Custom 3d Printed Rollers For Frieze Pattern Cookies-Books Pdf

Custom 3D Printed Rollers for Frieze Pattern Cookies
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These rollers are 4 to 12 cm high when stood on end as shown and are built from ABS plastic on a Makerbot. Replicator 3D printer 5 The STL files are freely available 3 so anyone can make copies of these rollers. Figure 2 Escher image Lizard 3D printed roller and sugar cookies. The Escher image in Figure 2 consists of black and white reptiles 6 A set of equivalent points are marked here. with circles The chosen point is the upper eye of a downward facing black reptile but any point would do. equally well to bound a fundamental region A square bounded by four of these points could be used as a rubber. stamp to produce the entire image using only translation Because a square rolls up into a thin cylindrical shell. that can be difficult to manipulate the marked rectangle consisting of two squares was input to the software to. produce the roller in Figure 2 It has two copies of the image and twice the diameter of a roller based on a single. square The black and white regions of the image are translated into two levels of depth A sugar cookie dough. recipe 7 was rolled out and imprinted with the roller to make the cookies shown. Figure 3 Escher Horsemen positive and negative 3D printed rollers and their Play Doh impressions. In a similar way a rectangular region from M C Escher s Horsemen shown in Figure 3 6 was fed to the. software to produce the two rollers shown The four levels of darkness in the image are translated to four. different heights in the rollers They differ according to whether the darkest parts of the image correspond to. mountains or valleys on the roller The user can adjust this by negating a parameter alpha described. below For this design it is apparent that the second impression is easier to recognize where the black lines. correspond to mountains on the roller which make valleys on the cookie surface. Figure 4 Hittite cylinder seal image with Janus 3D printed roller and Play Doh impression. There is a long history of carved stone rollers being used to make clay seals e g to authenticate royal. documents 8 Numerous examples of intricately carved Sumerian Hittite or Mesopotamian cylinder seals. going back to 3500BC can be found in museum collections An exhibition of ancient Near Eastern cylinder. seals at the Morgan Library in NY inspired us to begin this project Figure 4 shows a drawing of a Hittite. seal 9 a 3D printed roller we derived from it and a Play Doh impression from the roller In this case we. started from a modern line drawing of the seal which indicates the contours of the figures so our cylinder. recreates these lines However the original 15 century BCE seal in the Louvre Museum 10 on which the. drawing is based is actually a relief in the style of the following example. Figure 5 Mesopotamian cylinder seal modern impression and depth map reconstruction of one cycle. Figure 5 shows an interesting Mesopotamian seal from 3000BCE also in the Louvre Museum and a modern. impression taken from it 11 Some people promote this example as evidence that dinosaurs and humans once. co existed but we and no doubt the Louvre do not endorse that interpretation Unlike the line drawing of. Figure 4 this image makes clear the nature of the typical reliefs on ancient cylinder seals The depth of the. carving is a sculptural representation of the volume of the objects shown To recreate this style called intaglio. or counter relief we need to feed the software an image in which darkness corresponds to depth In modern. computer oriented jargon this is termed a depth map An approximate reconstruction of its depth map shown. at bottom right in Figure 5 was made by tracing the impression image using an air brush tool in a drawing. program Feeding it to the software allows us to produce the roller in Figure 6 and recreate the relief style of the. impression Close examination shows this is not a very exact reconstruction but it illustrates a possible. technique if one wishes to use this software to produce this style of roller. Figure 6 3D printed cylinder and Play Doh impression roughly reconstructing Figure 5. Figure 7 Greek Key mosaic unit drawing and resulting frieze pattern imprinted in Play Doh. Rollers for arbitrary frieze patterns can be produced by creating one translational unit of the pattern in a drawing. program and feeding the image to the software Figure 7 shows an example in which we created a mosaic like. image by stamping white and grey squares on a black background to create one unit of a classic Greek Key. frieze pattern The resulting roller has two heights of square tiles rising above the lower background of grout. between the tiles Some cookies made from this roller can be seen in Figure 1. The software implements mathematical transformations of linear scaling darkness to distance and changes of. coordinate systems An image file provides a table of data which encodes darkness as a function of x and y. position in the image The program transforms x and y to become angle and height in the cylindrical coordinate. system with the full range of x scaled to one complete revolution so the right and left sides of the image join. The nominal radius of the cylinder is chosen so that the ratio of the cylinder s height to its circumference equals. the ratio of the image height to its width Then this nominal radius is modified at each point by adding an. increment proportional to the darkness of the image at the corresponding pixel. An arbitrary parameter called alpha in the program can be specified by the user to scale this depth increment. Setting a larger value for alpha causes the white regions of the image to be carved deeper into the cylinder. leaving the black to protrude from the cylinder and make deeper indents in the cookies Setting alpha to zero. would give a smooth cylinder with no texture so is not useful Setting alpha negative reverses the direction of. the increment so white regions of the image protrude from the cylinder which gives a different visual effect. Figure 3 shows the effect of negating alpha Most users with drawings containing dark lines will find a positive. alpha works best but some experimentation may be worthwhile. Finally the cylindrical coordinates are transformed into 3D Cartesian coordinates which are exported to the STL. file The handedness is such that the image appears reversed on the cylinder so it is not reversed in the cookie. Circular caps are added at the top and bottom of the cylinder so the whole is a water tight manifold without. boundary which is necessary for the slicing software in 3D printers to operate correctly. To use the software for your own custom roller follow these steps. 1 Find or create an image file in any standard image format Degrees of darkness will correspond to desired. depth so it is best if it has good contrast If starting with a color image converting it to gray scale in an image. editing program will show how the software interprets it The X direction of the image will wrap around the. circumference of the roller and should be about 400 500 pixels wide for reasonable resolution without an overly. large file Rescale the image if necessary It may be useful to blur or smooth the image if it is grainy to avoid a. very jagged roller surface If working from an image of a frieze pattern or tessellation crop it to a rectangle that. repeats by translation as indicated in Figure 2 Save the file in any standard image format. 2 Download the software 3 and open it within Mathematica Execute the first block of code to import your. image Edit the lines which specify the desired cylinder height and the alpha parameter if you wish to change. them The height is specified in arbitrary units that are set in the 3D printer s software commonly inches in the. US and mm elsewhere Alpha is the fraction of the height that black is scaled to e g if alpha 0 1 the deepest. indents of a 2 inch high roller will be 0 2 inches A negative alpha results in dark portions of the image indented. in the roller and light portions sticking out Execute the second block of code to generate and display the. cylinder model You will probably need to adjust alpha and regenerate it until the texture scale looks. appropriate When the scale of the mountains and valleys looks OK for your application execute the final line. of code to generate the STL file Set the output file name as you wish but include the stl file extension. 3 Send the STL file that is generated to your 3D printer or an online 3D printing service bureau to produce the. cylinder You will need to specify what units inches or mm to use. 4 Make cookie dough according to your favorite recipe and roll the roller over it Cookies which rise. significantly during baking will lose some of the imprinted detail We find a simple sugar cookie works well If. desired decorate them further by filling the depressions or coloring the raised areas Bake and enjoy. Applications and Future Work, Math cookies are good any time but they may be especially valuable in providing a fun activity for a classroom. or math club Many aspects of symmetry and transformations may arise giving the teacher an opportunity to. informally introduce the underlying mathematical ideas The seven classes of frieze patterns as categorized by. their symmetries are a natural topic to discuss It should be pointed out that only the translation operation is. naturally captured by the process of rolling a cylinder Any additional symmetries of a frieze e g reflections. rotations or glide reflections can be incorporated by explicitly repeating the pattern on the cylinder surface. Custom rollers are also useful to ceramicists for applying repeating patterns in their work There is a sensual. pleasure to be found in working with clay and we find it is very satisfying to roll these patterns into Play Doh. and physically recreate both ancient designs and Escher s designs Our current experiments with rolling soft clay. impressing patterns and baking are very promising A software option which we have implemented but not. illustrated here creates a cylindrical hole along the axis of the roller This mimics certain ancient seals that were. drilled out so they could be worn as an amulet necklace But we implemented it also to allow ceramicists to. place a dowel through the roller to use as a handle for finer control of the clay. Future work could look at software techniques to automate parts of the process for example finding the optimal. value of alpha for a given image or cropping an optimal unit rectangle from a tessellation image A friendlier. user interface might make it easy to double or treble an image to avoid making a roller which is too thin to roll. easily as we did manually when using two squares in Figure 2 More advanced culinary research might model. the dynamics of how cookie dough expands when baked and adjust the roller height field to create imprints for. optimal readability after cooking, Conclusion, Carl Sagan once said that if you want to make an apple pie from scratch you must first invent the universe. We see a corollary that if you wish to make cookies from scratch you might first need to write software to. generate the cookie roller s geometry This paper provides that step so users can go on to use it in making their. own mathematically patterned cookies And more importantly we provide a general tool which others may use. in many creative mathematically themed projects We expect it will prove especially valuable to people who are. comfortable with image processing software but are unfamiliar with 3D design software We look forward to. seeing what others create using their own 3D printed roller designs. References, 1 Evil Mad Scientist Laboratories http www evilmadscientist com 2008 sierpinski cookies. 2 Thingiverse http www thingiverse com thing 3248, 3 G Hart http georgehart com rollers. 4 Wolfram Research http wolfram com, 5 Makerbot http makerbot com.
6 Doris Schattschneider M C Escher Visions of Symmetry Abrams 2004. 7 G Hart trilobite recipe http www georgehart com trilobites trilobite html. 8 Wikipedia http en wikipedia org wiki Cylinder seal. 9 Bill Schell http campus murraystate edu academic faculty bill schell civweb101 Hittites htm. 10 Louvre cylinder photo http www lessing photo com dispimg asp i 08021530. 11 Wikimedia http commons wikimedia org wiki File Uruk3000BCE jpg and. http commons wikimedia org wiki File Cylinder seal lions Louvre MNB1167 n2 jpg. For completeness the executable statements of the Mathematica code are given here To save space comments. and other material are omitted but the complete executable file in Mathematica s format can be downloaded 3. img ColorConvert Import SystemDialogInput FileOpen Grayscale. height 2 0, alpha 0 05, n m ImageDimensions img, rMid n height 2 Pi m. r Map rMid alpha height 0 5 ImageData img 2, xyz Table r j i Sin 2 Pi i n r j i Cos 2 Pi i n 1 j m height i 1 n j 1 m. surface Table xyz i j xyz Mod i 1 n 1 j, xyz Mod i 1 n 1 j 1 xyz i j 1 i 1 n j 1 m 1. bottom Table 0 0 0 xyz i m xyz Mod i 1 n 1 m i 1 n. top Table 0 0 1 1 m height xyz Mod i 1 n 1 1 xyz i 1 i 1 n. g Graphics3D EdgeForm Map Polygon Join surface top bottom. boundary which is necessary for the slicing software in 3D printers to operate correctly Process To use the software for your own custom roller follow these steps 1 Find or create an image file in any standard image format Degrees of darkness will correspond to desired depth so it is best if it has good contrast If starting with a color

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