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PMA Day Workshop PMA Day Workshop March 25, 2017 Presenters: Jeanette McLeod Phil Wilson Sarah Mark Nicolette Rattenbury 1 Contents The Maths Craft Mission ...................................................................................................................... 4 About Maths Craft ............................................................................................................................ 6 Our Sponsors .................................................................................................................................... 7 The Möbius Strip .................................................................................................................................. 8 The Mathematics of the Möbius .................................................................................................... 10 One Edge, One Side .................................................................................................................... 10 Meet the Family ......................................................................................................................... 11 Making and Manipulating a Möbius Strip ...................................................................................... 13 Making a Möbius ........................................................................................................................ 13 Manipulating a Möbius .............................................................................................................. 13 The Heart of a Möbius ............................................................................................................... 15 Other Investigations ................................................................................................................... 16 Notes .............................................................................................................................................. 17 The Menger Sponge ........................................................................................................................... 18 The Mathematics of the Menger Sponge ...................................................................................... 20 Levels of the Sponge .................................................................................................................. 20 The Menger Sponge ................................................................................................................... 20 The Strange World of Fractals ................................................................................................... 21 Fractals in the Wild .................................................................................................................... 21 Making a Menger Sponge .............................................................................................................. 22 Notes .............................................................................................................................................. 25 Origami ............................................................................................................................................... 26 The Mathematics of Origami ......................................................................................................... 28 Polyhedra ................................................................................................................................... 28 Construction ............................................................................................................................... 29 Units ........................................................................................................................................... 30 Making a Sonobe Unit .................................................................................................................... 31 Making a Sonobe Cube .................................................................................................................. 33 Notes .............................................................................................................................................. 35 Contact Us .......................................................................................................................................... 36 2 This work is licensed under a Creative Commons Attribution-NonCommercial- NoDerivatives 4.0 International License. For the terms of the license, please see https://creativecommons.org/licenses/by-nc-nd/4.0/ 3 The Maths Craft Mission 4 5 About Maths Craft Maths Craft is a non-profit initiative founded in 2016 and run by academics who are passionate about engaging the public with mathematics through craft. The inaugural Maths Craft event was the Maths Craft Festival held in the Auckland Museum in 2016, and sponsored by Te Pūnaha Matatini. Following the success of the Festival, where almost 2,000 people were entertained by the mathematics in craft and the craft in mathematics, we were inundated with requests to go on the road with our quirky brand of maths outreach. The result is Maths Craft New Zealand, a nationwide series of events supported by an Unlocking Curious Minds grant. This year, in addition to an even bigger flagship Festival at the Auckland Museum, there will be events in Wellington, Christchurch, and Dunedin, along with a series of workshops. Our aim is to show young and old alike the fun, creativity, and beauty in mathematics through the medium of craft, and to demonstrate just how much mathematics there is in craft. Hyperbolic crochet at the Maths Craft Festival in 2016. To see a list of upcoming Maths Craft events and to sign up for our email newsletter, visit us at mathscraftnz.org. 6 Our Sponsors In February 2017, Maths Craft was awarded a grant from MBIE’s Unlocking Curious Minds fund to take Maths Craft on the road. Thanks to Curious Minds and our generous sponsors, this year we'll be running events around New Zealand. 7 The Mobius Strip 8 9 The Mathematics of the Möbius Explorations into the Möbius strip serve as an excellent introduction to the field of topology. Topology (from the Greek τόπος, place, and λόγος, study) is the mathematical study of the properties of objects that are preserved through deformations, twisting, and stretching. (No tearing, cutting, or gluing allowed!) Topologists are interested in the geometrical properties of objects that are enduring and survive distortion. For instance, the roundness of a circle will not survive being stretched or twisted, but the fact that it has no ends remains unchanged. One Edge, One Side A Möbius strip is a surprising object; it’s a surface with only one edge and one side. To understand what this really means, let’s compare it to a similar yet more familiar shape: the cylinder. Take a strip of paper and glue the short edges together to form a cylinder. Put your cylinder on the table, resting on one edge. We’ll call this the bottom edge. A paper cylinder. If you place your finger on the edge facing upwards, the top edge, you can trace all the way around the edge and return to where you started without ever touching the bottom edge. The cylinder clearly has two separate edges. Now place your finger on the outside of the cylinder, about half way between the two edges, and trace all the way around the cylinder until you get back to where you started. Your finger will have stayed on the outside of the cylinder for the entire journey. To get to the inside of the cylinder, your finger would need to cross over one of the two edges. Our (unsurprising) conclusion is that the inside and the outside of the cylinder are separate, and so the cylinder has two sides. It all seems pretty straightforward, doesn’t it? But with one small alteration to the instructions above we can create a much more complex shape. 10 Make a Möbius strip by taking a strip of paper, bringing the two short edges together just as you did for the cylinder but give one of the ends a half-twist (that is, twist it through 180 degrees) before gluing the short edges together. A paper Möbius strip. Choose any point on the edge of the Möbius strip and place your finger on it. If you trace along the edge of the strip, you’ll see that you trace around the entire edge before ending up back where you started. There’s no top or bottom edge, it’s all one edge! Now if you place your finger on the “outside” of the strip, just as you did with the cylinder, and trace along the surface, you will soon see that your finger ends up on the “inside”. So you will arrive back at your starting point, but on the “inside” of the strip! If you keep going, your finger will eventually come back to where you started for a second time, and once again be on the “outside” of the strip. You have now journeyed over both the “inside” and the “outside” of the strip without ever taking your finger off the paper or crossing over an edge. This can only happen if the “inside” and the “outside” are not separate, forcing us to the curious conclusion that the Möbius strip has only one side. This is the perplexing property of the Möbius. But it is not alone. Mathematicians call shapes with this property non-orientable, and there are more of them. Meet the Family The Möbius strip was discovered by the German mathematician August Ferdinand Möbius in 1858. Having made his discovery, and in typical mathematician style, he then wondered about how a strip with two half-twists would behave, and how a strip with three half twists would behave, and
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