In order to make a monkey saddle I created the surface in Mathematica. I then exported it as a .wrl file and imported it into Cinema 4D. Once it was in Cinema 4D, like all surfaces, I made it the correct size, optimized the polygons and extruded them by 0.20 cm.

I then added an equation to the surface by punching it all the way through. This time with the equation I used Arial as the font instead of Times New Roman to hopefully avoid the issues we had with the formula when printing the hyperbolic paraboloid.

I printed the monkey saddle using the liquid printer and the formula in Arial font ended up looking great. The model can be found on Thingiverse here

# Helicoids

My next project (after finally finishing all the quadratic surfaces) was to make a helicoid. I spend some time on Mathematica creating different helicoids by changing the parameters of the formula. The helicoid is pararametrized by $$x=u\cos(t), y=u\sin(t),$$ and $$z=u$$, where $$u\in[-1,1]$$ and $$t\in[0,2\pi]$$.

Professor Denne and I decided to print two of the ones I created to start (we may print more!). I exported the following Mathematica files and imported them into Cinema 4D.

Professor Denne used these files (as well as others) to create another worksheet in Mathematica. It can be found here

After making them the correct size I optimized the polygons and extruded them by 0.20 cm (a process I can now do very quickly after all my practice with the quadratic surfaces). I then printed each of them on the liquid printer and had fantastic results!

The .STL and .form files for both of these helicoids can be found on Thingiverse.

Later on, I made another helicoid, this one with $$u\in[0.25,1.25]$$ and $$t\in[0,2\pi]$$. This model can be found on Thingiverse here.

# Klein Bottle

Just a really short post to share our general excitement over having just about completed all of the objects from Multivariable Calculus. We just have a few more to print out. We will spend our remaining week(!) printing out some interesting topological objects – many of these directly from Thingiverse.

We printed one such object today. This is the Voronoi Klein Bottle from MadOverlord on Thingiverse. We printed this on the MakerBot 2X with a raft but no supports. After a moment’s thought one can see that the print succeeds (despite the short horizontal lines on the design) because the Voronoi cells are small enough. Interesting! The black filament also hides a few rough spots on the print.

The Klein bottle is named after Felix Klein (25 April 1849 – 22 June 1925), a German mathematician who saw many connections between Group Theory and Geometry. It is a one-sided surface and is a generalization of a Mobius strip. (In fact, it is topologically equivalent to two Mobius strips glued together along their boundaries.)

There are many fabulous descriptions of this topological object, one of my favorites is The Adventures of the Klein Bottle found on YouTube (from the wonderful folks at the Frei Universitat in Berlin).

# Volumes by Slices: Iterated Integrals

I have modeled the the solid from Example 8 of Section 12.1 from Stewart’s Essential Calculus.  It is bounded by the surfaces $$z=\sin x \cos y$$, $$z=0$$, $$y=0$$, and $$x=\pi/2$$. The example demonstrates the strategy behind computing a double integral using Fubini’s Theorem.  I approximated the solid by eight slices in the $$x$$ and $$y$$ directions. In order to draw the correct splines in Cinema 4D, I had to use the correct parametric equations to plug into the inputs $$x(t),\, y(t)$$, and $$z(t)$$.  For the first object I held $$x$$ constant (approximating integration with respect to the $$y$$-variable). The parametric equations were $$x(t)=\frac{k\pi}{32}$$, $$y(t)=t$$, and $$z(t)=\sin(\frac{k\pi}{32}) \cos(t)$$ for $$k=1, 3, 5, \dots, 15$$.  I then created a slice in Cinema 4D by adding straight splines. I extruded each slice by 0.2, which is just greater than $$\pi/16$$ (the width of each slice).  I placed each slice so that it overlapped slightly with the next slice – this will allow the objects to be merged (via a Boole) and will prevent vertical lines from showing in the print after the slices are collected into one object.  I had to use a Boole with two cubes for the two approximations by slices because the solids each had two very thin edges (that is, I shaved off some volume from two of the edges).

I then repeated the entire process, but this time with a constant $$y$$-variable.  I also printed the smooth solid, which is the volume that is approximated by the slices.  In order to do this, I imported the solid from Mathematica and put equations on it like I have in other models. These models can be found on Thingiverse here, here, and here.

My next object was a bulge-head solid. This solid lies above the $$xy$$–plane, outside the unit sphere, and inside the cardioid of revolution given by $$\rho=1+\cos\phi$$. Professor Beanland had given us these equations, since he was really curious to see what the solid looked like. He’d nicknamed it the cone-head solid, but after printing we renamed it the bulge-head solid.

Since the outside of the solid was a cardioid of revolution, I decided to create the solid in Cinema 4D by creating two splines (one for the cardioid, the other for the hemisphere) and revolving each around an appropriate axis.  Professor Denne helped me to figure out which parametric equations to place into Cinema 4D’s inputs for a formula spline. These were $$x(t)=1+2\cos(t) + \cos(2t)$$, $$y(t)=2\sin(t)+\sin(2t)$$, and $$z(t)=0$$, where $$t=[0,\pi/2]$$.  For the spline that would later become the hemisphere, I used $$x(t)=\cos(t)$$, $$y(t)=\sin(t)$$, and $$z(t)=0$$, where $$t=[0, \pi/2]$$. I then used the Lathe Tool with an angle of $$360^\circ$$ to make the two boundaries of the solid.  I then put them into a Boole to make a union between the two boundaries.  I printed the bulge-head solid on the FormLabs printer using clear resin. When loading the object into the FormLabs software, we got a warning about the object’s integrity, but we decided to continue the print anyway. Later on we were worried that the object would use up too much resin and that it may have some problems on the surface (like the smooth strange bowl did).  It turned out that added a bit more resin mid-build, just to be on the safe side.  The solid looks pretty good right now because it only has a few pimples on the inside, but no significant lumps. The object is still hardening and once it’s completely dry we’ll remove the outside supports. This will probably leave a few pimples as well.

You can find this model on Thingiverse here.

# Quadratic Surfaces – Hyperboloid of Two Sheets

The last quadratic surface I printed was a hyperboloid of two sheets.

For the hyperboloid of two sheets I created the entire object from scratch in Cinema 4D using the same process I used to create the cone and other similar objects. For this surface I used the same formula spline as the hyperboloid of one sheet $$x(t)=cosh(t), y(t)=sinh(t), z(t)=0$$, and then rotated it to the correct orientation. I then used the lathe tool and rotated this spline 180 degrees since this was all it needed. Because of this I needed to use only 30 rotation segments for a total of 60 all around the object.

I also had to reverse the normals on half of the object to make sure they were all aligned with the other half before I extruded the surface. I optimized the polygons to be sure the edges joined up into one object. I then extruded the surface to create my hyperboloid of two sheets. I copied this and put equations through one of them. I also made sure to Boole the edges of the hyperboloid to make them flat for printing.

Here is a picture of the final object! It can be found on Thingiverse here.

# Quadratic Surfaces – Hyperbolic Paraboloid

The quadratic surface that gave me the most trouble was the hyperbolic paraboloid. This surface could not be created in Cinema 4D and had to be imported from Mathematica. When I imported the surface from Mathematica the center of the saddle had hundreds of little polygons that overlapped, which became a huge problem when I tried to extrude them to give the surface thickness.

I had to spend a long time experimenting to find the lowest number of plot points I could use in Mathematica and still get an accurate object. Once I had done this I did the same thing with the optimize tool in Cinema 4D to see how big I could make the polygons before the surface started to lose accuracy. The first time I went through all these steps the hyperbolic paraboloid I had chosen just didn’t work correctly. So, I went back to the beginning and created a new Mathematica file of a hyperbolic paraboloid, and spent some time deciding where to cut it off to create edges that were as straight as possible.

Once I had done this and imported the surface into Mathmatica I optimized the surface as much as it would allow and extruded it. Finally I had a surface I could print! I then added the hyperbolic paraboloid’s equation to the surface. Instead of just imprinted the equation, since the surface was so thin I punched it all the way through.

In order to print this surface I used the FormLabs liquid printer. When the object came out of the printer it looked great and only had a few minor flaws to fix after this first print. One of the issues was the size of the object; it was just a little too small. The other issue was that the 2 in the exponent of the equation didn’t quite form correctly because it was too small. The final issues was that the equation had a $$+$$ sign where there should have been a $$–$$ sign (oops). The equation was little too long with a 0 that was missing its center.  To fix these problems I rearranged the equation (and fixed the sign issues) in Adobe Illustrator and then punched it through the surface again. The second print on the liquid printed I made 1.4 times larger than the last print.

The final print still had issues with the formula but otherwise worked out well. We are currently looking into changing the font to see if that helps with this issue. This model can be found on Thingiverse here.

Using my experiences building this and the other quadratic surfaces, I’ve put together a set of instructions on how to build quadratic surfaces using Mathematica and Cinema 4D. This can be found here.

The next quadratic surfaces I printed were an elliptic paraboloid and a regular paraboloid.

For the elliptic paraboloid I imported the surface from Mathematica.

I then optimized the polygons, extruded them by 0.20 cm to give the surface thickness. After that I used the boole tool to make the edge flat and added an equation through the surface.

I created the regular paraboloid from scratch in Cinema 4D using the same process as the cone.  I used the formula spline $$x(t)=t, y(t)=t^2, z(t)=0$$ and then used the lathe tool with 60 rotation segments to rotate it 360 degrees. I optimized the polygons and extruded them to give the surface thickness. I also made sure to “boole” the edge to make it flat and added an equation.

I printed both paraboloids on the same build bed with the MakerBot 2X printer. They can be found on Thingiverse here and here.

The next quadratic surface I made was a cone. This is a particularly challenging object since the cone’s two halves meet at a single point in the center. In order to deal with this printing challenge, I originally attempted to create the object and add a certain amount of thickness to the middle so it would print correctly.

The first cone I made using a method similar to my other quadratic surfaces by using a spline with the formula $$x(t)=t, y(t)=1.5*t, z(t)=0$$ (it was important to use the * otherwise Cinema 4D did not multiply) from $$t=-2$$ to $$t=2$$ and rotating it with the lathe tool 360 degrees with 72 subdivisions. Then I optimized the object by 0.02cm. With the first cone I made I reversed the normals so they were facing outwards from the cone surface. When I extruded the surface by 0.25cm this automatically gave center of the object more thickness. Unfortunately I realized that while it gave the center thickness, it also offset the lines of the cone so they didn’t match up, which is not what we wanted. In order to fix this problem I did the same thing but did not reverse the normals (so they were facing the inside) and extruded the surface to give the cone thickness.

With this second cone I removed the center using the Boole Tool, and a cylinder of radius$$=0.25$$cm and height$$=2\cdot 0.25\cdot 1.5=0.75$$cm. We did this in order to preserve the lines of the cone and give it the support the object needed. After removing this cylinder from the center I then added one of the same dimensions in its place. We knew this print had a high potential to fail but decided to print it as a test to see what we might need to change in our design. We left it overnight and it was a huge mess the next morning. Clearly this design didn’t work and we needed to rethink it.

I decided that creating both sides of the cone in one object was just not going to work. Instead I decided to create both halves separately and then connect them using a cylinder made from a sheet of clear overhead plastic (an idea that came from Henry Segerman’s Calculus Surfaces). In order to do this I used the same formula spline except this time from only $$t=-2$$ to $$t=0$$ and copied and rotated it to create the second half. On one half I put the equation for the cone $$\frac{z^2}{4}=x^2+y^2$$ and immediately ran into trouble. I had used too many subdivisions (72) and the object was not accepting the Boole with the equation. After creating many different cones with different subdivisions I found that 60 worked. Once this problem was solved I added the equations to one of the halves of my cone and printed it. When I added the equation I put it all the way through the surface and not just imprinted on it since the object had very little thickness to it.

The print was successful but showed a few flaws in my design. One was the equations and how they were a little too big and how the fraction parts of it needed to be downsized to match the other parts of the equation. This was easy to fix in Adobe Illustrator by changing the font size in the fractions from 36 to 24. I also made sure the numbers in my equation were not italicized and just the variables were. The other design flaw I found was that the bottom of the cone was not completely flat and was angled from when I extruded the surface to give it thickness. To fix this I used a cylinder and “booled” the bottom of both cones to make them flat. I then reprinted my object and had great results. These objects can be found on Thingiverse here.

My next successful quadratic surface was an ellipsoid. This surface I simply imported from Mathematica and then added equations to it, using the same process as described in my post on the hyperboloid of one sheet.

The first ellipsoid I made in was $$\frac{x^2}{16}+\frac{y^2}{25}+\frac{z^2}{4}=1$$. When I put my .STL file into the MakerBot Desktop program I noticed that the program created supported that went up to the equation because it was on a curved surface and cut into the object. I decided to print the object with these supports and the equation on the side of ellipse and no raft.

The first print I did of the ellipsoid I cancelled the print early on so that I could inspect the sides. The surface looked a bit melty, where the filament had shrunk. We decided this was fine and to try to print it again. The second time I printed this object at about half way through the print it fell over and we found it covered in a stringy mess of filament.

Taking this failure and the melty-ness of the ellipsoid, we decided to create a new ellipsoid that was a little rounder. I followed the same process as for the first ellipsoid using the equation $$\frac{x^2}{4}+\frac{y^2}{6}+\frac{z^2}{3}=1$$

This time when I printed it I decided to put the equations on the top of the ellipse and use a raft. It printed perfectly.

After my success with my second ellipsoid I decided to try to print my first ellipsoid again and this time with a raft. The object never fell over and printed perfectly. These ellipsoids can be found on Thingiverse here and here.