## Mathematics Flat Out!

Dimensions are interesting things. Something with zero dimensions is a point, a dot in space (is that the same as no dimensions?). Something with one dimension is a a straight line (it has length). Something with two dimensions is a planar shape ( having length and width like a triangle, square or any polygon). Something with three dimensions is an object with volume (length, width & height, such as a pyramid, sphere or greater stellated icosahedron).

It is difficult to imagine, but there are far more than three dimensions – indeed many more dimensions are necessary for our understanding of modern physics. A little starter is at “Imagining the Tenth Dimension”, and that is only for the first ten dimensions! Watch it and learn that you need far more than three dimensions to exist in!

As we saw in class, you can make the most amazing 3-D shapes from flat patterns – we made the greater stellated icosahedron – which was won by Prathu. But for the purposes of our class, we need to be able to calculate the total surface area of a variety of polyhedra. For this, we need to be able visualise and draw the “nets” of polyhedra – what they look like unfolded and laid out flat. Some are easy to imagine – like a cube, others are far more challenging, like a icosahedron. Fortunately, there is a lot of help out there for you to develop this skill.

Probably the best site is Wolfram – but you do need to intall their free plug in to see them. I have linked to a list of good animations of unfolding nets here. Another site is Edumedia – you can only see part of what they do, as they only offer a free trial. Finally, the most complex yet is at Plus Maths – if you can follow all of that, you should be at university already!

But what fun is it to look at unfolding nets, when you could be making them for yourself? Here is an entire site full of paper models for you to make (yes, that is a challenge – make an amazing model polyhedra from that and you will get some bonus points – more if you correctly calculate the surface area – lots if you can do the volume! (I’m not sure I could, for some of those shapes!)).

The final concept I want you to look at is the relationship between Euler’s (pronounced Oiler’s) Rule and the concept of network traversabilitiy. Euler’s Rule links the number of faces, vertices and edges in the following formula:

Which tells you that if subtract the number of edges from the total of vertices and faces, you will always get the answer of two. This is useful, because if you can count the faces and vertices, you can easily calculate the number of edges (which can sometimes be difficult to count). Euler was a Swiss (Yay!) mathematician who also investigated the concept of “traversability” – which means given a network diagram – like a net of a polyhedra, the question was can you trace around it without lifting your finger or repeating a section. He discovered that, as long as each vertex has an even number of edges connected to it, or there are exactly two vertices with an odd number of edges connected, it is possible traverse. Here is a good explanation of the concept. You may want to think what this has to do with our study…

OK, and a final video for those of you who want a more visual presentation of the calculation of total surface area:

See you in class.

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### 8 Comments on “Mathematics Flat Out!”

1. Ellayne Garcia Says:

Hi Sir,

Is the exercises checklist due next week or the week after that?

2. Prathu Khairnar Says:

hey sir i tried this formula on a lot of things…and then counted the edges and vertexes (by changing the subject of v-e+f=2) and apparently it only works on symmetrical-3d shapes …

is that right or am i doing it wrong?

3. Jurem O Says:

I didn’t know any of this, but you explained everything to me clearly in the post. Thanks Mr G.

4. Prathu Khairnar Says: