I made/finished a paper shell for Build Day at pier 9 today!

Now I can have nice shapes and colors to look at when I sit at my desk. Yay shapes and colors.

This sort of shell shape is a Gnomon, or a shape that by adding a copy of itself becomes a scaled-up version of the same shape. Another way of describing this property is that the shape is self-similar, or a geometric fractal. If you zoom into a small part of the shape, you see a replica of what you saw before zooming in. Of course here there are inner and outer cutoffs: It is impossible to make a physical object that extends into infinity, but one can get the idea.

I find it interesting that a simple geometric shape like this can appear to be such a good geometric (and perhaps even temporal) model of naturally occurring sea shells. Also it is nice to look at.

Step 1: Model a Shell

A nice way to model a shell is in Blender, which has an array modifier which allows one to make spiraling shell shapes easily.

1. Make a base profile. I started from a circle.

2. Make an array of the profile, and adjust by scaling, translating and rotating the Empty. Make a spiral.

3. Extrude the profile and then snap the vertices of the extrusion to those of the next element in the array. This will make a continuos mesh.

4. Export just the base mesh (without the array applied) as an obj or stl. This will result in a mesh of just the base unit.

For a detailed description of this process see the other instructable I made on the topic here:

https://www.instructables.com/id/Geometric-Fractals-in-Blender/

Step 2: Unfold the Base Unit

Blender is really just a good way of visualizing what the spiral will look like. In fact only the base shape is necessary, it contains all the information on scaling and orientation for subsequent modules.

There are many ways one can unfold a mesh like this. If it is sufficiently simple one could even take each face and transform it to be flat in the horizontal plane, and then join up these "layed-out" faces along edges.

An easier way is to use software: Pepakura, 123D Make, or rhinoUnfolder (I made rhinoUnfolder, so it is a bit customized and hacky, but it works, sometimes). Pepakura is great but you need to buy it so that you can export DXF files. 123D Make gives tons of awesome connection options. If you code, rhino Unfolder can be fiddled with to do anything you can imagine.

Once you have chosen an unfolder:

1. Import obj or stl.

2. Unfold and make sure tabs are to your liking.

3. Export to a DXF and bring it into your favorite vector-drawing program.

NOTE: Another route is to apply the array modifier in blender and export an obj or stl of the full model. This puts more work into the unfolder, since you may have to manually specify where to cut the model, and there will be many faces. It also means all the tabs will be the same size, which might interfere with the scaling aesthetic/ gluing of the shell. But from there you can go straight to a inkjet printer or lazer cuter and make the model!

Step 3: Arrange Cut-File

1. Make sure the scale of the flat part is correct. A trick I like, which gives a good visual check on scale, is to bring the 3d model into the same file as the flattened parts. I scale the 3d model to my liking, measure an easily identifiable edge, and then make sure that the corresponding edge on the flattened parts is the same length, by scaling. Make sure to scale all the flattened parts together! Of course if you use 123d Make or pepakura these operations can be taken care of in those programs. The reason for this workflow is the large number of faces and edges that can make doing all the unwrapping in a single program a pain.

2. Take the base object and copy and scale and rotate, (sometimes these operations are combined into an 'orient' command in some softwares) to connect the units. This avoids unnecessary confusion with tons of parts, and makes neat cutout patterns.

TIP: Orient a copy, to get two connected units. Then orient those two units to get four, and so on. With five such operations you can get 2^5 = 32 units, which is likely plenty. This depends on the scale factor and your patience when it comes to assembling the thing.

3. Lay our your parts in the sheet size you have. This is where this rather manual workflow can become annoying: if its too big you either need to scale down, thus changing your final model size, or split up onto multiple sheets.

Step 4: Cut It Out!

1. Find a laser cutter, CNC paper cutter, or an inkjet-printer+exacto-knife+much+patience.

2. Find some good paper. Card-stock is nice, but depending on your scale thinner stuff might be preferable.

3. Cut it out! In this case I used an epilog laser cutter.

Score Lines: To get a score line, simply set the speed high and the power very low. Do some tests before cutting the whole thing.

Order of Operations: Make sure to do score lines before cut lines. Otherwise cut out parts might shift around, and then scores will be in the wrong place. On and epliog laser cuteter, one can specify this using color coding. Another route for laser cutting is to send two jobs, first just the score lines, then the cuts.

Tip: Also you could cut out other sheet materials like metal, veneer, very thin plastic, seran-wrap, chewing-gum wrappers, or mango skins.

Step 5: Fold and Glue Together

Much patience

Tips:

1. Imagine gluing it up before doing so. This can save some pain, for it is often easier to build the model one way than another. For these scaling objects it is sometimes best to start at the small end. In this case I glued a few big units of the black side, then, realizing I should start with the small pieces, switched to the tiny units. I actually glued the first 15 small units edge-to-edge. The thickness of the paper is sufficient to hold a tiny bit of glue. I cut of the tabs with scissors as needed.

An interesting effect is the stack up of error. Because the paper has thickness, but the 3d model was an idealized zero-thickness surface, an accumulated error occurred where the green paper became more and more off as I glued smaller and smaller units. Once the error got so bad that I thought it would be visually and structurally obvious, I simply cut the green spiral, added a little paper to take up the gap, and continued.

This is a problem I have encountered before with multi-colored paper models. Perhaps the solution is to distort the model to allow for thickness, or maybe even somehow incorporate thickness into the model in the first place.

Thats it! Now you can make lamps, shells, hats, etc!