Greater than one and less than infinity

A snippet from the IB Economics textbook: “greater than one and less than infinity”. Isn’t everything less than infinity?!

UPDATE: Apologies for sixth-former-me, I now realise that infinity is not a number, but a limit, and so while every number in the reals is less than infinity, a limit (including the PED, I suppose) may be infinity.

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Quarks, quanta… Oh, and a few other worlds!

Following a lecture by David Wallace from Balliol College, Oxford, on “The World(s) According to Quantum Mechanics”, I thought I’d share the beduzzling theorems and problems it presents.

As a quick introduction,  even though light travels in waves (that’s why it can’t go round corners, remember?), it’s also a kind of particle that comes in little chunks, or quanta, that we call photons. You can’t get half a photon, so the smallest bit of loght you can get is a photon. Pretty simple.

That part makes sense. What surprised the scientists when they were working with these tiny particles, though, is what happens next.

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Why I Love the Icosahedron

An icosahedron is any three-dimensional shape with twenty faces. However, we usually use the word to mean a regular, convex icosahedron, and this is the one that I like the most. You can see what it looks like in the picture, courtesy of Wikipedia:

IMG_0502

This kind of icosahedron is made up of twenty equilateral triangles the same size. If you look at the vertices (corners) of the shape, you can see how each one is made up of the points of five triangles. They seem to meet in a kind of pentagon-star.

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Smaller on the inside

This week, I’ve been reading about the hyperbolic plane. I became curious about this peculiar concept in mathematics after reading a short parody of Alice’s Adventures in Wonderland, in which the Mad Hatter’s tea party took place at a hyperbolic table.

The basic idea is that the distances in a hyperbolic plane are longer when they are closer to the boundary. When we illustrate this in the normal Euclidian geometry that everyone learns in school, a regular shape such as a regular pentagon looks distorted and irregular, since the side nearest the boundary will appear much smaller to make up for the fact that distances near the edge are actually much larger.

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“The imaginary part of a complex number is real.”

This is what our maths teacher told us when we first learnt about complex numbers. You can imagine our class’s bafflement when asked to explain what it meant: what did it mean? We guessed pretty quickly that it was something philosophical – maybe it was telling us that imaginary numbers do exist, even though they’re intangible… or maybe it was even deeper than that…

But this is maths. Mathematics, the most logical, clear and abstract of all the sciences. The polar opposite of philosophy. We were wrong. Why would a maths teacher have a “deep statement” pinned above his whiteboard? Unless it wasn’t really deep…

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Introducing… hyperspace!

Do you ever wonder if there’s something else out there – another world, a world different to our own, something bigger than this? As much as this may sound like the sort of ‘transcendental universe’ you might discuss in one of the more exciting R.E. lesson, in maths it is a real possibility.

Recently, I’ve been dipping in and out of Martin Gardner’s Colossal Book of Mathematics. If this book doesn’t live up to its title, I don’t know what does. Weighing roughly the same as a small elephant, it covers topics from recreational mathematics but also from more advanced areas , such as the concept of hyperspace. See the Resources section of this blog for a more detailed review.

When you start learning about shapes, you learn about the first three dimensions: 1D is limited to straight lines, 2D is flat, and 3D is made up of points in all three directions.

It’s perfectly reasonable to imagine, then, that shapes can extend into a fourth dimension. But how would this look? Can we even picture it?

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