Bringing the Platonic Solids to Life
Monday, 13 July 2020 | Hannah Hoskins
As part of my interactive Mathmagics show, I was looking for something new for the younger pupils - Years One and Two in particular - for whom numeracy tricks and stories would be too advanced and who are often unable to enjoy visits from those working with the older pupils. I wanted to be able to provide sessions for everyone, irrespective of their experience and abilities.
I'd created Treasure Hunts - so many steps forwards, backwards and sideways before... "You've found the Treasure...!" - when in fact it was nowhere to be seen... until someone spotted it, usually an apple, above the intrepid "hunter", reminding us that in fact we all live in three dimensions, not two....
We'd enjoyed stories of bold ants circumnavigating spheres - and Pythagorus welcoming his guests with his Cup of Fairness - an unexpected stream of falling water, a guaranteed cue for hilarity... but I wanted something more sophisticated... something for which I could use as many volunteers as possible and control the mood of the audience - if they were noisy, quieten them; if quiet, excite them. You know what I mean.
And then I discovered Magnetic Polydron... Perfect...!
I first saw them watching the wonderful Selwyn van Zeller, whose fabulous "Maths in a Suitcase" and other events have thrilled and intrigued thousands of children of all ages across the country. Using Polydron, I saw him demonstrate various 3D solids, including a terrific demonstration of the strength differences between triangles and squares, cubes and tetrahedra, and I thought "I should get hold of some of those"...!
So I did.
I obtained a collection of Polydron, magnetic and clip-together plastic, enough to create all five Platonic Solids and more, and then I wondered: how can I bring these to life - and, in the process, involve as many helpers as possible?
One of the things that I love is that the Magnetic Polydron ("polydra"?) clip together so easily, accurately and satisfyingly. And I'm not just saying that ‘cos they've asked me to write this article: I find there's something very satisfying in the way that they spring together so helpfully to form just the shape that you're after, that it really is "child's play". This meant that my volunteers would always experience success, no matter what their abilities - and even if they "broke" the set-up (perhaps they'd grip them too hard and crush my carefully prepared set of three, four or five triangles clipped together), whatever they did, it was generally quick and very easy to repair. All I'd have to do was make a huge show of how brilliantly they'd done, even if all they'd done was to place a few triangles on top of one another. This truly was to be an act in which children of almost all abilities could participate equally, in front of their peers and teachers, with equal levels of success and praise - something very important to me.
(Indeed I've found Magnetic Polydron is used and and loved in many of the "special needs" schools in which I've performed, as well as in the mainstream.)
So, I'd found a great prop... Now, how to bring it to life...?!
What launched me was my discovery of "Plato": the net of half a dodecahedron can be laid out to look exactly like a two dimensional man: I say "exactly", well, close enough: two arms, two legs, a tummy and a head - it's even easier to see if you colour-code them. So that's what I did: clipped six pentagons together (for this I use the original non-magnetic Polydron), colour coded with matching arms, matching legs and differently-coloured head and belly. Easy to hold up and show them that it looks like a human - albeit a somewhat "floppy" one - and easy to show them that he's even a bit like me: two legs, two arms, a head and even a roundish belly..! I call him Plato, my friend Plato, because he looks like... a plate..!
Then, I explain, as I clip his pentagons together further, he can become three dimensional, so now he's no longer flat, no longer Plato, he's more like.....? "Bowlo.!" they all cry. (Very useful for catching things.) "And here's his friend, Holo" I say, picking up his counterpart, already in a bowl shape, with a hollow pentagon at its centre, through which I can peep..!
"The question is: would Bollo and Hollo fit together...? What do you think...?"
And then we're off: I switch to the magnetic equivalents and bring up my first two volunteers. Each holds one of the six-pentagon bowls and then, slowly, carefully (and pretty easily) they clip them together...
I hold up the dodecahedron as a sign of the amazing job they've just done and call for them (and for their shape) a huge round of applause...!
And they beam.
But that's not all... Just as they expect to sit down again, they're asked to stay as we investigate this solid more closely: "What a gorgeous object. It looks like a......" I say, encouraging everyone to fill in whatever they see it resembles. (I do this for every solid as we proceed - after the round of applause for making it, of course.
Who cares what they think it looks like..? Whatever they think is correct.. This is creativity - in maths - though the "diamond" suggestions for two tetrahedrons together usually prompts fun replies of "if only"...!)
So, with the volunteers who've created this amazing shape still at my side, we investigate in more detail: we observe that all the sides have equal length, then, together, as a full year group, we count out loud the faces around each point. With growing excitement, we realise that these numbers are always the same... We've discovered something amazing about this gorgeous shape: it follows a pattern, it's called a Platonic Solid...! (At which point I share that that's why he was called Plato - and who the real Plato was...!)
Then we name the solid, using Plato's Ancient Greek, of course (which any Greek speakers love) - with Simpsons-related cheap gags to be had with the "Doh!" in dodeca and great fun later when they name a cube an "exihedron" (because exi is Greek for the six faces that it has - it makes sense)... "So what's it. called.?" "An exihedron..!" they cry. "Er, no.... It's a cube.!" (Classic ‘undercut'..!). (There's also fun to be had with "ochto", meaning eight, as in Octopus, Octagon and October, the eighth month...!)
And so, we've created and named the first shape: a dodecahedron.
"Would anyone like to make another?" The hands shoot up...!
Thus we create all the other solids, with different volunteers for each, starting with the tetrahedron, clipping together four equilateral triangles - red, blue, green and yellow - with a great round of applause for its creators, as I hold it up, followed by an even bigger boost as, counting together, we discover that this too follows the rules of Platonic solids....!
...then the cube, which they construct easily with no help from me...
...the octahedron, who becomes what my daughter calls Mr Chatterbox - more on him below...
... and finally the complex, gorgeous and, by that time, very impressive icosahedron at the end.
We also make some shapes that don't make the grade:
- three red and three green triangles, clipped together effortlessly in a beautiful shape ("which looks like a...?") "but..." we discover, as our count of the faces- around-each-vertex, which started so promisingly, suddenly goes awry "...is it Platonic..?" I ask doubtfully (a "leading question, I know, but they love getting it right...) "Noooo!" they all cry... and we feel the disappointment as we put it to one side - though the builders still get their round of applause..! It's still a gorgeous object.
- and shallow saucers of five green and five red triangles either clipped directly to each other or connected by five alternating blue and yellow squares, each of which also makes a lovely shape ("a bit like a....?") but, again, "they're not Platonic..!"
(As you can see, I also love the clear colours, which I carefully choose to make the faces easy to count - we even get five times table practice, when we finally construct our icosahedron from five red, five green, five yellow and five blue: complex to look at, yet simple, easy to build and very beautiful.)
Thus, with two volunteers for each, including the non-platonic options, and group- participation in counting, naming and deciding whether or not they're Platonic, finally we've created all five of the Platonic Solids.
Dozens of volunteers, loads of successes, huge involvement, great fun, emotional ups and downs - and a line up of beautiful shapes to show for it....!
And then, the Grand Finale: "Who'd like to make another one...?" I cry (as some teachers anxiously wonder how much longer this will go on for...) and, of course, all hands shoot up...!
I choose one - ask them to stand. Then my tone turns serious.....
"So... You'd like to make another one, would you...?" "Yes"
"Are you sure..?"
"And.. you'd be prepared to work quite hard at that..?" "Er, yes...?"
At this point they're beginning to look a little nervous, so I reassure them: "I thought you would. You look very hard-working. I'm sure you'll do a great job..."
They cheer up.
"So, you'd be prepared to work very hard, would you?" I prompt. "Yes"
"And.. for as long as it takes..?"
"Oh yes," they nod.
"Well, I'm afraid I've got some news for you..." I pause dramatically...
"You see: you could try making another one today, a shape that fits all these rules, another Platonic Solid... In fact, you could spend all afternoon trying to make one... but you wouldn't manage...
"You could try for the rest of this week, but you wouldn't manage.... "For the rest of this term...
"This whole year...
"In fact, you could try... for the rest of your life...!" (they imagine playing with Polydron into their dotage...) "but, I'm sorry to say, that you simply... won't... manage.... (sit down)....because, Ladies and Gentlemen... what Plato discovered - over two thousand years ago in Ancient Greece - is that, if you want to make 3D solids that follow his rules (equal length sides and the same number of faces around each vertex), it's only possible to make.. five....
"And you, Ladies and Gentlemen, have just made all five of them...!
"Ladies and Gentlemen, I present to you: The Platonic.. Solids..! Give yourselves a round of applause...!"
And so, over the course of a magical twenty minutes Year One or Two have now created and become experts in all the Platonic Solids, something that even some of their teachers may not have heard about - yet! They've interacted with me and with each other, used the Polydron and created some beautiful shapes.
Dozens, of all ages and all abilities, have volunteered and played a part. They've learned a little history, some mathematical methods and even some Ancient Greek along the way - and are now in a position to explain all of this, if they'd like to, to Years Three, Four, Five, Six and beyond..!
I love the thought of that..!
Of course, I can then give the teachers cut-out nets for the pupils to use and colour and talk about fair dice of 4, 6, 8, 12 and 20 sides and, for the more advanced, invite them to record their observations and discover equations relating the numbers of sides, faces and vertices, if they wish, but, whatever they choose to do now, they have had a magical experience and, hopefully, will see the world of shapes and colours around them in a whole new way...
And to be frank, I couldn't do this without my Magnetic Polydron.
The beauty of them is: they're quick, fun, satisfying, clearly coloured and easy for even the least dexterous to handle successfully - and that's for doing this activity, something very specific. Simply playing with them to see what shapes they can make and how their magnets interact together, there's far more that they can do. They're quick and easy for me to set up too, especially the magnetic ones, and for my audiences to use. In fact, without them, I don't think I could do this act, which I'm so proud of and enjoy so much, an act which has gloriously filled many an afternoon or morning session and delighted so many teachers and pupils along the way. So thank you, Polydron..! I couldn't do this act without you.
Oh, and Mr Chatterbox:
I discovered him late one evening in my hotel room. I was practising in front of the mirror the night before my first performance of this routine, perfecting it, rehearsing and laying out the Polydron on the bed, working out just how to set them up so I could work quickly and efficiently through all the shapes and options... and I found that, if I put my fingers and thumbs though four neighbouring holes in an octahedron, I could partially pull them apart on one side.. to make an opening and closing mouth - and thus my talking co-star was born...
On carefully counting the faces around each point, as normal, with the children - "One... two... three... four", "One... two... three..." "FOUR!" he suddenly bellows, springing into life.... with a very loud voice, huge enthusiasm and no manners whatsoever..!
And from then on there's no stopping him: neither the children nor I can get a word in edgeways as he happily volunteers "FOUR!" in answer to almost every question..!
"HELLO DUNSTABLE..!" he cries, "I LOVE MATHS..!" "What...?" "MATHS..!" "Yes, I know you..." "I DO...!" "Well, I" FOUR....!" "Yes, well, thank you Mr..." "HOORAY...!" "Yes, I.." "I LOVE MATHS..!" "Ye' you said.." "YAY MATHS..! YA- HAY..!"
Ah Mr Chatterbox, what would I do without you...? I love a spot of anarchy... Don't you..?!
David Hall May 2020
For information on David's inspirational Mathmagics sessions and other wonderful, interactive Workshops and Shows, visit www.davidhallworkshopsandshows.co.uk.
For Selwyn's "Maths in a Suitcase" and more, visit www.mathsinasuitcase.co.uk.