Seifert Surface 3D Print

Seifert Surface on Trefoil

Today's adventure is printing a Seifert surface on a trefoil knot! I'm sure you all know what that is (\sarcasm) If you want to know more about this surface, there is an explanation at the end.

When I was a first year in college, I did an independent study on knot theory and really loved it. For that project I made a model of Seifert surfaces on both a trefoil and figure 8 knot out of a wire hanger and panty hose. I wish I still had a picture. I have always wanted to carve a larger, more sturdy version out of wood, as a tiered cupcake stand. I got one step closer to this goal today; I 3D printed the thing!

When I read this post by Laura Taalman and Jonathan Gerhard, I realized I too could 3D print these surfaces! Laura and Jonathan used a wonderful program called SeifertView to model the surface of their knot. I downloaded it and got it running on my mac through wine bottler.  This program is so much fun! Check out some screen shots:

Trefoil Surface in SeifertView

Close Up

Laura and Jonathan did get a special version that would export to stl (although I only ever got it to export to .obj) in order to 3d print the surface. Clearly I needed this special version of SeifertView as the original doesn't have many export options. So, without ever meeting either of them, I emailed Jonathan (thinking it was Laura, I guess I didn't read the post that closely) and asked for it. He responded within a few hours! 

I then followed the suggestions from Jonathan and brought the .obj file into blender, and thickened the mesh using the solidify tool (here is an awesome tutorial on the tool) and exported as .stl.

Blender

From there I imported into makerware and printed! (thingiverse)

In Makerbot Makerware

Printing with rafts and supports. 0.3mm, 10% infill

Final Product! Would print at 0.2mm next time

About the Surface:

A knot in mathematics is not so much like the knot you tie on your shoe laces, rather it resembles more the knots of celtic art work. In fancy terms, a knot is an embedding of a loop in 3 dimensional space. An "untied" knot is just a circle. The simplest knot you can make is the trefoil. Here is a drawing of one:

Trefoil Knot

A knot surface is a surface whose boundary, or edge, is a knot. So for the unknot (or circle) the knot surface is a disk. The interesting question here is, what would a surface look like if its boundary were a knot. One option for a surface whose boundary is the trefoil knot is a strip with three half twists (pictured below).

Surface on Trefoil

The problem with this surface is that it has only one side! If you were an ant, placed on this strip, you could walk in a straight line and eventually end up under where you started without ever crossing the edge. See Vi Hart's video for more explanation.

The question remains: does every knot bound a surface that has two sides? Herbert Seifert found a solution to this problem. He devised an algorithm that would define a two-sided (orientable) surface on any knot!

The culmination of this one quarter was proving that the Seifert surface on a trefoil knot is isomorphic to a punctured torus. It would be really cool if I could somehow print the steps in the dissection proof... A project for another day.

Punctured Torus

The Trials and Tribulations of 3d Printing

It has been a while since I posted. I have been trying to get this weather box off the ground, but keep running into issues with the 3d printing! My colleague and I have a Makerbot Replicator, 5th generation for the Computer Science department. The "smart" extruder keeps getting blocked! It's been quite the journey with this thing.

Makerbot Smart Extruder

I managed to get a piece of the box printed (I am going for a modular design, tinkercad link here, thingaverse link to come).

The raspberry pi fits nicely!

Temperature and humidity sensor fits in the side. Wires feed into interior through the top

You can see the ribbon leading to the raspi camera. Also the wires on the right leading to the temp sensor.

The makerbot refused to print after this though, partially because we tore off some of the heater block insulation when trying to unclog the extruder. You can see the result of this on the above pictures. There are large, chunky imperfections in the print that happens when the temperature is not stable. So I ordered some kapon tape and ceramic insulation to re-insulate the block. Got to it today!

I decided to insulate a bit more than it was originally. The block still moves freely in the cage (which is super important because the smart extruder has a sensor that tells when the build plate has touched the print head).  I haven't tried to print with this yet as we somehow managed to lose the set screws that hold the thermocouple and heat sensor in place. Should have replacements by Monday though.

Meanwhile, our school's science department has two of the same makerbots and has been experiencing similar problems. They even ordered two replacement extruders, just to have them clog on the first print! We managed to convince them to give us one of their machines and all of their print heads to try to fix them. Less than a day with them and I got two of the extruders working! Without even taking them apart! Used the general idea from this video to unclog them. It didn't work 100%, I had to take the extruder off and shove the filament through with the catching clasp open (spring loaded tabs that stick out the side) to get one of the clogs out.  Just to make sure the extruders were really working I printed these:

Tetrakis Hexahedron and Rocking Trefoil Knot

Thanks Math Grrl for the designs! (thingiverse link)

While I did get two print heads working, I managed to shear off the tip trying to screw it into the heat block. Not sure how I'm going to fix this one.... I will let you know.

Perhaps I have been doing too much crossfit...

The last smart extruder had already been taken apart. The teacher who did it managed to break all but one of the clasps! I took it apart again because it was having trouble catching the filament. I think it might work now. I didn't get a chance to check it. Will try on Monday. 2 out of 4 is pretty good for one day.

I thought this might be a good teaching moment. This is the partial tear down of the smart extruder. the silver mass on the left is still together. It is the heat block, heat sink, cage and print head. The colored wires connecting to the circuit board lead to the heat sensor while the (difficult to see) black wires attach to the thermocouple. The steel cylinder is the gear that feeds the filament down into the heat block. The spring loaded plastic piece in the middle is the filament clasp.  Sometimes your filament will make it into this clasp, but won't be caught by the loading gear. If this happens, I recommend taking the extruder off while hot (be careful not to touch the heat sink or the heater block as they are obviously very very hot), open the clasp by squeezing the tabs on the sides and shoving the filament in. I did this today and it totally cleared the blockage. There are a few more pieces to this extruder, but they are hard to see here.