The Klein Bottle Beach House

by Burkard Polster and Marty Ross

The Age, 15 July 2013

Two leisurely weeks of school holidays, and what better place to relax than on the Mornington Peninsula? There's hot springs and wineries and beautiful beaches. And, there's a very mathematical beach house.

There is no apparent end to Victoria's mathematically inspired architecture. In previous years we've written of many impressive buildings, including of course Federation Square and RMIT's Storey Hall. Earlier this year we took a tour of Healesville Sanctuary's spectacular Costa building.

This week we’d like to introduce you to Rye's stunning Klein Bottle House. Constructed in 2008, this award winning beach house is the creation of Melbourne architects McBride Charles Ryan. (The lovely photographs are by architectural photographer John Gollings.)

The house is absolutely amazing, but where's the maths? Where's the bottle? Who is Klein? We have some explaining to do.

The Klein bottle is named after its discoverer, the 19th century German mathematician Felix Klein. It is a famous mathematical surface, a close relative of the Möbius strip, which we recently investigated. Below is a photo of a glass Klein bottle (available from the very quirky ACME Klein Bottle Company.)

Klein bottles needn't be made of glass. The discerning footy fan can acquire snug and stylish Klein bottle beanies. (The example below was created by our very own slave-knitter, also known as Grandma Maths Master.)

Now for some details. To make a Klein bottle begin with a rectangle and glue two opposite edges together to make a cylinder. Bend the cylinder around so it passes through itself, and finally glue the two circular ends together. (The pictures below were created with the 3D visualisation software JavaView.)

There is an obvious objection to the above procedure: since when are we allowed to have a surface simply cut through itself? Mathematicians are not at all bothered by this, blithely referring to self-intersecting surfaces as immersed. However, we admit that such surfaces are somewhat dissatisfying. It also turns out that for the Klein bottle these self-intersections are unavoidable. (Well, they're avoidable if you have a fourth dimension handy: a case where the cure is probably more puzzling than the disease.)

A second method of creating the Klein bottle demonstrates that it has a Möbius strip hidden within it. Recall that a Möbius strip is constructed by taking a long rectangular strip, giving it a half-twist and joining the ends together. The resulting surface has a single edge. Now, if we widen the Möbius strip in just the right manner, the edge will close in on itself and the end result is our Klein bottle.

It follows that the Klein bottle is one-sided just as is the Möbius strip. A bug crawling on the “inside” of the Klein bottle will wind up on the “outside”.

What about the Klein Bottle House? On the face of it, a one-sided self-intersecting surface is an unlikely model for a beach house. If nothing else, the heating costs would be astronomical.

Fortunately, the Klein Bottle House is not really a Klein bottle. Nonetheless, having viewed the many pictures of this stunning house, and this virtual fly-though, we can attest to its very Klein bottle-ish feel.

The architects Rob McBride and Debbie-Lyn Ryan were apparently inspired by origami Klein bottles (as they discuss, here and here).

So, our hopes of living in a real Klein bottle house have been dashed. But your Maths Masters have no quibbles with McBride's and Ryan's very stylish approximation. We'd both be delighted to move into one, preferably somewhere on the beautiful Mornington Peninsula.

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 (and iPads) with a hammer.