In the last article for the release of my liberty,
writes Doctor Gulliver, who was under some trouble at the time as you may recall,
the emperor stipulates to allow me a quantity of meat and drink
sufficient for the support of 1728 Lilliputians.
Some time after, asking a friend at court how they came to fix on that determinate number,
he told me that his majesty’s mathematicians,
having taken the height of my body by the help of a quadrant,
and finding it to exceed theirs in the proportion of twelve to one,
they concluded from the similarity of their bodies,
that mine must contain 1728 of theirs, and consequently
would require as much food as was necessary to support that number of Lilliputians.
By which the reader may conceive an idea of the ingenuity of that people,
as well as the prudent and exact economy of so great a prince.
The prince of Lilliput was the not the first nor was he the last ruler
to have bad advice from his scientific council.
The mathematicians of Lilliput were not right.
Indeed they were right in their geometry, yes, we discussed the geometry last time,
and I think all will agree that they understood Euclid
and they understood proportionality
and they understood scaling, in respect to geometry perfectly.
It is quite true that the man mountain Gulliver, being 12 times as high,
and proportioned similarly in all parts to a Lilliputian,
would be 12 x 12 x 12 as great in volume.
Or 1728 and the same amount greater in weight,
since he was made of the same material, bone and flesh.
Similar proportions in all respects.
That is the assumption of the text.
But where they failed of course to be quite right was when they said -
consequently he will require 1728 times as much food.
The "consequently" is very much open to question. Why is it consequent?
Euclid said nothing whatever about eating and drinking of spheres or polygons.
That's a physical problem. The biological problem,
a problem with energy, chemistry, the shuffling of atoms.
And we have to study that in a much more detailed way.
Now, that has been done. We shall do some of it here today and see it ourselves.
And I think we can come quickly to the conclusion that the Lilliputians
were in error on that small point. But that's not the interesting thing.
The interesting thing is how remarkable would be the result,
that is the real result in the real world,
which I think goes far to illuminate how we understand ourselves,
the animal world, our structures and so on.
Let us first look at a simple example if we have the map, Bill, please. Thank you.
Just to use a very early example of which is a quite familiar geometry.
I'll mention it quite quickly.
Here we have these two countries, I've drawn it with the shrunken Atlantic.
Possibly we're drifting that way again, I don't know.
And with the shrunken Atlantic, so you can see the two simply
in their proper proportions, and their proper scale.
Now notice if I ask how much coastline or edge, perimeter,
Britain has for each unit of area. There's really a good deal.
Nobody can be very far away and every square of area can possess
an edge on the sea in quite small squares.
But if I make squares of the same size in this big country,
I leave a great section of the country far from the sea.
The squares in there, of a size to fit neatly in Britain,
would be nowhere near the sea, they have no coast at all.
This region contains for example the state of Nebraska,
like Switzerland without any sea coast.
I will refer to it from time to time.
Western movies are often made in Western Nebraska
and Eastern Colorado, far from the sea.
And here we have the United States and Britain drawn on a scale more nearly equal
and now you see the same effect begins to appear.
Even though the shape is quite different, still it would be true
if I make squares which have edges that are coast, many squares,
little pieces of coast, many squares would be far from the sea here as here.
Indeed the larger the area, the less surface, the less edge it has per unit area.
Let us do that a little more formally on the board.
And with our blocks. And we'll see how it can work.
Let us first work, I shall first work and discuss the problem solely in the flat plane.
Just to make it easy enough.
Now, clearly the world is not flat, the world is not flat in any part.
It is our symbols which are flat.
Even paper, as we've seen, has plenty of thickness and plenty of scale.
But it's worthwhile to work first in our mind in two dimensions,
just because that's easier than three, so I'll speak of a flat world
as I speak of a flat map. In reality we shall very soon have to go over to the solid bulk.
In looking at the flat, then you can see that around a block, there are four edges,
around a square, the face of a block. There are four edges.
Every square owns the same four edges.
And only if I amass many together, do I begin to see that there are
some blocks, like Nebraska or like Switzerland, which are isolated.
Let us write down what that leads us to.
If I have a square. I'm not sure how you write what you call the distance round,
the region around, I will call it the coastline if you like.
If you more... a fancy word is perimeter.
Now the perimeter is proportional to the size of the edge.
If that is L, the perimeter of a square is 4 L.
If it's a circle, it's a different formula but it's still proportional to L.
If I take two blocks, make one just double the size of the other,
you would quite agree I get twice the perimeter but the shape remains the same.
That's the proof.
So perimeter is proportional to L, a straight line curve.
On the other hand, the area, all that, is proportional to L squared.
Again the same argument would work.
So you see if I simply consider bigger and bigger L's.
One, two, three.
I would get one, four, nine.
And clearly this is going up faster than this.
This is going up only in a straight line this is going up as the square.
So the amount of area per unit perimeter is always increasing, the bigger the object is.
Without any change of shape.
And of course for a smaller object, what happens?
Just the reverse. The more perimeter for the little bit of area.
And the same thing happens in the three dimensional world, as I say,
where we'll now discuss the surface which is
proportional to the square of the dimension.
And the volume which is proportional to the cube of the dimension.
Again, this goes up faster than that.
And indeed if you wish you may work out the ratio of these two and see how it goes.
And you'll find that the ratio, there's more volume, more surface per unit volume,
as you get smaller and smaller, just in the ratio of the sizes.
One power cancels out and the others remain.