You might have come across the Jitterbug Earth model I made for the Remix competition a few weeks ago https://www.instructables.com/id/Buckminster-Full... this took a geometric representation of the Earth and introduced some hinges so that the shape could expand and change

In researching and thinking about that model and where it could go further I found some really interesting papers and ideas that I'd like to share along with some of my models I used to try and wrap my head around the ideas as I think they've been under-utilised and might lead to some interesting practical (and entertainingly impractical) applications

## Step 1: First Step in Doing Anything Cool - Find Someone Smarter Than You Who Has Done All of the Hard Thinking

Researching the Jitterbug transformation I came across a really cool paper that takes the Jitterbug idea and runs with it https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa...

Go download it - it'll have your head sparking with ideas for days. Joseph Clinton worked with Buckminster-Fuller and has a fascinating career in his own right https://en.wikipedia.org/wiki/Joseph_Clinton and does a nice talk about these ideas and more at https://m.youtube.com/watch?v=J19eV7Z1x3M

## Step 2: Tessellations and Adding Moves to the Jitterbug - Different Shapes and Connections

In that NASA paper Clinton describes different ways of expanding tessellated shapes as a potential way to pack down and unfold structures - what was called "The Geometric Transformation Concept"

This concept was then described in both 2D and 3D

A tessellation is essentially any arrangement of shapes that fit neatly together to cover a surface - this could be regular like a checkerboard pattern with squares or more complicated with a mix of shapes

Transformations were described as adding animation and expansion to a tessellation. You can imagine that if you have an area covered in square tiles. If you were to rotate the square tiles a quarter turn (what he called a Face transformation) as in sketch (1) but keep the connections or relationships between corners labelled A you'd get an expansion.

In sketch (2) I've added 12 oval marks to show where you could place hinges between corners to achieve this

The paper then build further on this by showing that you needn't keep the relationship between immediately touching corners - as in sketch (3) you could have 12 struts that connect corners not to their immediate neighbour but one further along (what was termed an Edge transformation)

You can go further and further with this idea with other shapes and possible connections between them (Vertex transformation)

## Step 3: Taking This to 3D

The Clinton paper described this tessellation idea and applied it not just to shapes covering a flat plan but also to 3D shapes and set out a whole series of possible transformations

e.g. Face Transformations:

- tetrahedron -> octahedron
- octahedron -> icosahedron -> hexoctahedron (the original jitterbug)
- hexahedron -> hexoctahedron
- icosahedron -> icosadodecahedron
- dodecahedron -> icosadodecahedron

You can see from his diagram and the resulting models that the Face rotation translation is the original Jitterbug while changing the connections between corners gives a whole new scope

## Step 4: Applying These Transformations to 3D Shapes

Thinking about these movements I sketched up a few different variations on a hinge to hold the angle constant throughout the rotation and settled on a short 3D printed section with 2 bolts which the faces would rotate around. I later found the same hinge had already been suggested in an academic paper examining the jitterbug maths and similar models (available as a free pdf at https://www.sciencedirect.com/science/article/pii... if you want to see more)

## Step 5: Tetrahedral Transformation

Translating this idea into a laser cut tetrahedron with 3D printed hinges (sized for 3mm plywood and M5 bolts - you'll need either short (6mm or thereabouts bolts) or to add washers to outside as the inside angle is quite tight)

## Step 6: Edge Centred Octahedral Transformation

... and an edge transformation laser cut octahedron with 3D printed hinges (sized for 3mm plywood and M5 bolts)

## Step 7: Edge Centred Tetrahedral Transformation

... and lastly a laser cut edge transformation tetrahedron with 3D printed hinges (sized for 3mm plywood and M5 bolts)

## Step 8: Relationships Between Shapes

The thing that has me equal parts fascinated and puzzling over these shapes and transformations is how the dihedral angle in one shape becomes a vertices angle in the next shape up leading to a series of relationships between them. An octahedron is made up of two transformed tetrahedra... and so on and so on

If I go a bit crazy and end up doing alchemical and astrological experiments and muttering about celestial spheres you'll know where it started

## Step 9: Practical Applications?

I'm not sure if there are any established practical uses of these - I've seen a few sculptures and models yet but it seems like there should be more. Hopefully readers might come up with some interesting ideas

The closest thing that came to mind after making the edge transformation tetrahedron was that the resulting truncated tetrahedron is used for the cargo boxes in the sci-fi movie Silent Running e.g. cargo box image

## Discussions