Pentagonal Scutoid that Tiles with Itself

thingiverse

As a math enthusiast, I recently heard about scutoids, which are 3D shapes involved in the packing of epithelia cells. They have been generating quite a bit of buzz since they were introduced in an article in Nature: [Scutoids are a geometrical solution to three-dimensional packing of epithelia](https://www.nature.com/articles/s41467-018-05376-1). I printed mathgrrl's version of [Customizable Pair of Packable Scutoids](https://www.thingiverse.com/thing:3024272). These are great, but I was a little disappointed that I couldn't tile more copies of them. I should have known this because the top and bottom faces are regular hexagons and pentagons, and regular pentagons don't tile the plane. This is my attempt at making a scutoid-inspired shape that tiles with itself. It's based on a squashed pentagon tiling of the plane. The tilings on the top and bottom are just shifted from each other. Each unit has two mid-level vertices on opposite sides, so they're sort of double scutoids. In addition to the top and bottom squashed pentagons, there are four curved pentagonal surfaces on the sides and two triangles. As I was walking home later, I wondered if this could be made with planar faces. I came to the conclusion that it should probably work, since this shape is a deformation of a cube with opposite corners cut off. I gave it a quick try, but haven't gotten a version that tiles with itself yet. Let me know if you figure it out. I modeled these in Fusion 360 using the Patch option in the Patch workspace. This allowed me to draw all of the edges first, and then let Fusion fill in the faces. After the faces were created, I used Boundary Fill to create a solid object. (If you're trying this, make sure to have Preferences->Design->"Allow 3D sketching of lines and splines" checked.) It was my first time using these tools in Fusion, and I was pleasantly surprised with the results. The designs are very small, so you'll probably want to print at least 200% (the pictured ones are 200% and they're still pretty small). I'll probably actually reprint at 400% and try out some colors. I'd love to see your prints if you make some.