Making Your Own Sense

Reflections on maths, learning and maths learning support, by David K Butler

Art

I believe maths is something that can provide a source of enjoyment, a way for people to express themselves, and a way to bring people together.

So, I create mathematical artworks, which are usually collaborative and constructive in nature. Find out more about them below.

My Piece of Pi 2012 to 2018

About the artwork

My Piece of Pi was a collaborative artwork designed by David K Butler and drawn by members of the community.

On International Pi Approximation Day (22nd of July) from 2012 to 2018 we encouraged people to help draw and decorate more than 1000 digits of the decimal expansion of the number pi in chalk on the North Terrace footpath and the University of Adelaide grounds. This artwork is called My Piece of Pi.

Artist statement

This artwork speaks about the wonder of maths as a world of its own and the wonder of maths as a human endeavour.

The number π is part of the very fabric of the universe: it is there in every circle, and appears in the mathematical descriptions of many seemingly unrelated things. Yet it is impossible to write down exactly as a number. It cannot be written as a fraction, and its decimal expansion continues forever with no comprehensible pattern. This artwork highlights this paradox: a number so very fundamental that is so very unknowable. The digits will be written on the ground where people often walk emphasising the pervasive fundamental nature of π. But the day chosen to create the artwork is Pi Approximation Day, emphasising that however many digits we draw it is still only an approximation.

On the other hand, even though π is a fundamental universal constant, it is still a human construction. Humans had the idea of π and thousands of years of human ingenuity have gone into understanding the number. Indeed, we are only able to know as many digits of π as we do because of complex mathematical theories developed in the last century. Members of the community will be engaged to help draw the numbers, to emphasise the human element to mathematics. Each participant will choose their own style, colour and decoration and the total effect will highlight that our knowledge of π is the creative work of hundreds of people over time.

Finally, the collaborative nature of the artwork has further meaning: mathematics and art are not just reserved for so-called ‘mathematicians’ and ‘artists’. In fact, these pursuits are part of being human and any person can engage in them. Observers of this artwork do not have to be merely observers. Instead, they can participate in this mathematical artistic expression, and then stand over one of the digits and claim: here is My Piece of Pi.

Video of the 2012 My Piece of Pi

Random Journeys

About the artwork

Used public transport tickets were attached to wooden boards according to simple rules in order to create a randomly-generated cellular automaton. The four finished boards are currently displayed in Level 3 East Hub Central on the City East campus of Adelaide University, near the Maths Learning Centre.

Artist statement

This artwork is about the fundamental struggle in mathematics between randomness and determinism and between simplicity and complexity.

Public transport tickets of the same shape were added to each panel according to four simple rules, but the choice of where to place them was left up to the people who participated. This interplay between simple rules and random behaviour has resulted in complex maze-like structures, each of which is different and yet somehow similar – each is different because of the randomness, but each contains similar structures because of the common set of rules. Moreover, while the tickets appear to be the same shape, the slight random imperfections in their shape have caused unexpected perturbations in the regular pattern produced.

When looking at the artwork, viewers are encouraged to look past the randomness to discover the rules used to construct it, and to notice the common structures in the maze that were created as the tickets were added. Thus the viewers participate in the fundamental tasks of the mathematician – to discover simplicity behind complexity, and to find the deterministic beneath the random.

Photos of construction 2012

The Sierpinski Sponge

About the artwork

The Sierpinski Sponge is a fractal that lives in 3D space. It is like a triangular pyramid, but it is made out of smaller pyramids, which themselves are made out of smaller pyramids, which themselves are made out of even smaller pyramids, and so on forever.

In 2011 and 2014, we built models of the Sierpinski Sponge out of paper and sticky-tape. Four small pyramids were joined at the corners to make a Stage 1 model. Then four Stage 1 models were joined to make a Stage 2 model. Then four Stage 2 models were joined to make a Stage 3 model, and so on.

You can see Stage 0, 1, 2, 3 and 4 hanging in the ceiling of the Maths Learning Centre on Level 3 Hub Central, City East campus of Adelaide University.\

Photo galleries on Facebook and construction videos on YouTube

Stage 6 Flythrough 2011

Space-Filling / Mind-Filling

About the artwork

Space-Filling/Mind-Filling is an artwork designed by me, with many students and staff of the University of Adelaide assisting in its construction. The artwork is built from shapes called Silver Rhombic Dodecahedra, which have been made from origami units folded from A4 paper according to a design by American origamist Nick Robinson. Special thanks go to Nicholas Crouch, Daniel Kon and Carolyn Haese for especial effort in construction.

Currently the artwork is on Level 3 of the Barr-Smith South building on the City East campus of Adelaide University, just next to the Polygon lecture theatre.

Gallery of construction up to July 2017

  • A window with posters on it and through the window in the corner is a small pile of silver rhombic dodecahedra made of white paper.
  • A window with posters on it and through the window in the corner is a pile of silver rhombic dodecahedra made of coloured paper.
  • A small room with one wall a window with posters on it. In the corner is a pile of silver rhombic dodecahedra made of white and coloured paper.
  • A small room with one wall a window with posters on it. In the corner is a pile of silver rhombic dodecahedra made of white and coloured paper.
  • A small room with one wall a window with posters on it. In the corner is a pile of silver rhombic dodecahedra made of white and coloured paper.
  • A small room with one glass wall. Around the edge of the floor is a pile of silver rhombic dodecahedra made of white and coloured paper.
  • A small room with one glass wall. Around the edge of the floor is a pile of silver rhombic dodecahedra made of white and coloured paper.
  • A small room with one glass wall. Around the edge of the floor is a pile of silver rhombic dodecahedra made of white and coloured paper.
  • A small room with one glass wall. Around the edge of the floor is a pile of silver rhombic dodecahedra made of white and coloured paper. In one corner, the pile reaches up to the ceiling.
  • A small room with one glass wall. Around the edge of the floor is a pile of silver rhombic dodecahedra made of white and coloured paper. In one corner, the pile reaches up to the ceiling.
  • A small room with one glass wall. Around the edge of the floor is a pile of silver rhombic dodecahedra made of white and coloured paper. In two corners, the pile reaches up to the ceiling.
  • A small room with one glass wall. Around the edge of the floor is a pile of silver rhombic dodecahedra made of white and coloured paper. In two corners, the pile reaches up to the ceiling.
  • A small room with one glass wall. Around the edge of the floor is a pile of silver rhombic dodecahedra made of white and coloured paper. In two corners, the pile reaches up to the ceiling.
  • A small room with one glass wall. Around the edge of the floor is a pile of silver rhombic dodecahedra made of white and coloured paper. In two corners, the pile reaches up to the ceiling.
  • A small room with one glass wall. Around the edge of the floor is a pile of silver rhombic dodecahedra made of white and coloured paper. In two corners, the pile reaches up to the ceiling.
  • A small room with one glass wall. Around the edge of the floor is a pile of silver rhombic dodecahedra made of white and coloured paper. In two corners, the pile reaches up to the ceiling.
  • A small room with one glass wall. Around the edge of the floor and filling most of the back wall is a pile of silver rhombic dodecahedra made of white and coloured paper. In two corners, the pile reaches up to the ceiling.
  • A small room with one glass wall. Around the edge of the floor and filling most of the back wall is a pile of silver rhombic dodecahedra made of white and coloured paper. In two corners, the pile reaches up to the ceiling.
  • A small room with one glass wall. Around the edge of the floor and filling most of the back wall is a pile of silver rhombic dodecahedra made of white and coloured paper. In two corners, the pile reaches up to the ceiling.

Poster describing the artwork

A poster of mostly text. Space-Filling/Mind-Filling by David K Butler The shapes you see here are silver rhombic dodecahedra. They are dodecahedra because they are solid shapes with straight edges and twelve flat faces -- the "dodeca" refers to the number twelve. They are rhombic because each of the twelve faces is a rhombus -- a four-sided shape whose four edges are all the same length. They are silver because the diagonals of each rhombus are in the "silver" ratio -- the same ratio as the sides of a so-called silver rectangle, which is the shape of an A4 piece of paper. There are a lot of interesting things to know about silver rhombic dodecahedra: They are space-filling, which means they fit together without gaps to fill space. Sometimes this action is called tessellating. This artwork demonstrates their ability to tessellate by fitting them together to fill this corner of the campus. The fact that the faces are silver rhombuses means that they can be folded from an A4 piece of paper. Origamist **** has designed a method of folding units so that they can be slotted together and these are the units we have used to build each dodecahedron. All their faces are the same, but they are not regular. A regular polyhedron has the same structure of faces meeting at each vertex, but our dodecahedra have some vertices with three rhombuses and some vertices with four rhombuses. There are in fact only a small number of regular polyhedra. If you draw in the short diagonal of every face, you can cut along these lines to slice off all the four-rhombus vertices. The shape that results is a cube. The six pyramids leftover fit neatly together to make an identical cube. If you draw in the long diagonal of every face, you can cut along these lines to slice off all the three-rhombus vertices. The shape that results is a regular octahedron (which has eight faces that are equilateral triangles). The eight pyramids left over fit neatly together to make two tetrahedra (which each have four faces that are equilateral triangles). This dodecahedron can be cut into twelve identical square-based pyramids. It can also be cut into four identical parallelepipeds -- a shape like a slanted cube. The fourteen points with coordinates (\pm 1, \pm 1, \pm 1) and the rearrangements of (\pm 2, 0,0) form the fourteen vertices of a silver rhombic dodecahedron. This rhombic dodecahedron is one of only two known to exist. The other is the golden rhombic dodecahedron whose rhombic faces have their diagonals in the golden ratio. When you start looking closely at this shape and all its properties, you will find that there is more and more to see. You may find that questions pile up in your mind, not just about this shape, but others, and they will spread to fill the corners of your attention. This shape isn't just space-filling, but mind-filling, just like the best possible mathematics.

(PDF version of artwork description poster.)

Video of how to fold the rhombic units

Crochet coral

About the artwork

In 2011, I was asked to consult as a mathematician on the Royal Institution of Australia’s Hyperbolic Crochet Coral Reef. As part of that project, I learned to crochet and spent many hours creating corals and other creatures based on various iterative rules. The string hyperbolic quadrics which I also designed were incorporated into the exhibition.

The first set of corals I made were donated to the Adelaide Reef, but there is a small collection of corals that have been made since then stored in the Maths Learning Centre at City East campus of Adelaide University. The string quadrics are also in the Maths Learning Centre.

Prezi of corals for the Adelaide Reef

This prezi displays photos of the corals I made for the RiAus Adelaide Reef in 2011 and 2012.

Photo gallery from the coral exhibition

A man in a black t-shirt (me) stands in the corner of a room where many crocheted corals in various shades of red are attached to the wall, as well as pink ones on threads hanging from the ceiling.
A round hyperbolic quadric is made of red string drawn through the holes in a plastic box and a small girl with curly blonde hair reaches inside to feel the crocheted coral attached there.
A saddle-shaped hyperbolic quadric is made of red string drawn through the holes in a plastic box. Crocheted coral in various shades of brown is attached on the wall behind.
A room with grey carpet and walls has a large amount of crocheted coral in a large variety of colours and shapes arranged in reefs on its floor and walls. The photo is taken from behind a clump of coral on the wall and looks down into the room.

Maths Sheep Play Sheep

About the artwork

Maths Sheep Play Sheep is a picture book that I wrote and illustrated in August 2023 in honour of Children’s Book Week. It is inspired by Where is the Green Sheep? by Mem Fox and Judy Horacek.

The book

You can download a PDF of the book here, and see and hear me reading the book aloud below.