Custom 3D Printed Hyperbolic Tiling Bowl

At this point, we're all familiar with flat, 2-dimensional, Euclidean space. It's what we see every time we use a piece of paper or a computer screen. The internal angles of a triangle add up to 180 degrees, for a given line and point there exists exactly one line passing through the point and parallel to the first line. You can fit 4 squares cleanly around a point like so.

If you're doing geometry on a sphere, then the internal angles of a triangle add up to greater than 180, there exist no parallel lines, and you can fit 3 squares cleanly around a point like so.

In hyperbolic geometry, you can fit as many squares as you please around a point, as long as it's more than four, like so. There exist infinite lines parallel to a given line and through a given point. The internal angles of a triangle always add up to less than 180 degrees. In a lot of ways, it's the opposite of spherical geometry, and it's super weird.

I was talking to Marek14 and ZenoRogue (mentioned later in this tutorial) about this strange type of geometry, and I was inspired by their work with digitally searching for and rendering hyperbolic tilings. Without them, this wouldn't have been nearly as doable.

I set out to design a 3D printed bowl with a hyperbolic tiling design inside of it, and so I did. I've provided both a way to make all the files involved (which allows for more customization), as well as downloads for the files that I made at each of their relevant steps (if you want to skip some steps and start working on the printing).

3D printer

Filament

Fusion 360 or equivalent subscription(Only if you want to customize the bowl)

First thing, you'll want to consult this handy catalog of hyperbolic tilings made by Marek14 and ZenoRogue. You can download pretty much anything from there that you think looks cool. I chose one of these, although I can't remember which.

This step is lengthy and involves more customization than some people probably care about. I've attached the file I got out at the end of this, if you want to skip ahead to Step 3 where you do some CAD with it.

The .tes file format is instructions as to how to build the hyperbolic tiling, which is very far from what a CAD program is set up to interpret. Because of this, you have to do some rather interesting things to turn it into something usable. First, we need to plug it into the video game and geometry research tool that the file format was made for: HyperRogue. This game was made by the aforementioned ZenoRogue.

Install the game, and move your tiling to the "tessellations" subfolder.

Open up the game and go through the following button sequence:

main menu > special modes > experiment with geometry >

projection > projection distance > Klein model > go back > go back >

{This makes it so that the tiling will look right when we project it onto the bowl later}

geometry > hyperbolic > load from file > ../ > [the name of the file you chose] > [press enter] > go back >

pattern > single color > [pick any color] > go back >

{If you want to spend some time here, there are a lot of settings to mess around with. HyperRogue is a great way to learn about hyperbolic geometry.}

take screenshot >

format > SVG >

sight range settings > draw range based on > draw based on size in the projection (generation)

From there you can tinker with the screenshot settings even more, but the rest of the defaults are perfectly fine for this project. You can now take the screenshot and save it wherever you like.

You might note that this is also still not anything like a CAD file. That's okay. This last part of the file conversion is the easiest. You can turn the .svg into a .dxf for free with CloudConvert or some other generic file conversion service of your choice.

Make a new project in Fusion 360. If you're already a bit experienced in Fusion, you can make your own custom bowl and skip to Step 5. Just make sure that the inside of the bowl body is a hemisphere, and that there is also a separate, hemispherical surface body right there.

In the drawing above, I made a sketch. The minimum thickness of the bowl's wall will be 3.5mm, and the radius of the opening will be 50mm. My 3D printer can do 45 degree overhangs, so I set the dimension at the bottom to that. Adjust these as you see fit.

Using the revolve tool in the surface menu, revolve the inner surface of the bowl to make a hemisphere. We'll model the rest of the bowl later, it's not important at the moment.

Create a construction plane at the top of the bowl. In my design, it's parallel to the XY plane and the distance between them is the bowl's radius plus its thickness, which is 53.5mm.

Create a sketch on that plane, and import the .dxf file that we made in Step 2. If your computer is anything like mine, this will take up to a couple minutes.

Scale the drawing so that it's a bit bigger than the hemisphere, centered so that it's extending past the circle about a millimeter or two. This part doesn't have to be exact, just make sure that the hyperbolic tiling extends a small amount past the circular border of the surface body.

Finish the sketch.

Select the region between the tiles of the tessellation, and then extrude it, cutting away parts of the hemisphere. This might also take some time to compute. Afterwards, you should be left with a whole bunch of surface bodies, each in the shape of a distorted polygon.

Revolve the sketch from earlier about the Z-axis so that there's a bowl. This is what we're going to merge the tiles onto.

Use the thicken tool, and select every single one of the little surface bodies. Then, thicken them towards the inside of the bowl. I did a distance of half the thickness, or 1.75mm. The 3D model is now finished, and you can export it as an .stl and 3D print it without any supports. I've also attached the .stl that I made. I've had lots of fun with switching filaments in between layers of the print to create a bowl with different colors.