexpected use of color. As described by Ferrier, “TheFauves explored the spectrum; for them, colors were not onlymere stimuli on the retina but could also express feelings.”[11] The most famous Fauvist was Henri Matisse, and otherwell-known members of the group were Henri Manquin,Albert Marquet, Georges Braque, and Andre´ Derain.Fig. 6. Colors and associated luminance. (Source: http://www.workwithcolor.com/color-luminance-2233.htm.Artists have long recognized that color and luminanceplay different roles in visual perception. One of the goalsof Fauvism (1905-1907) was, according to Douma, “to givecolor greater emotional and expressive power.” [10] Doumaalso observes that “Neuroscientifically, Matisses paintingswork like a black and white photograph.” And that is exactlywhat I needed to 3D print a portrait.The portrait that I focused on for this project is GenieMask, seen in Figure 7, by Andre´ Derain. Clearly, as withmost Fauvist paintings, we do not have realistic colors beingused for this human face. But, the luminance of the variouscolors in this painting line up with pretty well with thedistance from the virtual “lens” that is viewing the portrait.This is apparent from the grayscale version of Genie Mask,also in Figure 7.So, I was back in business, and I expected that it wouldnot take long to find a JPG of this portrait that I could use,modify my Mathematica file to import the image, createf(x; y) and render it as a 3D image, export the result asan STL file, and 3D print the result. For the first two steps,I was correct, but the rest of the steps required extensivecomputations, and actually my computer ran out of memory.I needed a mathematical tool well-known to me to finish theproject.V. SAVING TIME WITH WAVELETSI’ve been working with wavelets since the late 1990’s.Inspired by the adoption of wavelets by the U.S. FederalBureau of Investigation (the FBI) for compression of finger-print images, I learned the basics using publications writtenby Strang [16], Mulcahy [14], and eventually Daubechies [9].After teaching introductory wavelets materials to undergrad-uates, Steve Schlicker and I co-wrote our own introductionto wavelets for undergraduates [1], [2], and I began leadingundergraduate research projects that applied wavelets in avariety of ways (see https://sites.google.com/site/aboufadelreu/). So what are these wavelets?At their simplest, a wavelet family is a set of linearindependent functions that can be used for analysis, in thesense of taking other functions and writing them as a linearcombination of wavelets functions. Or, more accurately,approximating functions with linear combinations, which isbasically a projection onto a wavelet subspace. For instance,the Haar wavelets are piecewise constant functions, andFig. 7. Original and Grayscale version of Demain’s Genie Mask.projecting continuous functions onto the Haar subspace leadsto examples like what is seen in Figure 8.With the type of discrete wavelet analysis being describedhere, the projection is created by computing a linear combi-nation of function values. For instance, with the Daubechies-4 wavelets, the “low pass” projection filter is computed inthis way:(Hs)k = h0s2k + h1s2k+1 + h2s2k+2 + h3s2k+3Fig. 8. Example of a projection of a continuous function onto a Haarwavelet space.h0 =1 +p34p2 0:48296h1 =3 +p34p2 0:83652h2 =3 p34p2 0:22414h3 =1 p34p2  0:1294There are also “high pass” filters which are computedby “differencing”, as indicated in this formula from theDaubechies-4 wavelets:(Gs)k = g 2s2k 2 + g 1s2k 1 + g0s2k + g1s2k+1:g 2 =1 p34p2  0:12941g 1 =  3 p34p2  0:22414g0 =3 +p34p2 0:83652g1 =  1 +p34p2  0:48296:The high-pass filters are good for, among other applica-tions, finding edges in images, and spikes in one-dimensionalsignals. For instance, a high-pass filter was used in the“Boston Pothole Project” that I completed with three studentsin 2011 [13].VI. CREATING A 3D VERSION OF Genie MaskThe filters above are for one-dimensional signals, but thereare versions for higher dimensions [6], and for my attemptto 3D print Fauvist paintings, I needed something that wouldwork on a two-dimensional image. For simplicity, I decidedto use the two-dimensional Haar low-pass wavelets filters,which basically smooth an image by averaging pixel valueson 2 by 2 blocks. So, we think of the image as a matrix ofgrayscale luminances (either in the 0-1 range, or the 0-255range, it doesn’t really matter). After dividing the matrixhorizontally and vertically into 2 by 2 blocks, we replaceevery block with one number which is the average of the 4values in the block. This creates a new matrix which has halfthe width and height, and creates a “smoother” version of thematrix. Applying this methodology several times (which is apart of what is called the “pyramid scheme” in the literature)leads to smoother and smoother versions of the 3D renderingof Demain’s Genie Mask (see Figure 9).Fig. 9. Rendering of output from multiple rounds of filtering of GenieMask matrix.This, in turn, led to the 3D print of Genie Mask, seen inFigure 10.In 2014, Carol McInnis, an artist that called herself“Fauvist”, posted the “Blue” portrait on the left in Figure11. I applied my methodology to this image, yielding the3D printed portrait also seen in Figure 11. (Neither Carolnor this image can be found on the Internet in June 2016.Also note that in the image that I used, I adjust the scalingof the x axis.)Fig. 10. Genie Mask as a 3D print.Fig. 11. Another example of starting with a Fauvist image, converting tograyscale, applying wavelet filters, and then 3D printing.VII. FINAL COMMENTSThis is not a very sophisticated application of wavelets,but it accomplished what I wanted, which is usually all thatyou need in a mathematical modeling project. An interestingdirection to pursue for a future project is to apply theDaubechies-4 wavelets, either the one dimensional in twodirections or the two-dimensional version [3], to see how thatwould smooth the original image. One advantage of usingthese wavelets is that they are applied to 4 by 4 boxes ratherthan 2 by 2, so that fewer rounds of smoothing would beneeded.Some other directions to pursue: Is there a good “Fauvistfilter” to apply to school portraits? If not, is it possibleto create one? And, of course, would this approach workwith other Fauvist paintings, particularly those which are notportraits?REFERENCES[1] Aboufadel, E. and Schlicker, S.: Discovering Wavelets, Wiley, 1999.[2] Aboufadel, E. and Schlicker, S.: “Wavelets”, in Encyclopedia of Phys-ical Science and Technology, 3rd edition, Robert Meyers, ed., Elsevier,2001.[3] Aboufadel, E., Cox, A., Vander Zee, A: “Bivariate Daubechies ScalingFunctions”, Available on the arXiV, March 2001. http://arxiv.org/abs/math/0103198[4] Aboufadel, E., Krawczyk, S., Sherman-Bennett, M.: “Talk to the Hand:Generating a 3D Print from Photographs”, Available on the arXiv,February 2015. https://arxiv.org/abs/1502.02871[5] Aboufadel, E.: “3D Printing A Pendant with A Logo”, Available on thearXiv, July 2015. https://arxiv.org/abs/1507.03102[6] Addison, P.: The Illustrated Wavelet Transform Handbook, NapierUniversity, United Kingdom, 2002.[7] Austin, D.: “How to Make A 3D Print”, American Mathematical So-ciety web site, March 2014. http://www.ams.org/samplings/feature-column/fc-2014-03[8] Cheng, E.: How To Bake Pi: An Edible Exploration of the Mathematicsof Mathematics, Basic Books, 2015.[9] Daubechies, I.: Ten Lectures on Wavelets, SIAM, 1992.[10] Douma, M.: “Color Vision and Art” and “Causes of Colors”,Online exhibits, 2008. http://www.webexhibits.org/colorart/fauve.html and http://www.webexhibits.org/causesofcolor/mind.html[11] Ferrier, J-L.: The Fauves: The Reign of Colour, Terrail (Paris), 1992.[12] Livingstone, M.: “What Art Can Tell Us About The Brain”, Videoavailable on YouTube, October 2009. https://www.youtube.com/watch?v=338GgSbZUYU[13] Mackenzie, D: “REU Students Apply Wavelets to Bostons Pot-hole Challenge”, SIAM News, June 2012. http://www.siam.org/news/news.php?id=1983[14] Mulcahy, C.: “Plotting and scheming with wavelets”, MathematicsMagazine, 69, 1996, pages 323 343.[15] Segerman, H.: “3D Printing for Mathematical Visualization”, TheMathematical Intelligencer, Volume 34, Issue 4, 2012, pages 56 62.http://www.ms.unimelb.edu.au/˜segerman/papers/3d_printed_visualisation.pdf[16] Strang, G.: “Wavelets and dilation equations: a brief introduction”,SIAM Rev., 31, 1989, pages 614 627.[17] Whitfield, S.: Fauvism, Thames and Hudson, 1991.

3D Printing & the Artists Known As the Fauves

Abstract

Similar works

Full text

Available Versions

Scholarworks@GVSU