Category Archives: 3d Printing

New Torus Link, Improved Visualizations, and Cinema 4D Problems.

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Written by Hillis Burns, Shannon Timoney, Hall Pritchard (students in Math 383D Knot Theory Spring 2023).

We created the T(2, 8) torus link (or 821 link) using Cinema 4D. The equations for this two component link are x = Cos[t]*(3+Cos[4t]), y = Sin[t]*(3+Cos[4t]), and z = Sin[4t]. The second component is created by the equations x = Cos[t]*(3+Cos[4t+Pi]), y = Sin[t]*(3+Cos[4t+Pi]), and z = Sin[4t+Pi].

We also created the T(3, 3) torus link (or 632 link). With the  632 link, each of the three components goes around the longitude once and goes around the meridian once. The equations for this knot are x1 = Cos[t]*(3+Cos[t]), y1 = Sin[t]*(3+Cos[t]), z1 = Sin[t], x2 = Cos[t]*(3+Cos[t+2*Pi/3]), y2 = Sin[t]*(3+Cos[t+2*Pi/3]), z2 = Sin[t+2*Pi/3], and x3 = Cos[t]*(3+Cos[t+4*Pi/3]), y3 = Sin[t]*(3+Cos[t+4*Pi/3]), z3 = Sin[t+4*Pi/3].

Image showing the T(3,3) torus link lying on the torus.

The T(3,3) torus link shown lying on the torus.

After creating the T(3, 3) or 632 link, we wanted to build a model that helps demonstrate what the torus link actually is. We did this by first opening back up the T(3, 6) or 632  in Cinema 4D. Then we created a torus surface and rotated it 90° so that the torus link sat in the right position on the torus surface. Then we changed the radius of both the meridian and the longitude so that the 3d model was in a presentable format. Our final model which is shown above gives a good physical representation of how a torus link is constructed.

We also created the  T(3, 6) torus link (or 633 link). With the 633 link, each of the three components goes once around the longitude and goes three times around the meridian. The equations for this knot are x1 = Cos[t]*(3+Cos[2t]), y1 = Sin[1t]*(3+Cos[2t]), z1 = Sin[2t], x2 = Cos[t]*(3+Cos[2t+2*Pi/3]), y2 = Sin[t]*(3+Cos[2t+Pi]), z2 = Sin[2t+2*Pi/3], and x3 = Cos[t]*(3+Cos[2t+4*Pi/3]), y3 = Sin[t]*(3+Cos[2t+4*Pi/3]), z3 = Sin[2t+4*Pi/3].

While using the Cinema 4D software, the biggest problem we had was fixing the join at the end of two strands. In Cinema 4D, the join will sometimes not look correct. In order to fix this, we will first decrease the period of the parametric equations in order to make the join fully noticeable. We decreased it from 2π (~6.28) to 6.275.

Image showing the two ends of the torus link with too many points highlighted

Too many points are highlighted near the ends of the torus link.

We then try to highlight all the points at the end of the knot, in order to use the “stitch and sew” function. We came across problems when we accidentally highlighted other points not at the end. This is shown in the figure to the right. This would happen more often when we increased the sample size to a number that was higher than necessary (>300). This is the case in the image below. By decreasing the sample size, it made it easier to highlight just the end points of the knot, as shown below. At this point we were able to successfully use the “stitch and sew” function.

Image showing points highlighted along the ends of the torus link.

This image shows that just the end points of the two ends of the torus link are highlighted. This allowed us to successfully use the Stitch and Sew function to join the ends together.

Overview of Torus Shapes, Knots, and Links

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Written by Hillis Burns, Shannon Timoney, Hall Pritchard (students in Math 383D Knot Theory Spring 2023).

Image of a torus with a 3 component link on it

The T(3,6) torus link.

The torus is the surface of a donut in 3-dimensions. A torus knot/link is a knot/link that can be moved to lay on the torus surface in R3.The image on the right shows a link being wrapped to lie on a torus; this is the T(3, 6) or 633 torus link. Knots are also commonly described in knot tables using the notation, Crnj. The crossing number is denoted by Cr, the number of components by n, and the certain configuration is j. As seen in this image, the torus has two key circles: the longitude, which wraps around the long way of the torus, and the meridian, which wraps around the short way. These are illustrated in the image below.

Image of a torus showing the longitude and meridian curves

A torus with some Longitude and Meridian curves highlighted.

The notation for torus knots is T(p, q); The knot wraps around the longitude p times, while it wraps q times around the meridian. The two figures are from a Mathematica file which visualizes torus links, and this website. This Knot Plot website also has a neat table showing many of the torus knots and links.

Using a Mathematica file provided by Professor Denne, we were able to start creating the T(2, 4) torus link (or 421) in Cinema 4D. This is a two-component link where each component goes once around the longitude and twice around the meridian, as illustrated below.

Image showing the T(2,4) torus link

The T(2,4) torus link.

The parametric equations that create the link are x =  Cos[t]*(2+Cos[2t]), y = Sin[t]*(2+Cos[2t]), z = Sin[2t], with t going from 0 to 2Pi. However, for this link there are two components. Thus, a second equation was needed for the second component. To create a torus link like this, the second equation must be rotated 180 degrees to fit with the first curve. To do that, we added Pi to the trigonometric equations of the first sweep: x = Cos[t]*(3+Cos[2t+Pi]),  y = Sin[t]*(3+Cos[2t+Pi]), z = Sin[2t+Pi].

We also decided to create the T(2, 11) torus knot (also known as 111) in addition to the links in Cinema 4D. This is a knot (one component link) where the curve goes twice around the longitude and 11 times around the meridian. The topmost figure below shows the original image of the torus knot that we created. The knot does not look smooth, as Cinema 4D only evaluated a few points along the parametrized curve. However, after adding more sample points we were able to make the torus knot smoother. The progression of sample points, from 20, to 50, to 100, to 200, is shown from top to bottom:

T(2,11) with 20 sample pointsT(2,11) with 50 sample points

T(2,11) with 100 sample pointsT(2,11) with 200 sample pointsWe also had to adjust the radius because the loops were too close together. In order to spread out the components, the radius was changed from 2 to 3.

Next, we created a T(2, 6) torus link (or 621) in Cinema 4D. With the 621, each of the two components goes once around the longitude and three times around the meridian. Using the Mathematica file, we knew that the equations for this link were, x = Cos[t]*(3+Cos[3t]),  y = Sin[t]*(3+Cos[3t]),  z = Sin[3t].  The second component’s equations, again rotated by Pi, consist of, x = Cos[t]*(3+Cos[3t+Pi]), y = Sin[t]*(3+Cos[3t+Pi]),  z = Sin[3t+Pi].

3D Printed 7_1 Mosaic Knot

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Written by Sion Jang and Charlotte Peete (students in Math 383D Knot Theory Spring 2023).

The mosaic number for the 71 knot is six, meaning that it cannot be created on a grid using mosaic tiles smaller than 6 x 6. We are creating a 3D version of 71 mosaic knot using Cinema 4D as our main design program. We created this knot using the same method as our Trefoil knot. However, we made changes to the Chamfering process, the diameter of the tube, and the distance between some of the over-strands and feet.

When creating this knot with the same process as the Trefoil and Figure-8 knots, we came across a few problems.

The arrows show the three feet where the z-coordinates were changed.

Figure 1: The red arrows show the three feet where the z-coordinates were changed.

Since the 71 knot has more crossings than either of the other two knots we created, we found that the close proximity of the feet would lead to self-intersections. We first changed the diameter of the circle from 6 to 4 mm. While this change helped to solve the intersection problem, the feet still seemed to be close together. Aesthetically, we still weren’t satisfied with how crowded the feet looked. So, we changed the z-coordinates of the horizontal feet to create more space between the adjacent vertical feet. The arrows in Figure 1 point to the three feet for which we changed these coordinates. Figure 2 shows an overhead view of the spacing between the feet with these changes.

Overhead view shows spacing between alternating feet.

Figure 2: Overhead view shows spacing between alternating feet.

The biggest challenge we came across with this knot was figuring out how to properly curve vertices without distorting the rest of the knot. Our original method of Chamfering did not work because there wasn’t enough space between the curves of the knot and the feet. To fix this problem, we added an additional point next to each vertex of the over-strand immediately before the foot. These points were added as close to the original vertices as possible.

Figure 3 shows our final product.

Finished 7_1 mosaic knot.

Figure 3: Finished 7_1 mosaic knot.

 

3D Printed Trefoil Mosaic Knot

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Written by Sion Jang and Charlotte Peete (students in Math 383D Knot Theory Spring 2023).

Mosaic knot theory uses a combination of the following eleven tiles to create a knot or a link representation on an nxn grid. These tiles are shown in Figure 1.  As explained in the Knot Mosaic Tabulation paper by Hwa Jeong Lee, Lewis D. Ludwig, Joseph S. Paat, and Amanda Peiffer, the mosaic number of a knot K is the smallest integer n for which K can be represented on an  mosaic board. This mosaic number is a knot invariant and can be used to distinguish between two knots.

Tiles used to create mosaic knots

Figure 1: tiles used to create a mosaic knot or link.

The Knot Mosaic Tabulation Paper provided the minimal grid mosaic diagrams for all 36 prime knots of eight crossings or fewer. The mosaic number for a trefoil is four, m(31)=4. Thus, a trefoil cannot be created on a 3 × 3 grid, and the minimum grid needed is a 4×4 grid. We created a 3D version of 31 Trefoil mosaic knot using Cinema 4D as our main design program. To create this knot, we first drew a coordinate plane onto the diagram of the Trefoil knot, as shown in Figure 2.

Mosaic trefoil knot with coordinates in xy-plane

Figure 2: Trefoil mosaic knot with coordinates in xy-plane.

We did this so that we could insert integer coordinates into Cinema 4D that would allow for accurate spacing in our knot. Our final knot was scaled to be around 7×7 cm without the circle sweep, which is the thick tube around the curve.

One of the main challenges we encountered in constructing this knot was figuring out how to represent the over and under-crossings. Using this blog by Laura Taalman as inspiration, we decided to create “feet” for our knot. The distance from the center of each foot to the center of the over-strand is 1 cm. We also put a .125 mm space on either side of the over-strand between the legs to show that the strands are not connected. Figure 3 shows a close-up representation of one of the feet and legs on this knot.

Close-up image of the legs and feet of an under-strand.

Figure 3: The foot and two legs create a strand which crosses under a different strand of the knot.

The coordinates we chose made the knot have an angular rather than curved shape, and we manually curved the knot in Cinema 4D using the “Chamfer tool” once our knot was constructed. Figure 4 shows our knot before we curved the edges. Figure 5 shows our knot after we curved the edges. For the different curves, we used different values of the Chamfer to get the desired look.

Top view & angled view of the trefoil knot showing the sharp corners.

Figure 4: Top view & angled view of the trefoil knot before curving the corners.

Top view & angled view of the trefoil knot after curving the corners.

Figure 5: Top view & angled view of the trefoil knot after curving the corners.

The Figure-8 mosaic knot

FIgure 6: The figure-8 mosaic knot.

Using this same process, we created 41, the Figure-8 knot. The final product is shown in Figure 6.

Modeling and 3D-Printing Equilateral Stick Knots

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Written by Aidan Aengus Kitchen, Arun Ghosh, and Alex Wolff (students in Math 383D Knot Theory Spring 2023).

Brief Math Background:

Image showing knots with 3, 4, 5, 6, 7 crossings, and a few 8 crossing knots.

Knot table. Image from https://knotplot.com/zoo/

A knot is a simple, closed curve in space. This means that it forms a closed loop and does not intersect itself. The figure to the left illustrates the simplest knots with 0 to 8 crossings (from https://knotplot.com/zoo/).  The knots are labelled with the crossing number and a subscript which is a number that distinguishes between different knots with the same number of crossings. A polygonal knot is composed of a finite number of edges or straight sticks. For an equilateral stick knot, the edges have the same length.  We constructed equilateral stick knots representing the knots shown above. In the models we made, we used the minimum number of sticks necessary to construct the knots.

Clayton Shonkwiler is an Associate Professor of Mathematics from the University of Colorado. His primary research is using geometry to solve topological and physical problems. His recent work published in 2022, New Stick number bounds from random sampling confined polygons, looks at equilateral stick knots. The paper focuses on finding the upper and lower bounds for stick number and also gives the coordinates for constructing the knots in 3 dimensional space. These coordinates can be found in following GitHub repository: https://github.com/thomaseddy/stick-knot-gen/tree/master/stick_number/mseq_knots

The table below displays the crossing number of the knots we constructed, as well as the number of sticks we used for each one.

Knot Crossing # # of sticks used
31 3 6
41 4 7
51 5 8
52 5 8
61 6 8
62 6 8
63 6 8
71 7 9
72 7 9
73 7 9
74 7 9
75 7 9
76 7 9
77 7 9
949 9 9
K11a1 11 12
K12n63 12 11

How We Built the Knots:

For each knot, we first retrieved the coordinate data from the GitHub repository and saved it in a text file. Then, we imported the data into Cinema4D by creating a linear spline object, opening the Structure window, clicking on “Import ASCII Data” and navigating to the text file. In order to close the knot, we had to add a point at the origin and connect it to the last point in the imported data using the Spline Pen.

Image showing the 3_1 equilateral Stick Knot.

3_1 Equilateral Stick Knot.

Then, we resized the spline to 6-7 cm for each dimension, and swept a circle of radius 0.3 cm around the spline, creating a tube around the knot. We did this to make the knots 3-dimensional, because we cannot 3D-print a spline (a 1-dimensional object in 3D space). To round the corners, we used the Chamfer function. Finally, we exported the models as .stl files and 3D printed them. In total, we designed all of the equilateral stick knots with three, four, five, six, and seven crossings. Additionally, we modeled knots with higher crossing number, such as 949, k11A1, and k12N603. The images above and below depict the 31 and 61 equilateral stick knot models.

An image of the 6_1 equilateral stick knot

6_1 Equilateral Stick Knot

Challenges and Observations:

  • Scaling
  • Self-intersections
Image showing self-intersections in the 6_1 knot

Self-intersections in the 6_1 knot

Initially, we did not scale our models to the appropriate size (suggested 6-7cm, or palm size). We also moved vertices around in the original 63 model, so the sticks lengths were altered, and we had to redo it. Additionally, the radius of the sticks was too small to neatly 3D print a label on the physical knot, so we decided to tape the labels on after the knots were printed. One interesting observation we made was increasing the radius of the tubes created self-intersections between the “tubes” that were not evident in the spline. The images above and below highlight self-intersections in the 61 and K12n630 models.

Image showing self-intersections in the K12n630 knot

Self-intersections in the K12n360 knot.

Creating a 3D-printable Lorenz attractor

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The Lorenz attractor is a set of chaotic solutions of the Lorenz system which, when plotted, resemble a butterfly or figure eight. The Lorenz system is a system of ordinary differential equations first studied by Edward Lorenz in the 1960’s. It is notable for having chaotic solutions for certain parameter values and initial conditions. 

In Winter 2015, my colleague Professor Greg Dresden used the Lorenz attractor as an example in his course on Partial Differential Equations. At that time, I worked with Dave Pfaff in the IQ center at W&L to find a way to 3D print a physical model of the solution curves. Dave also created model of the solution curves that could be viewed with 3D glasses in our stereo 3D lab. To create the models we followed three steps:

Step 1:  Create points along the solution path (the Lorenz curve) using Mathematica. Here is the Mathematica code Greg developed for this purpose.

(* Here are the differential equations *)
leqns = {  x'[t] & == -3 (x[t] – y[t]),
y'[t] & == -x[t] z[t] + 28 x[t] – y[t],
z'[t] & == x[t] y[t] – z[t] };
(* Here, I define two paths, p1 and p2, which start at slightly-different initial values. *)
p1 = NDSolveValue[{leqns, x[0] == z[0] == 0, y[0] == 1}, Function[Evaluate[{x[\#], y[\#], z[\#]}]], {t, 0, 30}];
p2 = NDSolveValue[{leqns, x[0] == z[0] == 0.03, y[0] == 1},  Function[Evaluate[{x[\#], y[\#], z[\#]}]], {t, 0, 30}];
(* Let’s look at a plot of these two paths, to verify that they seem correct *)
pic1 = ParametricPlot3D[p1[t], {t, 0, 30}, AxesLabel -> {x, y, z}, PlotStyle -> {Red, Opacity [0.5]}];
pic2 = ParametricPlot3D[p2[t], {t, 0, 30}, AxesLabel -> {x, y, z},
PlotStyle -> {Black, Opacity [0.5]}];
(* Finally, we export them as two separate files *)
tableout1 = Table[p1[t], {t, 0, 30, 0.01}];
tableout2 = Table[p2[t], {t, 0, 30, 0.01}];
Export[“TableOut1.xls”, tableout1]
Export[“TableOut2.xls”, tableout2]

Step 2:  Use Excel to tweak the data into a form we can use.

  • Open the files in Excel.
  • Insert a column before the three columns of numbers, this is column A. Make the first row 1, and the second row 2 in column A. 
  • Highlight these two entries in column A, then drag the box down to row 3001. The numbers in column A will automatically fill in the correct numbers.
  • Highlight all of the numbers in all four columns. Click on Save As, then save the file as a Tab Delimited Text (.txt) file. 

Step 3:  Create the thickened Lorenz curve in Cinema 4D.

  • Open Cinema 4D
  • Construct the spline with the Lorenz data.
    • Left click and hold on the Spline button on the top row, then select the Linear (spline) button. Click once on the view-port to add one point.
    • Go to the Object Manager. On the right side, click on the Structure button. You’ll see the Point 0, with X, Y and Z coordinates. 
    • Click on the File button above the Object Manager. Click on Import ASCII Data… Open the .txt file created above.
    • Click on the 0th point (the one you added initially) and Delete it. The remaining points are from the Lorenz data (see figure.)
    • Click on the Objects button on the right side of the Object Manager. 
  • Thicken the spline for 3D printing.
    • Left click and hold on the Subdivision Surface button on the top menu, then select the Sweep button.
    • Left click and hold on the Spline button on the top row, then select the Circle button. Under the Attribute Manager, make the radius of the circle 3mm (0.3cm).
    • In the Object Manager, move the Circle and the Spline under the Sweep. The Circle should be above the Spline (see figure). The thickened spline is now complete, and the Lorenz curve can now be exported (as a .stl file) and then 3D-printed.

A thickened Lorenz curve in Cinema 4D is shown below on the left. Two curves can be added to the same plot in Cinema 4D and given different colors.  Dave Pfaff printed two such curves using the Project 260 3D printer at WLU. (This printer uses a gypsum-like powder hardened via a laser then finished with superglue.)  The Project 3D printer can print in color (using inkjet cartridges). Dave also designed a stand for the curves. This gives the beautiful model shown below on the right. Cinema 4D also allows the user to animate the curves. Dave created such an animation, which nicely shows the chaotic nature of the Lorenz attractor. The initial points of the two curves were very close, but in the long term, the curves diverge.

 

 

Five Intersecting Tetrahedra

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One of my favorite mathematical models is the Five Intersecting Tetrhadra model. I first became aware of this model from Thomas Hull, a mathematician at Western New England University. He has written extensively about mathematics and origami (check out his book Project Origami). In particular he has written fantastic instructions for creating this model using modular origami.

dodec1tetThe five intersecting tetrahedra model is based on the dodecahedron. This one of the five classic regular polyhedra consisting of 12 pentagonal faces and 20 vertices. Take 4 vertices in the dodecahedron which are the same distance apart. These form the 4 vertices of a regular tetrahedron, as shown on the right (figure from Tom). Since the dodecahedron has 20 vertices, we can inscribe 5 such tetrahedra inside the dodecahedron.

The origami version comes about by making a thin frame for each tetrahedron. Provided the frames are thin enough, they won’t intersect each other. Instead they will form a marvelous interwoven pattern. As shown on the left, I have successfully followed Tom’s instructions for making the origami version of this model. 

The next natural question is whether or not the model can be 3D-printed. The answer, is of course, yes! I used crsfdr’s model Interlocking Pyramids  from Thingiverse to print the model on the UPrint SE printer at W&L.  The photo at the top shows the finished version. The one to the right shows the model just out of the printer before the supports have been removed. Since the supports are made of a material that dissolves, the UPrint was the perfect printer for the job.

My colleague Professor Michael Bush has used the Five Intersecting Tetrahedra model when teaching introductory Group Theory to undergraduate students. Indeed, this was the motivation for 3D-printing the model as the origami version is not really robust enough to use in a classroom setting. The model is a great tool for discussing the rotational symmetries of the dodecahedron (or its dual the icosahedron). Roughly speaking, the rotational symmetries of the dodecahedron act in a natural way on the five tetrahedra giving a permutation representation of the symmetry group. After some noodling around this allows one to see that this group is the alternating group \(A_5\). (Michael usually suppresses details about the faithfulness of the representation at this level.)

 

More quadratic surfaces

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In July it was time to get back to designing and 3D printing some more math models. I had previously seen some beautiful models printed by Dave Pfaff on the Series 1 Pro, using its ability to print in a spiral. I wanted to print a large hyperboloid of one sheet using this printing technique.

So I went back to Cinema4D to design the model. I used the Function tool to make the hyperbola \(\frac{x^2}{4} -\frac{y^2}{9}=1\), which has asymptotes \( y=\pm \frac{3x}{2}\). I then used the Lathe tool to revolve the hyperbola around the \(y\)-axis and create a hyperboloid of one sheet. I knew, from the asymptotes, that the surface would have an angle  greater than \(45^\circ\) angle with the \(xz\)-plane. (Recall that in Cinema 4D, the \(y\)-axis points upwards where the \(z\)-axis usually points in mathematics.) Having a steep angle with the plane is important feature when 3D printing in a spiral. Since there won’t be any supports printed, a steep angle increases the chances of a good print.

At this point I used the surface I had created to make two different kinds of models. For the first model, I simply added a top and bottom to create a closed volume.  For the second model, I thickened the surface using the Extrude tool (2mm), again creating a closed volume. The extruded surface had raised top and bottom rims, which meant the model would not sit flat. So I selected all the point around the top (respectively bottom) rim and brought them down (resp. up) an appropriate distance. Apart from this, this thickened model needed very little doing to it. I then created a third model by adding equations and removing them from the thickened surface. This ended up being quite involved as I used the Bend Tool to make sure the equations sat nicely on the surface.

hyperboloid-tallIn order to get a spiral print, I took the first closed model, and adjusted the print settings on the Series 1 Pro printer. The top and bottom layers of the model are not printed; instead the nozzle goes around in a spiral, printing the surface 1 layer thick. Initially, we had trouble with both the fan and print speed settings. This meant the filament was not setting at the bottom of the model, leaving a gaps and giving a ragged appearance. After we slowed the print speed down and made sure the fan turned on after 1mm (rather than after 10mm), the prints turned out beautifully. The small model is about 13cm tall, and the large one about 20cm tall. These models can be found on Thingiverse here.

The next set of models was the hyperbolic paraboloid again, but this time I wanted to surface to look more like a saddle shape. To achieve this I went into Mathematica and use ParametricPlot3D command to plot the surface \(z=x^2-y^2\). Instead of displaying it inside a box, I had Mathematica just show the part of the surface inside the circle \(x^2+y^2=1\). To do this, I used the RegionFunction command.

hyper-paraboloid-form1  hyper-paraboloid-drying

I exported the Mathematica file as a .wrl to Cinema4D. Once there, I extruded the surface (again 2mm thick), then had to spend a significant amount of time repairing the mesh. To do this, I used the Optimize function, and also went around the surface repairing the mesh by hand (removing points, lines and faces, then using the Fill Polygon Hole tool to repair the gaps). I made two copies of the model, one with equations and one without. (I also built the equations in Cinema4D and extruded them. I then removed them from the surface using the Boole tool.) I 3D printed the resulting models on the FormLabs Form 1+ printer in clear and grey resin. The surfaces printed very well, though there were a fair number of supports to remove.

hyp-paraboloid-roundThese models can be found on Thingiverse here.

Flowers everywhere

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IMG_4709In early May, the IQ center at W&L was filled with 3D-printed flowers. Dave Pfaff and his work study students printed flowers in many bright colors on the Cura 3D printer. They are from Super Flowers found on Thingiverse. The fine filaments are created by printing in the air. That’s right, the printer puts down a single layer of material, then returns to the center. Since there are no supports underneath, a “droop-loop” of filament is created.

IMG_4708The vases were also designed by Dave Pfaff. He started with a flower shape on the base, then expanded and twisted the shape around creating the vase shape we see. Fabulous work!

Museum Magic

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IMG_4499Recently, I had a wonderful two days visiting Ricardo Nemirovsky and his team in San Diego. I started by visiting the wonderful Fleet Science Center and being the mathematician for their Meet the Mathematician event on the Sunday afternoon. (I even got my own poster – wow!) Ashanti Davis met me and showed me around the Taping Shape* exhibit for which I had designed the 3D-printed mathematical models.

IMG_4526    IMG_4515

IMG_4506These photos don’t really capture how marvelous it was to walk inside the topological shapes. How often does a mathematician get to explore the interior of a torus, or walk down the leg of a pair of pants?!? The lighting kept changing color as well, adding to the experience.

I was able to see my 3D-printed models in action; anything from little kids throwing them around, to big kids and grandparents building more complex topological shapes.

IMG_4522     IMG_4524

IMG_4518The folks at the museum had set up a tank with soapy water, and the frame for the Schwarz P surface could be lowered into and out of the soapy water by folks using a wheel. The frame worked perfectly, beautifully showing the Schwarz P surface.

I spent my hours in the museum talking with people who stopped by about the math of the Taping Shape exhibit and of the 3D printed models. Using zome tools and another tank of soapy water, I was able to demonstrate just a few of the many different shapes that soap film (a.k.a. minimal surfaces) take. The kids experimented with their own zome tool shapes. They created some bizarre models giving interesting soap films. Many of the soap films we created showed the classic angles where three or more soap films joined together. (Math in action really does work!) It was a wet and fun time, and my hands ended up very, very clean. Thanks go to Ashanti for setting things up and keeping me company during much of the afternoon.

IMG_4502   IMG_4535

IMG_4545A number of my designs had been 3D-printed into giant sized models, which was great to see. On the final day of my visit I was able to meet with the entire Informath team. I was also able to hang out with Bohdan Rhodehamel and see his lab. He was responsible for 3D-printing and then assembling the models for the Taping Shape exhibit. I wrapped up my trip by giving the math department’s colloquium on Mathematics and 3D Printing at San Diego State University.

 

*The Taping Shape exhibit is part of the InforMath project funded by the National Science Foundation (DRL-1323587).  (The InforMath Project is a partnership between San Diego State University and several museums at the Balboa Park, including the Rueben H. Fleet Science Center .)