Bridges 2026 preview

 

These are the steps to making a compact dodecahedron (see the compact octahedron shown here) from faces made from 5 coffee stirrers. This is new work and the start is to make the reciprocating frame sides using the black jig and a cad image showing the right placement of sticks relative to the jig. An stl for the jig is here.

 

All the sides, and 

The sides made into a dodecaheron with a spare side

 

The dodecahedron

There are 2 types of dodecahedron available from these sides. I've made the version shown in purple above, which has the edge joins near the corners. As shown below in "net" and "link" versions, the links created are different.

Hi

Last year I attended a maths and arts conference called Bridges. I'm a retired engineer and dabbling in the maths and arts fields is fun, interesting and challenging for me. It's lucky that I've been an engineer and now live in a time where there are affordable computers, 3D CAD packages and 3D printers. This means I can use and access some powerful artistic tools. 

At the end of last year, the chance came up to present at a local maths conference - it had a Maths and Arts lecture stream and a Maths and Art exhibition and I came up with some new work for that. It's not officially published as there are no conference proceedings, and there was no requirement to submit a paper in conjunction with the conference. That was good as it was easy to get a foot in the door and make the presentation, but not good as no legitimate publication resulted. I've unofficially published anyway, just parking the presentations along with the 3D printed parts required to replicate my work which on thingiverse here and here. 

Now all the rush to produce the work for the conference is over and so is a summer holiday season. Hopefully also the horrible hot summer weather and bushfires are over also! Anyway, its February now and I've sort of been spat out into the new year and need to work out what I want to do with it. The chance to present at and attend Bridges 2026 is there, and having presented at the local maths conference there's some material I can write up. But but when I looked at the deadline for papers it was an oh sh!* moment. The deadline for standard 8 page papers is already past, and if I want to write a short paper or run a workshop then I need to write and submit in February. 

Meanwhile, after not thinking about maths stuff for a bit over a month, I have returned to it now and am starting to see the work I did in late 2025 from a different perspective and am making and calculating new arrangements of parts from the jigs I made last year. An example is shown above.

There are some nice templates for short papers on the work already done. I had heard of Doris Schatschneider before Bridges last year and met her briefly at the conference. She has written a nice paper on some of M.C. Escher's commercial artwork and its in the monumentous and free Bridges archive here. Four Pages doesn't sound like much but they are four squishy pages which use a small font (1800 words or so) and still manage to fit in some nice illustrations. For the workshop papers somebody I know from Melbourne is the chair of the sessions, and I attended 2 workhop sessions last year. It cannot be that hard!

So the plan for now is to write a 4 pager and have it ready to go, and then work on a workshop session and paper. If I'm not happy with the workshop paper I will be able to fall back on the 4 pager.

Wish me luck! 

Update, Feb 9, 2026

While watching Dog Park on ABC iview,  I finished the second dodecahedron version I'd mentioned and sketched above. pics are below. 

The 2 types of dodecahedron which can be made from the same side.

Several 5 stick bands can be seen in these structures which are topological links. On one, the bands are outside the pentagon shape and an the other they are inside.

New Dodecahedron with side

New dodecahedron

Janet Frame Fan Box Set

 

Halfway through glueing

Aaaaargh! A bad start, with 2 misprints. Ok, actually not too fussed about this.

Alan and Janet Box Sets

Obverse

Side view 1

Side View 2

Alan's books - faded except for behind the box set cover

Hi

A few weeks ago, I came across a Paladin edition of part 3 of Janet Frame's autobiography called "The Envoy from Mirror City at a free book library, and re-read it, then later came across part 1 (To the Is-Lands) in the  same edition and read that too. At about this stage I decided to lean in to these findings and bought part 2 (An Angel at my Table) through ebay. 

I really like Janet's autobiographies, especially The Envoy from Mirror City in which Janet, in her early 30's finally breaks strings tying her to New Zealand and heads to Europe to finally be herself, be something new and escape her previous life. I had something similar happen to me - I spent a year overseas in my mid 20's which shaped my life - I am reminded of the McGarrigal's song "Talk to me of Mendocino" with the line "In New York state I came of age, when first I started out from home"

Anyway, the ebay book arrived, and then I decided to really lean in, and started to make a "Fan box set" out of the 3 books. The idea and template for this was the Alan Marshall autobiography box set from Cheshire which was lurking on my shelf - covers faded and slightly battered. I copied the style of the box set cover which is slightly quirky. 

I didn't go too far with all the details but made the cover from old manilla folders which I pre-cut to A4 size before printing. The yellow part needed to be printed 3 times - on the first print there were a few typos, the second ended up on the same side as the first, and the 3rd - voila I finally got it right, and the misprints are a feature, not a detraction. The slightly remedial printing from my old laser printer looks good too.

 It all came together quite well in the end. I quite like this sort of art, which is a sort of punk, with no need for technical perfection when there is a bit of originality.

Regards Steve Nurse 

Australian Maths Society Conference 2025

 

Jig used to make fractional polygon / reciprocal frame side.

 

A sculpture at Bridges Eindhoven

One of these gentleman was its maker. Unfortunately I couldn't find out the artist's name.

Here's one I made myself, reciprocal frame polyhedron  

In the middle of last year I went to the Netherlands and England,  and visited the Bridges Maths and Arts Conference 2025 in Eindhoven, and was inspired by a sculpture, a dodecahedron from one wooden piece repeated 60 times. On my return I made my own version, and did that with frames made from 5 craft sticks per side. Right from the start I made the pentagon faces using jigs to keep the shapes regular, and then joined the faces at edges to make polyhedrons. I had made “Stick Bombs” (without knowing that word for them) before but was now able to call them “reciprocal frames”. These were in the shape of star or fractional polygons, with stick endpoints pressing against each other and able to clamp edge–joiners. Initially the joiners were cut up bike tyres which have good friction properties!

Two types of tetrahedron made with coffee stirrers and bike tyres. A six sided figure substitutes for a triangle in these tetrahedra leading to the 2 different "2n" construction options. In the first, 2 polygon vertices are connected to each edge. In the second, one vertex is connected to each edge, and the other is used to connect a truncation side. (The second tetrahedron only has a demonstration truncation side)

 

Exhibition piece, the 0.7m diameter truncated icosahedron. 

The structural part of the sides are surrounded by cantilevers. The connectors used at edges are drinking straws.

Dodecahedron from decagons. This has 2 stick endpoints per side meeting at each edge.

Truncated octahedron.

Later I refined my techniques and used longer, narrower coffee stirrers as a material. Their flexibility let me make polyhedron sides from a central fractional polygon with cantilevers radiating out. They can make uniform sides which can be joined using drinking straws at stick ends. Soon I established an architecture where sides held by reciprocal forces could be made without glue and combined to form platonic solids. Key to these structures were “2n” sides where (say) a side designed to represent a triangle would have 6 sides with either 2 pairs of coffee sticks joining on each side or 1 pair of coffee sticks per side, and another pair donated to join truncation sides.

It was with this second technique that I made a large (70cm diameter) truncated icosahedron – this has radial six-stick frames as the 20 triangular faces and five-stick frames as the 12 pentagonal  truncations. This is my largest coffee stirrer creation and it didn’t need glue for several months when it sat in my lounge room. However it was later accepted for The Mathematical Art exhibition at the 2025 Australian Maths Society conference, maybe 10k from where I live. With the sides glued, I was able to disassemble it and take it by bike up to the exhibition. 

2 sticks - per side tetrahedron next to a 2-triangle cell net showing complete triangle edges.

 

Developing the figure of eight knot from the tetrahedron schematic

After further experimenting and shape making, I eventually found an interesting equilateral triangular configuration. This uses 2 coffee stirrer sticks per side. The four stick ends are distributed 2 to one triangle side, and one each to the other two sides. With cells of two triangles from these triangular frames, polyhedra could be made, and a series of stick joiners was designed to accurately set up the triangular sides. These joiners hold sticks at their crossing point.

 

2 stick per side tetrahedron and icosahedron

 

Net and knot for the icosahedron shown above

The first polyhedron made from sets of triangular cells was a tetrahedron whose sticks form a figure eight knot but look somewhat wild. It helps to put the construction on a 2d plot of its side to work out where the edges are and how the tetrahedron is formed. With that figured out, the construction method was applied to 2 icosahedra and 2 octahedra. Each of these shapes has an associated cell coverage and a knot or link which can be represented on paper.  These platonic solids (tetrahedron, icosahedron, octahedron) are part of the “deltahedra” family, the set of polyhedra with equisized, equilateral triangle faces.

Once I discovered the relationship between deltahedra and knots, I was able to submit an abstract (pitch to talk at) the conference where my large icosahedron was due to be exhibited. I had already pitched and had a talk on craft stick polyhedra accepted, and my 2^nd pitch was accepted too.

 

There was a photo shoot for the artists of the mathematical art show, this was the group shot

 

and all the artists present had their photos taken with their work.

The conference time rolled along and I worked quite hard at preparing conference material – the deltahedral knot family presentation was first and I had to have enough material to present before I was able to present it! I had previously worked out the angles and stick lengths associated with 2 styles of joiners, and that geometry was presented along with the spreadsheet where I’d put all my knot / polygon data. As well, with the discovery of the term “deltahedra” came the “8 convex deltahedra – and I aimed to cover all of them in my talk. My girlfriend gave me a critique of my just-finished presentation a few days before it was on and I managed to get the first presentation across the line with about 12 attending.

 It payed to be social and sociable at the maths conference. I let my 2 topology contacts at Monash University know I was presenting and later met one of them for the first time. I also chatted to people at registration which resulted in a topology researcher attending my talk. Also I signed up to attend an education afternoon featuring maths construction kits. There were only a few there but that resulted in another good contact.

All packed up! By the end of the talk following mine, most of the coffee stirrer polyhedra were broken down into sides and ready to be carried home by foot and car. Joel is in the patterned shirt at right - the shirt was a conversation starter and he was the sole audient at my 2nd talk.

Some comic relief! I made this slightly less serious item using bicycle spokes, 3d printed joiners and coffee stirrers. No extra thoughts about knots required!

At the art exhibition, I had spotted Matt (at right) knitting something mathematical at one of the plenary lectures, and saw and chatted to him a few times 

One of the plenary lectures with serious, difficult maths and discussions of topology in molecules.

 

 

My second talk was on the Friday afternoon of the conference, and I had to regroup after the first talk and bash out another presentation. This was at least for established material and I was a bit more relaxed and ready to present. However I ended up presenting to only one person who did, however show enthusiasm and talked about how he could engage his daughter in doing maths through the models I was showing. During the following talk I quietly packed down my models into sides and they were quite compact by the time I’d finished.

Thanks especially to Katherine Seaton! It is mostly through her Instagram posts that I knew about the AustMS conference and I was very pleased to participate and attend. 

My presentations and 3d printable files used to make my models can be downloaded for free from the Thingiverse website. This is not an academic website, just a very large repository of 3d printable and associated files. See

Here for the for craft stick polyhedra and

and Here for the deltahedral knot family

 

I’m still working on the deltahedral knot family, and aim to work out knots for all the 2d and 3d shapes and deltahedra up to 6 sides. As well I’m interested in how the deltahedra relate to each other and would like to show both shapes and knots, and compare their relationships. I’d also like to get the work published but that will take time.

Regards Stephen Nurse.