Archive for June, 2010

Now imagine the whole diagram being rotated out…

June 24, 2010

Now imagine the whole diagram being rotated out of the page about an axis that connects the centres of the two circles. The line will sweep through a plane, and the circles will sweep through spheres.
So in three dimensions inversion turns an infinite plane into a finite sphere in a one-to-one mapping of every point. This is where computer graphics comes to the geometer’s aid.
Drawing three-dimensional objects is quite difficult, but for computers the task is almost as easy as drawing a two dimensional object.
The computer simply projects the object onto a plane, in an act of transformational geometry called perspective.
Figure 8 shows a computer’s rendering of a plane and its image (a sphere) after inversion.
Each star on the plane corresponds to its inverted image on the sphere; you can see how the angles are preserved under the transformation.
There are various ways of mapping a plane on to a sphere (or vice versa) but there is no way of doing this without producing some kind of distortion or another; Mercator’s projection of the globe on to a flat map, leads to a diminutive Africa and an exaggerated Greenland.
Inversion has the advantage of preserving circles and angles, and hence proportion locally if not globally.
The upper pole of the sphere is the centre of the inversion and the increasingly small stars that approach it correspond to the increasingly large stars lying at greater and greater distances on the plane.
To produce this picture a computer program included a specification for both the position of the eye (the view-point) and the plane on to which the three-dimensional configuration is projected.
It also had to solve the problem of “hidden lines”: how do you tell the computer not to draw those lines that the eye would not be able to see?
Computer graphics can produce an image in a variety of ways, on a video screen, for example, or directly on to film.
For Figure 8, I used a plotter, a mechanical drawing board whose moving pen is controlled by the electronic signals coming from the computer.
Watching it in action is like seeing some fantastic automatic sewing machine, dropping and lifting its pen to draw and terminate the thousands of lines with a speed that belies its great accuracy.
Having created a plane pattern of tessellating hexagons and hexagrams (stars) and having programmed the computer to invert it, I watched the pen race round the board and wondered, would it draw anything sensible? Would it draw a sphere?
I knew it must in theory, but to see it happen was still a lovely surprise.
From its beginnings more than a century ago, modern mathematics has aimed at a total abstraction in the interests of rigour and distillation of its logic.
One consequence of this has been a distrust of geometric intuition and the downgrading of geometry by educationalists over the past 50 years or so.
This demise of the oldest discipline in the world is clearly absurd and ripe for serious reconsideration.
To begin with, geometry has more value than simply as a means of acquiring a grasp of mathematical concepts.
The applications of a disciplined spatial intuition to art and design, and to the study of natural morphologies in every conceivable science is so great that perhaps we might think of geometry as a semi autonomous department of mathematics with different as well as overlapping purposes to abstract mathematics.

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Inter-City gets the six-degree bends…

June 10, 2010

Inter-City gets the six-degree bends
BRITISH Rail are to develop an electric version of the 200 km/h high-speed train (HST), which runs its Inter-City 125 services.
Officially the electric HST is still intended as a stopgap for the much-delayed advanced passenger train.
But in practice this is not so much another nail in the coffin of the advanced passenger train, more shovelling the earth on top.
In a reversal of policy, British Rail has also accepted that conventional trains can take bends faster.
British Rail’s standards for the maximum speed that trains could round bends were based on tests done on a branch line in North Wales with a tank engine in 1949.
After these tests the railway engineers laid down a maximum sideways acceleration equivalent to tilting the track by 4 degrees.
Following recent tests with a 176 km/h train, however and the engineers have agree that the speed through bends can be increased to 6 degrees of track tilt ” or “cant deficiency”, in the railways’ jargon.
Trains with a maximum speed of 176 km/h (compared with the present maximum of 160 km/h will take bends faster and will run from London to Liverpool and Manchester next year.
The following year British Rail’s London Midland region hopes to run electric HSTs, but these are now the subject of a row between the region and engineers at BR’s headquarters.
The headquarters’ engineers want to develop an electric version of the HST, which will involve considerable redesign and could take several years to complete.
The London Midland region, however which has to run the trains, wants them in service as quickly as possible.
It would dearly like (as one railway manufacturer has proposed) to rip the diesel engines out of an existing HST and replace them with electric engines, and then use this train for tests.
Meanwhile, back in the sidings at Derby the APT is still being modified, after its last run before Christmas when some bolts worked loose on its bogies. BR says it hopes to start test runs from London to Glasgow again “soon”. Mick Hamer
Tell-tale antibody that marks diabetics
A BREAKTHROUGH may be at hand in combating one of the most perplexing diseases of adolescence, juvenile diabetes.
American doctors may soon be able to spot potential victims before the symptoms develop.
Juvenile diabetes is an ailment that strikes its victims before or during adolescence, destroying their bodies’ abilities to produce insulin and the hormone that enables the body to use and store sugar.
About one in every 350 American children suffer the disease, which appears to run in families.
Doctors put the life expectancy of sufferers at about 40 years ” even if they have daily injections of insulin.
Until recently and researchers have thought that juvenile diabetes developed suddenly. Now it seems that the disease builds over a period of years.
Last week, a group from Boston’s Joslin diabetes centre announced that it had devised a blood test that can identify children most at risk.
If the test proves successful, it may enable physicians to start therapy for the condition before the diabetes causes irreparable damage.
The test, which at present can only be carried out in half a dozen laboratories around the world and seeks to identify an abnormal antibody in the blood of likely sufferers.
The antibody was first identified through studies of one pair of identical twins and another of identical triplets.
One of the triplets developed diabetes at the age of 13 while another was diagnosed as a juvenile diabetic at 21.

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But if it has that civil capability it…

June 7, 2010

But if it has that civil capability it is all too easy to turn it to less than peaceful purposes.
And a country with a nuclear power programme can have nuclear weapons for a modest extra cost.
Such arguments may seem trivial when bodies as thoughtful as the Church of England have taken it upon themselves to ponder the issue of the use of nuclear weapons; but the bomb has not gone off and when it has receded into the background we will be left with the demand for electricity. One day that demand will begin to grow.
Even if it does not we will need to replace ageing power stations.
The atom is there and when short-term oil surpluses vanish there will be pressure to build more nuclear power stations. How do we do that without spreading too widely the ability to make bombs?
Probably by encouraging the world’s wealthy nations to build nuclear power stations so that other countries need not follow suit.
Then the others will be free to burn the oil and coal that might otherwise be consumed in the nuclear nations.
There is a pressing case for the nuclear nations to deny fissile materials to would be customers.
It matters not that those customers are today seen as perfectly reasonable countries without weapons intentions who should not be denied the benefits of modern technology.
Unfortunately and today’s reasonable government may well be ousted by crazed fanatics ” 50 years ago in Germany, for example.
This area of business is so important that the nuclear nations should throw overboard all thought of evenhandedness. They should not try to be fair to other countries.
It isn’t fairness that the world needs.
While the US and USSR continue to play their game of trying to reduce the numbers of nuclear weapons and the battle must continue to prevent the spread of nuclear weapons to seemingly peaceful nations.
That means forgetting about the quick bucks to be made from selling nuclear technology.
With so much talk of reducing the number of bombs and the weapons implications of nuclear power seem to have been forgotten.
The trade in plutonium between Britain and the US has shown that two countries that proclaim their peaceful intentions as loudly as any are not beyond suspicion and that it is difficult to believe assurances that nuclear power stations have no weapons connection.

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