Behold the tesseract

In my teens I was fascinated with a mathematical construct called a tesseract. This is a four dimensional hypercube. A cube is constructed in three dimensions from six faces, each a square. A tesseract is constructed in four dimensions from eight faces, each a cube.

Funnily both my youthful introductions to tesseracts were from fiction. The first was in Madeleine L'Engle's wonderful children's classic (still very readable for adults) A Wrinkle in Time. I know some people don't like this book because it has an underlying religious message, but for me it is one of the best children's science fiction stories ever. In it, a tesseract is a gateway for interstellar travel.

But much more informative is Robert Heinlein's brilliant short story, And he Built a Crooked House, which appears in the collection The Unpleasant Profession of Jonathan Hoag. Dating back to 1941, this story tells of an architect who builds a tesseract house. Now, clearly you can't build a 4D structure, so what he built was an unfolded projection. We're all familiar with the way we can draw a cube on a 2D piece of paper. Similarly you can make a 3D projection model of a tesseract.

The best known version of this is a cube with another cube inside it. Each vertex of the inner cube is linked to the equivalent vertex of the outer cube. The eight cubic faces are the six shapes surrounding the inner cube, plus the outer cube and the inner cube. You might say 'That can't be right, those six surrounding shapes aren't cubes,' but remember this is just a projection. Look at how you draw a cube on paper. Most of the faces aren't squares in the projection.

You can also unfold a tesseract from four dimensions to three. If you make a 3D cube out of paper, you can open it up into a sort of cross shape and lay it flat as 2D. Similarly, you can open a hypercube out and 'lay it flat' in 3D producing a sort of cross shape of cubes. This is what the architect built in the story. But then, as a result of an earthquake, the house folds into a true hypercube. This means that various transitions from room to room are weird - and some of the windows look out on different universes.

We can't really make a hypercube, but here's the next best thing (thanks to Ramon Quintana for pointing this out). I don't know if it's correct, but it's certainly beautiful. Here is a 2D rendering of a 3D projection of a hypercube, being rotated on various axes:

Why I hate opera

If I'm honest, the title of this post is an exaggeration to make a point. I don't really hate opera. There are a couple of operas - notably Monteverdi's Incoranazione di Poppea and Purcell's Dido & Aeneas - that I quite like. But what I do find truly sickening is the reverence with which opera is treated, as if it were some particularly great art form. Nowhere was this more obvious than in ITV's 2010 gut-wrenchingly awful series Pop Star to Opera Star , where the likes of Alan Tichmarsh treated the real opera singers as if they were fragile pieces on Antiques Roadshow, and the music as if it were a gift of the gods. In my opinion - and I know not everyone agrees - opera is: Mediocre music Melodramatic plots Amateurishly hammy acting A forced and unpleasant singing style Ridiculously over-supported by public funds I won't even bother to go into any detail on the plots and the acting - this is just self-evident. But the other aspects need some exp...

Is 5x3 the same as 3x5?

The Internet has gone mildly bonkers over a child in America who was marked down in a test because when asked to work out 5x3 by repeated addition he/she used 5+5+5 instead of 3+3+3+3+3. Those who support the teacher say that 5x3 means 'five lots of 3' where the complainants say that 'times' is commutative (reversible) so the distinction is meaningless as 5x3 and 3x5 are indistinguishable. It's certainly true that not all mathematical operations are commutative. I think we are all comfortable that 5-3 is not the same as 3-5. However. This not true of multiplication (of numbers). And so if there is to be any distinction, it has to be in the use of English to interpret the 'x' sign. Unfortunately, even here there is no logical way of coming up with a definitive answer. I suspect most primary school teachers would expands 'times' as 'lots of' as mentioned above. So we get 5 x 3 as '5 lots of 3'. Unfortunately that only wor...

Why backgammon is a better game than chess

I freely admit that chess, for those who enjoy it, is a wonderful game, but I honestly believe that as a game , backgammon is better (and this isn't just because I'm a lot better at playing backgammon than chess). Having relatively recently written a book on game theory, I have given quite a lot of thought to the nature of games, and from that I'd say that chess has two significant weaknesses compared with backgammon. One is the lack of randomness. Because backgammon includes the roll of the dice, it introduces a random factor into the play. Of course, a game that is totally random provides very little enjoyment. Tossing a coin isn't at all entertaining. But the clever thing about backgammon is that the randomness is contributory without dominating - there is still plenty of room for skill (apart from very flukey dice throws, I can always be beaten by a really good backgammon player), but the introduction of a random factor makes it more life-like, with more of a sense...

Search This Blog