Tag Archives: norms

Jenga blocks and Gowers’ hyperplane space

This post is an attempt to communicate some of the feel of Banach space theory to those who aren’t familiar with it. I once tried to explain my research to a six year old using Jenga blocks, but fortunately only got as far as the triangle inequality. Near the end of my Phd, at my supervisor’s suggestion, I started to explore the complicated Banach space that is Timothy Gowers’ solution to Banach’s hyperplane problem. These experiences inspired the following explanation of one relatively simple observation (that I included as an example in my thesis) through the delightful medium of building blocks.

Our object of study are towers of good old-fashioned building blocks. Each block has a number written on its side, so each tower built from these blocks gives a sequences of numbers (x_1, x_2, x_3, \ldots). These don’t have to be positive natural numbers, but you won’t lose much by pretending, in this post, that they are. There are innumerably many different brands of towers, but we’ll concentrate on one particular brand: the ‘Gowers Towers’. Let’s say the number written on each block represents how heavy the block is, and is inversely proportional to the length of the block. So we’d represent the sequence (0, 1, 5, 0, 3, 5, 5, 1, 10) with the Gowers Tower pictured.

It’s worth mentioning that the Gowers Towers include every individual tower of finite height that you can build with your unlimited set of Gowers branded building blocks (and lots of infinite height, but you don’t really need to worry about those here).

Let’s pretend we’ve got a measure of the instability of a tower (the norm of the sequence), and whenever we increase the instability beyond a certain threshold, K, the tower collapses.

Blocks with higher numbers are heavier, as well as narrower and perhaps inherently more unstable. How the blocks of different weights at different heights affect the stability of the Towers of Gowers is extremely complicated. However, the towers do have some nice, intuitive properties.

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Getting into Norms: Part II

In Part I of Getting into Norms, I talked about three different ways of measuring distance (I also considered  the accuracy of a series of guesses to be a ‘distance’). All three of these were norms, but there are many ways of measuring distances that aren’t norms.

So to study norms, mathematicians must define them really rigourously, using something known as axioms. These are the basic assumptions and definitions of mathematics. Once we’ve made these assumptions we can prove what has to follow from them.

We can think of norms as a measure of distance from the origin. If you think about it in this way, the following seem quite obvious, and appeal well to our instincts. A norm satisfies the following three axioms.

  1. Distances are always positive!
  2. If the distance from your location to the origin is zero, then you must be at the origin. Or alternatively, if two points are separate then the distance between them isn’t zero. Conversely, the distance from any point to itself is zero.
  3. Taking a detour is always longer than travelling in a straight line. This is the triangle inequality: the sum of the length of any two sides of a triangle is longer than the length of the third.
  4. Now we come to axiom four. This one is tough to describe in words. Here goes. If you walk a pace forwards and then take another in the same direction, then you will have walked twice the distance of the original pace. Also it doesn’t matter whether you take a pace forwards or backwards: they will give you the same distance.

When mathematicians want to be precise, we use symbols. The distance between points x and y is written as \|x-y\|. The distance from x to the origin is \|x-\underline{0}\|=\|x\|. We say that \| \cdot \| is a norm if whenever we pick vectors x and y, and a number \lambda, then the following axioms hold:

  1. \|x\| \geq 0.
  2. If \|x\| = 0 then x = \underline{0}. And visa-versa.
  3. \|x+y\| \leq \|x\|+\|y\|.
  4. \|\lambda \cdot x\| = |\lambda| \|x\|.

These four conditions should match with our verbal descriptions above. You may recognise them from this blog’s exquisitely hand-drawn logo.

They were pretty trivial intuitions, once we thought of \|x\| as being the distance of a point x from the origin (the origin above \underline{0} is underlined to distinguish it from the normal 0, though we don’t choose a different notation because the origin behaves a lot like the number zero). Continue reading

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Getting into Norms: a technical postscript

For mathematicians’ eyes only. The post doesn’t require much theoretical knowledge to understand, but I haven’t given many definitions.

In the last two posts I’ve been talking about an example I made with four-dimensional vectors a, b, c such that \|a\|_1 > \|b\|_1 > \|c\|_1, \|b\|_{2} > \|c\|_{2} > \|a\|_{2} and \|c\|_{\infty}>\|a\|_{\infty}>\|b\|_{\infty}. Finding it was more difficult than I at first expected, so I thought I would write the investigation up, which happily gives me an excuse to introduce a useful inequality.

My first thoughts were to choose something like c=(10,0,0,0) and a=(6,6,0,0) or a=(4,4,4,0), and then I’d got stuck choosing b. So I decided to try to prove the opposite.

First of all, it’s not possible to create such an example in two dimensions. Continue reading

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Intransitive votes

In the previous post I gave an example of students deciding who best guessed some lecturers’ ages.

I chose the numbers carefully so that under three reasonable methods of measuring:

Method  First place  Second place  Third place
Method a Adam Beth Charlie
Method b Beth Charlie Adam
Method c Charlie Adam Beth

This is actually almost identical to Condorcet’s voting paradox:

Voter  First preference  Second preference  Third preference
Voter 1  A  B  C
Voter 2  B  C  A
Voter 3  C  A  B

If three people in an election vote for candidates A, B, and C this way, then even using a method that takes account of all the preferences in one of the Condorcet voting system leads to a deadlock.  Continue reading

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Getting into norms

This blog is named “Out of the Norm”, and the logo is the list of axioms that a norm satisfies (with some extra graffiti), but there’s a fair chance that it’s just mathematical gobbledegook to you. So what is a norm? At its heart, a norm is essentially a measure of distance, though not necessarily a spatial distance. Here are a couple of examples to give you some informal sense of what a norm is, and how they might be used.

Distance in space

If I asked you to go find some buried treasure 3 units of distance East, and 4 units North, how far would you have to travel? That is, from the origin (0, 0) to the point (3, 4) on the graph or ‘map’ below?

It’s not a trick question: the distance as the crow flies, by Pythagoras’ theorem, is simply \sqrt{3^2 + 4^2}=5.

But what if it were a trick question? Let’s say you were actually a taxi driver in some Manhattan-like city with square blocks (each a single unit in size), and were taking a passenger three blocks East and four blocks North (perhaps to the buried treasure, if you insist). Then the shortest route would be seven units long (7=3+4; there are many different shortest routes).

Two of the shortest routes

So now you’ve seen two different notions of distance in 2-dimensional space: the first very familiar; the second probably less so. Just to reinforce the point, you can do the same thing in 3-dimensional space. The distance as the crow flies directly to a point 5 units above the unnecessary treasure, that is, the point (3, 4, 5), is approximately seven units: \sqrt{3^2+4^2+5^2}=\sqrt{50}. However, our taxi cab driver has to drive around the buildings and then take some sort of car-lift, and travels 3+4+5=12 units.

Diagram not at all to scale.

You can extend this to as many spatial dimensions as you want, though as far as I know it’s impossible for humans to imagine a diagram with the point (3, 4, 5, 2) hovering somewhere. But that doesn’t mean the notion is useless.

Guess the age of the lecturers

The next example has been used by one of my former colleagues to introduce norms to his students. Firstly, he gets the students to each write down a guess of the ages of all the lecturers in the maths department, and writes the guesses up on the board. Then he reveals the true ages of the lecturers, and asks the students how they should decide who has won. Here’s some completely made-up numbers:

Let’s say there are four lecturers: a PhD student, a lecturer, a senior lecturer, and a professor, who are 25, 35, 45 and 60 years old respectively. We’ll write this as the vector (25, 35, 45, 60). Now the three students make their guesses: Adam guesses in order 27, 36, 55 and 66: again, write this as a vector (27, 36, 55, 66). Beth guesses (26, 35, 53, 51), completely misjudging the age order of the older pair. And, because there’s always one in every class, Charlie guesses (27.2, 34, 56.4, 62.4), which reminds us that we needn’t work only with whole numbers.

The students decide to judge the results Continue reading


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