Seeking the Perfect Möbius Roller

I’ve been trying to perfect the process of making these things that I call “Möbius Rollers”.

Issue #1—Painting:

On the top one I used multiple coats of filler primer, sanded it with 100 then 220 grit sandpaper, painted it blue, taped it, painted it green and removed the tape. The paint job came out looking great.

The second one, I used wood filler on the rough areas, which saved me a couple of coats of filler primer. Painted it green, then taped it. But, the tape strips were too small, and didn’t stick well in some areas, so some of the blue paint got under.

Issue #2—Cutting:

There is a channel with ball bearings that ride around inside. In order to get this to print, I have to make the channel just below the surface. Then, I just cut the top of the channel open, so you can see the balls roll around inside.

On the top one I tried different bits on my rotary tool, and when I tried the router bit, I thought it was working at first, but then it skipped all over, and tore the whole thing up.

The second one I tried a grinding wheel bit on a slow speed, and I got a cleaner cut, but it took a long time. Then, after it was open, I tried sanding the channel so the balls would roll smoothly, but I discovered that the channel is pinched in one area, and even a lot of sanding would not open it up. And, even still, I did not like the way the cut came out.

Issue #3—Model:

On my old printer the channels came out wide enough, I don’t know if it’s the slicer, or the printer, but I need to either figure out different print settings, or maybe modify the the model to make the channels bigger. Also, I want to find a way to make the plastic thin enough along the outside of the channel, so that I can pull it open like a zipper.

3D Printed Halftone

I had an idea to 3D print a halftone image by making a grid of holes where the larger holes would be brighter halftone pixels, and the smaller holes would be darker ones, and then I’d light it from behind. I tried several approaches, including one performed after I made this video, and they all failed. Blender 3D was not up to the task of doing a giant boolean operation, and the P5.js SVG library was not up to the task of drawing all the outline squares that I needed. So, I resorted to printing a background (just a large, white rectangular slab) and switching filaments to print the halftone pixels on top in a different color (black squares).

I have a lot of ideas for variations, like a non-solid background, and slicing a larger image and printing it out in pieces. Ultimately, I still want to try my original plan, so maybe Inkscape, or learn some Fusion 360 to see if it can do it, and then maybe openSCAD if that fails.

Here are the software tools I used:

Here’s the Javascript program, if you want to give it a try yourself: http://hipnerd.com/wp-content/uploads/2018/05/3D-Half-Tone.zip

Amazon Associate Link: Made with Hatchbox PLA

Golden Rectangle Icosahedron in Blender 3D for 3D Printing

 

I found this Golden Rectangle Icosahedron while reading about Icosahedrons on Wikipedia: Regular Icosahedron and wanted to make a 3D print of it. I realized that I could print it with a minimum of supports if I printed it standing on three corners.

Constructing the figure is super easy by making a golden rectangle, duplicating and rotating it two times. The problem that I encountered next was getting three corners resting on the XY plane. You’d think there’d be an easy to use tool or two, or maybe a plug-in, to do this in Blender, but haha–no. After a bunch of googling and wasting time trying to figure out how to do this with Blender tools, I landed on the idea of using duh-duh-duh MATH!

The process is simple, and can be generalized to any object. It requires only simple and familiar Blender tools: moving the origin of an object, translating the object, and rotating the object.

The basic steps are:

  • Move the origin of the object to one of the vertices
  • Move the object to 0,0,0
  • Get the coordinates of the next vertex and find the angle to rotate around the Z axis align it on one of the major axes
  • Get the coordinates of the vertex again and calculate the angle to rotate to the XY plane
  • Get the coordinates of the last vertex and find the angle to rotate around the axis in step three to the XY plane

I had a lot of trouble getting this to print on my new Prusa I3 MK3. Here’s some troubles I ran into, and how I resolved them:

  • After a week of printing fine, I started to notice that first layers were sometimes failing. The solution was to preheat the bed for just a couple of minutes before starting the print.
  • I never really had the Z-axis set correctly because the Prusa’s built-in first layer calibration routine is not good, and makes you guess too much. I used the instructions in this thread: life adjust Z – my way (sic), and everything was so much clearer and easier.

Once I had the printer dialed in, I had to experiment a lot with Slic3r settings. I think the most important three settings in getting this to work were enforcing support for the first 40 layers, setting the XY separation to 5%, and setting the infill speed to 375 mm/s².

Model on Thingiverse: https://www.thingiverse.com/thing:2822976

As far as the video itself goes: The Blender session was complete garbage, and I discovered that it’s really hard to edit that type of video. Also, voice overs are hard. Also, the painting was long and unnecessary (and badly done to boot!).

My New 3D Printer

Just got a Prusa I3 MK3 printer. I ordered the MK2 in June 2017, switched my order to the MK3 in October, which added a few months on to my delivery time. Finally arrived, February 2018. I’m super happy with it so far. Leagues beyond my Tronxy, but I’m keeping it and will still use it.

Here’s a time-lapse video of me assembling it:

The Dilettante – More On Infinity Part 2

Part 2 – Use this one weird trick to show that the number of points in a plane is the same as the number of points on a line, and 3D space, and any number of dimensions

You have a point on a plane, and it has an X and Y coordinate, such as:

point_on_a_plane

You can take the X and Y coordinates, and combine them to make a single, unique number by doing the following:

Take the first decimal place of the X coordinate, and follow it with the first decimal place of the Y coordinate, in this case: 0.32, then the second decimal place of the X and Y coordinates: 0.3250, then the third, fourth and so on to as much precision as you care to have: 0.32502689… this, like any number, can be represented by a point on a line. You can do this for every point on the plane and each point will map to a unique point on a line. Similarly, you can take a point on a line and pair up the numbers to make a unique point on a plane. So, now you have a 1:1 function that maps all the points on a plane to points on a line and vice versa.

Now you can add a third dimension.

X:0.07982
Y:0.92580
Z:0.10458

Combine the decimal places in triplets, so in our example the result will be: 0.091720954885208…

You can keep adding dimensions, and no matter how many dimensions you have, you can use this trick to map all the points in 1:1 fashion to a line.

The amazing conclusion is that the number of points on a line of any length is the same as the number of points on a plane, which is the same as the number of points in any dimensional space.

No Way Dude

The Dilettante – More On Infinity

One other fascinating discovery about infinity in One Two Three… Infinity that was new to me is that the number of points on two lines of any length is the same. Also, that the number of points on a plane, and even three dimensional space is the same.

First of all, what I mean by “number of points… is the same” is what you would naturally think: They can be put in a 1:1 (one to one) relationship, for example:

Set 1: A, D, A, M, Z
Set 2: J, U, L, I, E

Set 1 has five letters and set 2 has five letters. We are only concerned with how many items there are in each set, and not what the letters are. The fact that the letter ‘A’ is repeated twice in set 1 and that neither set has any letter in common are unimportant. To see if the two sets are the same size, or if one is bigger than the other, we pair off the items in the two sets in any order we choose:

J - Z
U - M
L - A
I - D
E - A

We find that both sets are the same size because they can be put in a 1:1 relationship.

Simple enough. So, here is the mind blowing visual proof that two lines have the same number of points:

lines2

mind blown

In this diagram the two lines of different length AB and AC are joined at A. The line CB connects the endpoints, and every line parallel to CB, such as DE, connects a unique point on AB with a unique point on AC and vice versa. So, even though there are an infinity of points on both lines, they can be put in a 1:1 relationship.

In case that’s a little too informal, let me just add that CB and all its parallels are just graphs of a line function. You don’t even need to know exactly what the function looks like, just that the input is a point on one line and the output is a point on the other, and that for every input there’s a unique output. To me, it seems entirely counter intuitive, but the logic is inescapable, two lines of unequal length have the same number of points.

End of part 1. Coming in Part 2 – Use this one weird trick to map all the points on a plane to all the points on a line.

The Dilettante – Different Types of Infinity

Different Types of Infinities (Wherein I will try to convince you that there are different types of infinities, and that some are ‘bigger’ or ‘more abundant’ than others.)

I’m going to concentrate on showing that integers and real numbers fall into two different categories of infinity. This is an informal proof, meaning that it lacks the rigor that would qualify as a proof for a mathematician, but should (hopefully) be convincing to a layperson. If you are interested in delving deeper, I recommend starting with this video by Vi Hart.

The guy who did the breakthrough work in the mathematics of infinity was Georg Cantor. Before Cantor came along infinity was taken to be a single concept of numbers going on forever, but Cantor showed that the picture was much more complicated and weird. In fact, some of the leading mathematicians of Cantor’s day rejected Cantor’s work and pilloried him.

The Proof

First, integers can be seen as a special case of real numbers that have all zeros after the decimal. We typically ignore the decimal when writing down integers, but another way to write the number 5 is 5.00000000000000… and 3843974937 is 3843974937.00000000000000… (the ellipses “…” denote that the zeros repeat forever). You can see in this diagram that the integers are a subset of the rationals which are in turn a subset of the reals. So, although both sets are infinite, one is a subset of the other.

Now, let’s look at a line segment (i.e. a line of finite length) on the number line, even the smallest line segment we can imagine, will encompass an infinity of real numbers, but only 0 or a finite number of integers.

For example, this line segment from 3.10… to 3.20… contains 0 integers, but an infinity of real numbers, including the irrational number, π (3.1415926…).

Num Line 1

This line segment, from 1.0… to 1,000,000.0… contains 1,000,000 integers, and an infinity of real numbers.

Num Line 2

Cantor developed several proofs, including the diagonal argument, showing the one to one relationship between integers and rationals, and also how the reals cannot be put into a one to one relationship with the integers. There are plenty of websites and videos that explain this concept. So, I’m going to present my own ‘cinematic proof’:

Imagine your are in a desert that stretches to the horizon in all directions. Each grain of sand represents a real number, and you are searching for the sand grains that are integers. You are about to pass out from the heat when you see something in the distance. It starts as a dot wavering in the heat waves but eventually you can make out that it’s a person on camelback. Eventually, Omar Sharif is standing before you. He presents you with an easy to follow map to the location of all the integers in the desert, and a scoop that can be adjusted in size to scoop as many of the integers as you want. When you scoop the sand and sift out the integers, you wind up with a small pile, and the sand you discard is a dune that piles up to the sky. That is how the integers and real numbers relate to each other.