The 2×1 rectangle and Domes

March 26, 2012

Next week I am going to be at the Gathering for Gardner, an exciting meeting of mathematicians, magicians, puzzlers and others inspired by the life and work of Martin Gardner. This post is a version of the talk I will be giving.

The 2×1 rectangle is not one of mathematics most celebrated shapes.

Yet it is so much more flexible than the more common square.

Even better you can cut it in half on the diagonal to make a 2×1 right triangle,

which has the beautiful property that it is a 5-reptile. Five copies of it come together to make a larger version. Repeating this gives the Conway Pinwheel tiling, which has triangles occurring in an infinite number of directions.

Yet the 2×1 rectangle is a lot more common in life, just go into your local hardware store:

Using the diagonal cut triangle and uncut rectangles, Vinay Gupta designed the hexayurt,

a small house that can be built from 12 sheets, without waste. In contrast to geodesic domes, that cannot be made from sheet materials without making many cuts or wasting material. Here is one:

and a plywood one:

Hexayurts have become one of the standard accommodations at Burning Man:

or look at this map, the red dots show the location of the hundreds of hexayurts at last years event.

Vinay set me the challenge of making larger domes using these shapes. The hexayurt itself suggests that hexagons will be important, and we can put two 2×1 rectangles together to make a square. Squares and hexagons come together to form the truncated octahedron.

This obviously would not work as a dome, so we must cut it. There are two natural cuts that can be made. One perpendicular to the 4-fold axis, and one perpendicular to the 3-fold:

So we have two new larger domes, the tri-dome and the quad-dome:

What is really cool is that both of these domes were made for Burning Man last year:

Tri-dome:

Quad-dome:

One neat thing about the truncated Octahedron is that it is a space-filler. You can use them to tile 3d space. We can therefore bring quad-domes together to make even larger structures, like this one:

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Posted by gelada

The future we deserve

February 29, 2012

Telling the future is impossible, yet we have to do it every day. We need to be realistic at the same time as allowing some hope, as not trying for better things is one of the best ways to ensure that they will never come. I was fortunate enough to be part of a recent project to try to imagine what is possible and what is real in our coming future. The future we deserve:

The book grew out of a single tweet from Vinay Gupta, it is a mix of dreams, plans, fears and wild hopes, yet all carrying a sense of reality. Although it much be said that such is the pace in which predictions stale that the future today already looks different in many ways to the futures painted in this book. I am proud to have two essays included in this book. You can buy the book, take a look at the essays online, see further developments of the project, or discuss on twitter, to help us all build a future we deserve.

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Posted by gelada

Being wrong

February 24, 2012

I hate being wrong, ask anyone in my family, they will get that slightly weary look and agree (and I am not the only one). I have tried to counter this by learning and improving my knowledge, which helps me, but if I am honest doesn’t help my family. In addition I am a teacher and so, in many situations, could just fall back on authority. Yet in teaching I have realised something important, I actually like it when my students are wrong. I would not say it is the best situation, perhaps, but it is positive. The reason is simple: to be wrong you have to be engaged.

I was thinking about this as I read the post on The Renaissance Mathematicus  talking about the birth of HistSci Hulk, sworn enemy of anyone who is wrong about the history of science (a noble and dangerous quest). This might seem to be the opposite position to the one that I gave above. I have felt the sting of his corrections myself, luckily in private not public! It is not opposite, in fact it is the essential counterpart. Being wrong is positive, but only as it helps on the way to better understanding. Reading about how the concepts of gravity were starting to come together before Galileo, and that he did not experiment by dropping things from the tower of Pisa, does take one further. Yet this does not make the original story worthless. It introduces the idea of gravity, the sense there was a change in understanding and  Galileo, himself.  The correction builds far more happily on this knowledge than it would standing on its own. For this to be effective, of course, we have to accept that stories (especially much loved ones) can be wrong, and more to the point we ourselves might be wrong.

I believe this is actually the great strength of the scientific method, and mathematical proof. Not that they can be used to show things are right, not even that they can show things to be wrong, but that they give a framework to persuade someone they are wrong. They help to develop understanding faster and further.

So do not get embarrassed when you are wrong. Do not get defensive. Learn to embrace it, be grateful, admit it. Then you are learning.

“It is better to open your mouth and learn that you were a fool, than to remain silent and never know.”

Some other takes on the same idea come from the inventor James Dyson and the author Kathryn Schulz.

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Posted by gelada

Word powers of ten

February 2, 2012

How do we understand the number of words on the internet? Its hard to even grasp how many there are, and the number is growing so rapidly. Trying to understand a similar problem, the size of the universe (or just the observable universe) Charles and Ray Eames came up with the classic Powers of Ten video. Lets try the same for words:

1 (one) word
10 (ten) words a haiku, a sentence or a tweet

100 (hundred) words a paragraph, an abstract, a newsitem

1000 (thousand) words an article or blogpost

10,000 (ten thousand) words an essay or short story

100,000 (hundred thousand) words a book

1,000,000 (million) words an epic, Proust’s “A la recherche de temps perdu” is 1.5 million, the complete Harry Potter Saga is just over 1 million.

10,000,000 (ten million) words  an Encyclopedia, the 2002 Britannica is 44 million

100,000,000 (hundred million) words  a large Encyclopedia, like the Yongle Encyclopedia from fifteenth century China

1,000,000,000 (billion) words  Wikipedia (actually over twice that)

Then there is a gap…

10,000,000,000 (ten billion) words

100,000,000,000 (hundred billion) words

1,000,000,000,000 (trillion) words

10,000,000,000,000 (ten trillion) words

100,000,000,000,000 (hundred trillion) words gives you the internet in 2008

So perhaps soon the internet will surpass the work of a single man. The great french author Raymond Queneaux:

10,000,000,000,000,000 (ten thousand trillion, ten thousand million million, ten million billion) words  the word count (assuming 10 words per line) of the complete text of “Cent mille milliards de poèmes“

Having exploded outwards, it is not time to come back down, through encyclopedias, books and stories, back to tweets and the word:

1/10 (tenth) of a word a letter

1/100 (hundredth) of a word gives you a line segment which has an interesting property, it can itself be divided.

1/1000 (thousandth) of a word gives you a shorter line segment, allowing you to dive as deeply as you wish theoretically, in practice you will dive surprisingly quickly through atoms, protons, neutrons and quarks to the lower limits of our understanding.

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Posted by gelada

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