Why can you not buy a Tetrahedron?

Today, I have been on a very geeky (or possibly nerdy – you decide) quest.

It started with a very charming lampshade that I stumbled across in Homebase:

Icosahedron2.jpg

A delightful Icosahedral ‘pendant’. (All lampshades are seemingly called pendants now.)

Inspired by this, I set off to find more shapes, but instead found this object:

lamp1

(Picture from Homebase – I couldn’t find it hanging up.)

As the previous lamp had been called ‘Darwin’, I did not have high hopes for a decent name for this one. However, I was shocked by what was on the box.

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Ceci n’est pas un hexagone.

 

I admit it takes a particular type of nerdiness to know that this is in fact a Small Rhombicuboctahedron (imagine a cube that has been peeled along its edges), but nevertheless I felt somewhat outraged. What is there about this that is hexagonal? If you look at it from the top, it is an octagon, and although joining two triangles with a square does make a six sided shape, this is not composed wholly of those.

And thus my quest began. What other interesting shapes, ideally Platonic or Archimedean could I find?

Around the corner, I found the next:

Octagon1.jpg

In the same manner as before, I was preparing to (pretend to) get cross about this one as well, but on closer inspection:

Octagon2.jpg

It is definitely ‘Octagonal’. Points to Homebase.

Having exhausted their lighting department, I headed next door to Dunelm, where a cornucopia of exciting shapes awaited me.

They had:

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A Cuboctahedron tealight holder.

These utterly charming Dodecahedral lights, available in hanging:

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Or in table top versions:

Dodecahedron2.jpgDodecahedron3.jpg

I did count – there are 12 pentagonal shapes, meeting 3 to each vertex.

Although they look a little seasonal, snowflakes have six-fold symmetry, so it can’t be.

But it wasn’t just restricted to lighting!

You could buy these Octahedral flower pots:

Octahedron1

Or these (rather well disguised) cube decorations:

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I admit it – these are lights as well, but if you look carefully (for example on the left one), you can see that theses were made out of collections of eight copper strips crossing one another, taking advantage of the duality of the cube and the octahedron. In other words, this is made by forming a cube, then joining the centres of its faces to one another, and messing it about a little.

However, the most creative and impressive mathematical shape I found all day was this one:

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It isn’t obvious at a first glance, but this shape is a Truncated Icosahedron – pentagons surrounded by hexagons.

TruncatedIcosahedron2

A topologist would want to play football with this.

 

The perspective makes this look a little odd, but it was very pretty in the shop.

I would have ended my journey there, but had to go to Sainsbury’s for some shopping, so had a look in there as well.

At first, I was hopeful, and thought I had found another Dodecahedron:

NotDodecahedron1

Lots of pentagons. A good start. A moment’s thought revealed a problem:

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Hexagons again!

By this stage, the morning’s coffee was wearing off, and there was one platonic solid left to find – the Tetrahedron. I scoured the shelves for one.

A novelty Easter Egg box? No.

A beauty product? No.

Some strange health food? No.

I sought professional help. ‘Do you sell anything pyramid-shaped?’ I asked hopefully.

These paperweights were the best we could do:

SquareBased Pyramid

Not a single Tetrahedron in sight.

Although if you were to buy two of those, you could make an Octahedron….

Why can you not buy a Tetrahedron?