Chadstone – The maths fashion capital

It's a fine and fun tradition to head out to Melbourne's Chadstone Shopping Centre on a Saturday afternoon. You can shop for clothes, have a coffee, take in a movie, and top it all off with some mathematics.

Well, that last part may be news to some of you. Indeed, it was to us, too. But there, in the September 2007 issue of Journal of Mathematics and the Arts, was Chaddy's sculpture. It is titled Origin and was created in 1999 by Greg Johns, one of Australia's most popular and successful sculptors. (Many of the photos in this column are reproduced with Johns's kind permission).

Origin is intriguing and beautiful, but is it really mathematical? Let's begin by comparing Origin to two of Johns's other sculptures, on display in Melbourne and Adelaide (Johns's home town).

Though each has its own distinctive style, the sculptures also share some common features. To begin, all of the sculptures are constructed from the same basic building blocks: quarters or halves of solid rings with square cross-sections.

The building blocks are always "joined" at the square ends, but the blocks are permitted to overlap. That means that there are eight different ways to combine any two blocks into a combo-block, although one these "ways" will result in a complete overlap of the two blocks. Pictured below are the six essentially different ways of joining two quarter-rings. 

Union, the left sculpture above (and on display at 330 Collins Street, Melbourne) consists of four quarter-rings, all meeting at a common end. The right sculpture, Guardian Figure (Adelaide's Botanic Gardens), is composed of four small half-rings and two large half-rings. And, Chadstone's Origin consists of four small half-rings and four large half-rings.

A second, less apparent feature of all these sculptures is that they are symmetric. Union and Guardian Figure remain the same when rotated 180 degrees around a vertical axis. Chadstone's Origin is unchanged by a 180 degrees rotation around a horizontal axis.

These symmetries are not always easy to see, and it's not practical to go around physically rotating the sculptures (and the owners tend to disapprove). Fortunately, Greg John's website features some excellent animations, which highlight the symmetries. You can check them out here and here.

An interesting consequence of such a symmetry is that we seemingly get the same view of the sculpture from two different vantage points. (In the case of Origin, you have to stand on your head for the second view). But of course, the two views are not really the same, since the lighting and the background will differ. So, the symmetry actually demonstrates the importance of setting to our view of a sculpture.

This is all reminiscent of the dragon curves that we discussed in a previous column. Constructed from quarter circles, dragons curves live in a 2D planar world, but they also have their 3D counterparts, the logical limits of Greg Johns's sculptures. Below is one of these curves, brought to life by mathematician Chaim Goodman-Straus and artist Eugene Sargent, in sculpture form.

In summary, Greg Johns begins with a minimal set of building blocks and then explores the aesthetic possibilities when the blocks are combined in the most natural ways. It is indeed a very mathematical approach to his very mathematical materials, and we are more than happy to give Johns's sculptures the Maths Masters' seal of approval. Not that we imagine Johns has been losing much sleep awaiting our approval.

Puzzle to Ponder: Below is a Johns sculpture titled Squared Circle (Carrick Hill, Adelaide). What does it have to do with a cube?

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator.

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

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