Great Stellated Dodecahedron decorations

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I tried making a great dodecahedron as a kid – though no trace of it remains, sadly. The book I copied the net from, Mathematical Models by Cundy and Rollett, still lives amongst the cookbooks and its shapes and nets remain tempting.

UST [employer] allowed me to be in charge of this year’s Christmas decorations [why?], so I decided to outdo my 11 year old self and go a further stellation, and produce (with the help of some lunch-goers, motivated only by free sandwiches and love of polyhedra) a series of great stellated dodecahedrons to go on the tree. Unlike my childhood self, I didn’t try to scale up the tiny drawings of nets from the book, I just used one of the free nets online which, being a vector image, scaled up perfectly.

Don’t get this reference? Don’t worry.

Polyhedra seem to be at the edge of what is more simply defined by a process than by its geometry. In the case of the great stellated dodecahedron:

  • take regular pentagons, arrange such that one side of each of three pentagons are touching all the way along each side – you make a dodecahedron
  • extend each face until it meets another extension of another face (this a stellation – in 2D a pentagram is a stellation of a pentagon – you extend each side until it meets another side’s extension) – you should be imagining a spikey shape with 5 sided spikes on it here
  • stellate two more times (to the great dodecahedron then to the great stellated dodecahedron)

This ‘process rather than geometry’ paradigm seems to be increasingly important, with parametric design, and a wide range of ways to generate designs and search and order design spaces, becoming popular. This allowing humans to spend more time thinking about the aims of a design and its basic parameters, rather than its precise geometry. Thinking about an object like the great stellated dodecahedrom in terms of a process is an entry point to this, and there is a whole family of increasingly spikey shapes that result from increasing the number of stellations beyond the three used here. Other families occur when you stellate other shapes such as the icosohedron.

I digress – lets get back to making the polyhedra: one trial version is constructed from 5 sheets of thick sketching paper:


Trial run – 5 sheets of A4, 2 sheets of A4 uncompleted in background

Another, much less successful one, was produced from two pieces of thick paper (its net is in the background). The ratio of card thickness to length was too great, and it was uncrisp and fiddley.

Smaller polyhedra, paper too thick

Following these trials, a few improvements were made, mainly: make the thickness/size ratio smaller – so large sheets of thin paper, built in prettiness without using acres of virgin material and finding a better system than ‘glue dots’. Printing the net on the back of the paper is a huge time saver and it isn’t visible on the final product – it also makes production less frustrating. Improvements missed were using a laser cutter to get really accurate cutting and scoring and the development of a tab system to remove the need for glue or tape in construction.


Nets printed on back of old maps, no tessellation yet

For built in interest, reasonably stiff high quality paper and printed nets I used old maps, I used UST’s A0 plotter to print on the back of these – note to self, this is a nightmare, try to get a flatbed next time. By chance the maps I had spare were of South London and Elgin, Scotland. Local maps seem to be really popular, people love seeing places they recognise and have stories to tell about them (examples include holidays, birthplaces, homes, distilleries and others) which makes the making process much more fun.


A grid of spikes, the first half of the build process completed

Net cut out, pre-folded and after first round of joints taped up, creates the spikes, each spike has a triangular base, and these triangular bases form an icosohedron. Also for future reference, there is a real desire for normal instructions rather than laying down rules – in my case I said that ‘when cutting out the shape, whenever two long lines are coincident, or two short lines are coincident, you do not cut, when a long line and a short line are coincident you do cut’. This did not come across as clearly as I would have liked. However, an instruction like this, when seeing the final shape (where on the shape do long lines and short lines meet?) gives much more freedom and insight than ‘cut along these lines in this order’ type instructions. There were plenty of improvements on my making techniques during the session.


The final product

The making session.


Output of the workshop

There are a few more around the office where I need to chivvy to get them finished off.

For future Christmas decorations, these seem to work nicely, they are large enough for about 10 to fill a medium sized tree comfortably, and are (I think) inoffensive and just unusual enough to provide interest. However, not many guests seem to play with them or inspect them when they are sat at the sofas – I had thought that they might. On the other hand, the ones made at the lunchtime session have been named, so maybe they will be taken home to be used and enjoyed again? Hopeful.

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