Monday, September 8, 2014

What a Thousand Looks Like

Usually, multiplication is about generalizations of the relation between areas and lengths; but it can also be accomplished by nesting or iteration, which is sort-of how multiplication gets implemented in Alonzo Church's Lambda calculus and also in polymorphic type theory as the natural transformations
$$ \mathbb{N} = \forall X, (X\to X) \to (X\to X) $$
and so on...

Anyways, Paul collected some hyssop seeds
and expressed some interest in how many plants that makes.

So, just for fun and perhaps for reference, here are $ 1600 = 10 \times 10 \times 4 \times 4 $ black squares arranged in a square of squares; something between one and two thousand. Note that you can probably see the spaces between the squares. Note that you can't see the spaces between most of Paul's seeds.

This kind of visual counting can add interest to films and historical photographs involving well-organized collections of people and all sorts of other things.

There are easily two thousand people watching, from the further stands, looking at us

Another way to think about it: your fancy camera today probably boasts some megapixels per photo; which means you could capture a thousand people, dedicating a few thousand pixels per figure, with a camera you then hide in your pocket. If they'll sit still long enough.


Paul Stilwell said...

Thank you for the explanation. The thought did occur to me at the time that one way of getting a good estimate would be to count some in a small area, and then roughly multiply it to the full area...and it does work for this...but what do you do if the container is fuller? Meaning that one then has to consider the layers underneath surface area. Do you just calculate the depth and work that in?

Belfry Bat said...

Yes, one has to count by volumes if you have what looks like a volume... and now we're getting to where it feels like those fun-fair contests "how many jelly-babies are inside this tardis-shaped cookie jar?" I was never much good at those...

But, anyways, a thousand milliliters (one litre) fits in a box ten centimeters on each side, which isn't much, only we almost never see volumes presented in boxes! A litre of wine is usually stretched-out in one direction, and arranged circular, and inside the bottle is probably less than 10cm accross... but anyways, packing of noncubes into spaces more than two dimensional is a complicated problem!

The strange old man on the beach in Local Hero guessed he could hold about a hundred thousand grains of sand in his hand, but that was sand from his own beach; it could vary a lot either way with different sand, and different hands...

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