The Academy: Axiom 1

September 3, 2011

The rule

This post is not trying to do anything clever. It is making a statement that seems self-evident:

There are three ways to gain understanding of the world:

  • Personal experience
  • Systems of rules
  • Stories

All are equally important, and each has its strengths and weaknesses.

The important point is not the content of the statement but the stating of it. This is not just something that feels correct (to me) but something that feels fundamental. This mirrors one of the quests of mathematics to find the simplest statements on which to build the whole subject. I have my suspicions that the same thing would not work completely here, though writing the “Elements of the Academy” with this as one of the axioms might make a curious exercise!

This axiom maps onto the world of academia. The Sciences are primarily concerned with the use of rules to understand the world; the Arts centred on the creation of objects that attempt to transfer personal experience; and the Humanities write, dissect and try to understand the stories of the world.

All three areas, of course, do and should take advantage of the strengths of the other two methods as well as their primary concern.

The story

As a mathematician I obviously come from the grand tradition of finding rules to understand the world. For much of human history this was known to be rather limited in its scope. It was applicable to commerce, certainly; but also to questions of measurement, and to the study of the stars and music. Then, with the acceptance of arguments based on infinitesimals and the genius of Newton and Liebniz, the models of calculus opened up a vast array of phenomena to understanding through rules. It was so successful that many started to believe that it would eventually explain everything.

I do not believe this to be the case. Chaos theory shows that even perfect models can be severely limited by small, unavoidable, measurement errors. The work of Gödel and Turing shows that even in the purely theoretical world, there are unanswerable questions. Some even believe that as fundamental a system as arithmetic might contain contradictions. Before we even get to these hard limits we must deal with the soft limits imposed by the great ideas that we have yet to have.

Unfortunately, or fortunately depending on situation and personal preference,  the world offers many questions that we cannot answer with a systematic, rules based approach. Questions we cannot ignore. I wanted to define for myself the other options, and place them in some imagined framework.

The personal experience

I don’t believe I have said much here. It is, as I stated, self-evident. I also think it is important. It has been useful and practical to me. So, if you have managed to read this far, I thank you, but ask one further thing. Think about it yourself and see if it is a useful for you too.

Acknowledgements

This post grew out of a string of tweets, out of which grew very valuable discussion with  Colin Wright (@ColinTheMathmo) and Daniel Colquitt (@danielcolquitt), on twitter and elsewhere.

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Posted by gelada

Hexayurt dome details and models

August 7, 2011

People are now starting to build my tri-dome and quad-dome versions of the hexayurt, so it is time to give some of the technical details. To start, however, here is an application of the intermediate value theorem!

When I started working on the details for the tri-dome I realised I had made a bad assumption (thinking that the form was geometrically pure). This means that some of the details in my original write up were wrong. All a little embarrassing. Ironically, I might have missed a form that does actually work, had I not made the bad assumption. The shape, like the hexayurt, starts with a hexagonal based pyramid. In a traditional hexayurt this lies on top of a hexagon of vertical walls. Instead of this we attach a square to three of the edges and the classic hexayurt triangle (isocoles triangle with base and height the same length) to the other three. We can look at what happens as the pyramid is moved away from the ground, while the edges of the shapes rest on it:

This does not give a great building; there are holes. The holes are triangles and two of the sides have a fixed length. The final edge changes length, starting long, and ending short. We know we can fill the holes with classic hexayurt triangles. Two of the edges are the right length we just need the third. The length changes smoothly as we raise the roof, and starts shorter and ends longer than we want. Here we can apply the intermediate value theorem, so the correct length must be passed. As a mathematician I would stop there, the system works; however people are building the things…

So to calculate the correct angle for the square sides of the model we can look vertically down, calling the angle of the square face θ, (and assuming that the boards we are using are 8′ by 4′) needing as the classic maths problem asks to “find x”.In this case

x = 4 \sqrt{4 \cos(\theta)^2+1+2\sqrt{3}\cos(\theta)},

we want x = 4\sqrt{5} so:

4\sqrt{5} = 4 \sqrt{4 \cos(\theta)^2+1+2\sqrt{3}\cos(\theta)}

5 = 4 \cos(\theta)^2+1+2\sqrt{3}\cos(\theta)

0 = 2 \cos(\theta)^2 + \sqrt{3}\cos(\theta) - 2

Solving the quadratic:

\cos(\theta) = \frac{-\sqrt{3} \pm \sqrt{19}}{4}

Which gives an angle of about 49°, and the height of the roof (assuming 4′x8′ panels) is 8 \sin(\theta), just over 6′ at the edge and 10′ in the centre. We can use these, and useful facts about general tetrahedra to calculate all the angles between faces by using the lengths of their edges. If you want to follow the details yourself, you need to add vectors to get some of the edge lengths, then use the Cayley-Menger determinant to find the volume of the tetrahedron, and then the generalised Sine rule to (halfway down this page) to give the angle.

Technical details for TriDome: angles to nearest half degree, lengths to nearest inch (assuming 4'x8' panels). On the left the angles between faces and point heights, on the right lengths and angles of the base.

Technical details for QuadDome: angles to nearest half degree, lengths to nearest inch (assuming 4'x8' panels). On the left the angles between faces and point heights, on the right lengths and angles of the base.

Finally here are the hexayurt models (rhino 3dm and vrml formats) of the hexayurt, H13, TriDome, QuadDome, plus a couple of others, including a very large one.

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Posted by gelada

Will the next generation act?

July 21, 2011

Mathematics and policy need to meet in preschool

[A recent collaboration with Vinay Gupta, available as a pdf]

We are all products of our environment, so education is one of our best chances of producing a better human race in time to do something about our world’s plight. Our instinctive approaches to educating our children are rooted in our deep ancestry and our more recent cultural accumulations. As we see all around us, instinct and culture are failing us. Our inability to correctly model our world and act on our conclusions endangers us all.

Our ability to believe in our models rests firmly on our affinity for mathematics, yet centuries of breakthroughs in mathematical thought have not been broadly integrated into our culture. Although the fruits of pure mathematics – nuclear physics and digital computers and networking – more or less define the modern age our basic regard for the practice of mathematics has not increased in keeping with its importance, nor have our educational practices reflected the changing role of mathematics in the world. Cryptography is the backbone of all commercial use of the internet, and while hackers draw endless media attention, do you know the names Rivest, Shamir or Adleman?

Although mathematics is at least as old as agriculture our mathematical heritage is not as treasured as other cultural links with the distant past. Correcting our cultural bias against mathematics is an intergenerational struggle. In sport, art and music we encourage appreciation by non-practitioners, but interest in mathematics is expected to be confined to experts. Prejudices like if it’s not hard it’s not mathematics have interfered with our ability to appreciate or even identify mathematics.

Quilting and other forms of textile design, have some overt mathematics, counting and measuring, but making satisfying repetitive patterns uses the mathematics of symmetry. Tetris uses the tetrominos for pieces. Part of the satisfying regularity of the game is that the pieces aren’t arbitrary – all the possible shapes are there. Traditional card games lead to many areas of mathematics, but the deck itself is rather arbitrary – why four suits, rather than five? We need better artifacts to train thinking.

Games
Set In comparison to a standard deck, the Set card game is very ordered, having 81 cards (3x3x3x3). This forms a regular-yet-surprising deck, including every possible card for four choices of three options, and thus has the same sense of completeness as the Tetris blocks. Hands are matched all-same or all-different, and even very young children catch on quickly and can compete against adults!

Doodling You can make your own mathematical games on squared paper, or just play around with ideas. For inspiration you need look no further than Vi Hart’s videos.

Puzzles
Rubik’s Cube The ubiquitous Cube was the definitive puzzle of the 1980s. The 3x3x3 plastic puzzle encapsulates substantial group theory, and is solved by discovering or learning algorithms. Guides for learning how to solve the Cube have improved a lot over the years, it’s easier than ever to solve.

Penrose Tiles These two simple shapes fit together to produce an endless array of different patterns which never repeat and never run out. The puzzle pleases when decisions made earlier come back as you find you have to retrace your steps to continue laying the tiles. Beautiful patterns and shapes result.

Toys
Lego is the universal solvent for technical professionals. Everybody played with lego, and everybody describes how formative lego was in shaping their capacity to plan, execute and make. Modern lego has tended towards branding itself as a toy rather than a building system, but large boxes of basic bricks are still available. You can even bend it!

Zometool Want to see four dimensional space? This toy gets you about as close as is humanly possible, and you just have to build it. It is also brilliant for exploring three dimensions beyond the right angled system of Lego.

Polydron A simple idea, shapes that clip together at their edges forming a hinge. Mathematically they can look at how geometry jumps from two dimensions to three, what will you make out of them?

Meccano Another classic old toy that should not be underestimated. Metal and bolts vs. machined plastic. The long standing “Meccano people vs. Lego people” controversy can easily settled by buying both.

Scratch The easiest way for children to make software, taking their first steps into the source code that will run our lives. Scratch has excellent support for sound, graphics and even video, and is free.

Further Resources
Martin Gardner Ask mathematicians what got them into the subject as there is a very high chance that Martin Gardner will be mentioned. For years he talked puzzles, games and even broke new mathematical results in his Scientific American column. He left us with books stuffed full of curious intriguing and meaningful mathematics.

The Museum of Mathematics opens in 2012 in New York, this will be a mathematical wonderland, giving an intuitive glimpse even into many corners of mathematics. The website is packed with videos and resources.

Edmund Harriss & Vinay Gupta, Cloughjordan, 2011
with the kind support of Django’s Hostel

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Posted by gelada

Numbers are meaningless

June 28, 2011

Although not the nicest of men Francis Galton was also a bit of a hero of numbers, drawing them into the human domain, and developing ideas such as correlation. Unfortunately, an heir to Galton, Steve Jones did not employ the same subtlty in a recent article:

Type the phrase “scientists find the gene for” into Google and 68,000 results appear. Most of the hits are about human beings – which is a pretty impressive number, given that we have only 20,000 genes altogether.

Francis Galton: The man who drew up the ‘ugly map’ of Britain   Steve Jones

We have become used to numbers swirling about us, we talk so often about their power, that we forget that on their own they are meaningless. Meaning must be added, and we need to be careful when comparing, as he did, two numbers that come from different contexts.

We give meanings to numbers in many different ways, sometimes only using some of the abstract properties. House numbers make full use of the ordering on numbers, but No. 23 does not combine with No. 41 to make No. 64.  Yet think about how we teach number. Nearly every primary school has a number line, it might start with one apple, two bananas, three oranges. Yet, while one apple plus two bananas might be one smoothie, it is certainly not three oranges. So remember the old saying!

The ultimate form of abusing numbers is the bogeyman of numerology, diving into the abstract world of mathematics and jumping out again in different contexts to pull some conclusion out of thin air. So I might surprise you by coming to its defence.

A classic method is to turn a written idea or just a single word into a number. Then look up that number to see what other words come to the same value. Perhaps, for some, the meaningless of the number stage is preceisely the purpose. The ideas connected in this way will have no obvious connection. The game then becomes finding something that draws them together. Not to see something of cosmic significance, but to stretch the imagination and get creativity and thoughts flowing. Creating a space for creative randomness.

So to bring this to some form of a conclusion, remember to be careful with numbers; but do not be afraid of them. Think about, play with and subvert the meaning that they are given. Perhaps you might even get lucky in your random connections and realise something about yourself or the world.

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Posted by gelada

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