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Jig used to make fractional polygon / reciprocal frame side. |
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| A sculpture at Bridges Eindhoven |
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| One of these gentleman was its maker. Unfortunately I couldn't find out the artist's name. |
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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!
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| 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) |
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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. |
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| Dodecahedron from decagons. This has 2 stick endpoints per side meeting at each edge. |
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| 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.
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| 2 sticks - per side tetrahedron next to a 2-triangle cell net showing complete triangle edges. |
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| 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.
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| 2 stick per side tetrahedron and icosahedron |
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| 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 2nd pitch was
accepted too.
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| There was a photo shoot for the artists of the mathematical art show, this was the group shot |
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| 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.
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| 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. |
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| 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! |
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| 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 |
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| 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.
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