Moebius Transformations

Douglas N Arnold and Jonathan Rogness of the University of Minnesota have produced a short video entitled
“Moebius Transformations Revealed,” which has already been watched on YouTube by more than a million people.
It is a lovely way to present new concepts in mathematics.
[July 2012]
Posted in Geometry | 3 Comments

Cassini-Huygens landing on Titan January 2005

It was some time ago now, but the first landing of a space probe on the ONLY body in the solar system (besides of course earth) known to have liquids on the surface was a huge excitement. A friend of mine, Dr John Geake of UMIST, had devised a refractometer containing a prism made of sapphire. It was supplied to ESA by the Space Sciences Department of the University of Kent at Canterbury, because they had an interest in finding exactly what the liquid in the seas on Titan was made of. (We can be certain it isn’t water – far too cold!) The lander was released on Christmas Day 2004. If you want to know what happened next, watch this amazing NASA video (turn on your sound!):
You will see the misty brown atmosphere of Titan slowly resolve into the landscape and the point of impact will become larger and larger until … Anyway the rest of the mission was a total succcess … but we never found out what the refractive index was of the liquids in the seas!
[May 2009]
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Setting Up Your Own Business (in the UK)

Setting up your own business 
1. Decide what you want to do, and plan to do it well. If it is something that you love doing, then that’s perfect
2. Start small, start cheap. Do you need tools or equipment? Try to beg or borrow from relatives, or buy second-hand. Remember that in business, if you don’t owe anyone any money, you cannot become bankrupt! If you fail and just cease to trade without owing any money, you will be respected for it, and may have learnt valuable lessons.
3. Whatever it is, start doing it now! And do it for friends, for relatives, if necessary working for nothing. But don’t be embarrassed to ask for your expenses, they will expect at least that. If you do a good job, you’ll be asked to come back again, and the next time you might be able to ask for proper wages.
4. How much to charge? Think of the national minimum wage as a starting point – but many self-employed people get paid even less when they start out! Sometimes it is a good idea to ask the customer what they are willing (and able) to pay; the answer might be more than you expect.
5. Invoicing: Every time you are paid, write out an invoice which will contain a serial number, date, your name and address and telephone number, the customer’s name, the nature of the job, the amount paid, and “Received with Thanks”. Keep a copy for your records. 
6. Your Accounts: Start a sheet of squared paper or a computer spreadsheet with the heading BUSINESS INCOME. Head the columns: serial number; date; customer’s name and contact details; nature of job; amount paid. Also start a separate sheet or spreadsheet, headed BUSINESS EXPENSES with headings: serial number; date; supplier’s name; item; cost. Keep all receipts, and on each one write the serial number and the tax year (e.g. 09/10).
7. Your Profit: At the end of the week, subtract the total expenses from the total income. The result is the profit of the business, your wages for the work you have done. Have you made a profit? Great! Is the profit less than £15 (or whatever your benefits allow you to earn in a week)? No problem, but you are not in business yet. Keep going!
8. In Business! Now imagine that at last, you have made a good profit for the week. Great! You are now truly in business! To prove it, you must carry out the following steps:
(i) Contact the benefits office to stop your benefits. The reason is that you cannot be self-employed and unemployed at the same time.
(ii) Choose a unique business name, preferably one that includes your own name and describes the business. e.g. “Jill Smith’s Dog Walking Service.” Don’t include “Co” or “Ltd” – that comes much later. If you want to change the business name later, it is much easier if you haven’t spent money on having cards printed specially. 
(iii) Then look up the address of your local tax office, and write them a letter which simply says “With effect from [date] I am self-employed and trading as [name] as a [type of business].” Include your full name, national insurance number, and date of birth. They will send you a tax reference number. Keep it safe.
(iv) If you have another job, remember that you can quite legally be self-employed in one or more businesses and employed at the same time. All the taxman needs to know is your total earnings. Your employer may not be so happy though, especially if they think you are trying to compete with them!
(v) The first time 5th April comes round, that is the end of your first trading year. In the accounts sheets add up the income and the expenses, and calculate the profit or loss for the tax year. Made a profit? Fantastic!
(vi) Fill in the tax returns, either on paper or on, before 25th September. Make sure you have put aside enough money to pay the tax right away, otherwise you will be hit with a £100 fine and charged interest. But unless your earnings are over about £6000, you will not be due to pay any tax at all!
(vii) On 6th April, start new accounting sheets for the new tax year, and off you go again. Good luck!  
Some Basic Rules of Business
(1) Remember the saying “You are not in business unless you know what business you are in.” Stick to the business you know, and don’t be tempted to take on work you know to be beyond you. Remember no-one minds if you refuse a job. But if work is done badly your reputation will suffer.
(2) Don’t be tempted to take on too much – “I’m too busy” is something everyone understands – and respects.
(3) Treat your customers as you would wish them to treat you.
(4) The same goes for your suppliers. Never try to get cheap credit by paying suppliers late – only big companies can get away with that!
(5) Treat all other small businesses as allies, not as competitors. If you can’t do a job, recommend someone else, and tell them what you’ve done – they’ll remember you, and might return the favour one day.
(6) Never agree to a deal where someone else will be hurt. Trust is an element of every business; make sure you are worthy of that trust. If you are straight with people, people will be straight with you.
[March 2009]
Posted in Organizations | Leave a comment

How many colours are there in a rainbow?

In his book "Opticks" Sir Isaac Newton states that there are seven colours in the rainbow, and (Bk 1, Prop VI Prob II) suggests these might be related to the seven notes in an octave of music. Uncannily, even though he did not believe that the light was a wave motion, we now know that the wavelength of light doubles between the violet end (0.4 micrometres) and the red end (0.8 micrometres) of the visible spectrum, just as an musical frequency doubles from one octave to the next. The colours Newton lists are red, orange, yellow, green, blue, indigo and violet. However, the history books say that in fact Newton was partially colour blind, and had to get his servant to count the colours. He was particularly keen that there should be seven, because seven is a significant number in alchemy which Newton also studied (all his  papers on the occult were burnt by the Cambridge University authorities immediately after his death). In fact, it is practically impossible to distinguish indigo in practice, but Newton’s servant was unusually obedient! So seven there are, and seven there will always be.
[December 2008]
Posted in Optics | Leave a comment

Is cheese the basic particle of the universe?

"Three quarks for Muster Mark!
Sure he hasn’t got much of a bark
And sure any he has it’s all beside the mark. "
These immortal lines written by James Joyce in Finnegan’s Wake initiated the transatlantic dispute in the pronunciation of one of the fundamental particles of the universe.
Finnegan’s Wake was written in a cheerful Irish variation of English, the deciphering of which has kept the academics busy for decades. Now anyone who knows about life in Dublin about 1939 would know that it was common for beer to be taken away from a pub in quantities of quarts (pronounced KWORTS: 1 quart = 2 pints). So "Three quarts for Mr Mark" would have been a cry that James Joyce may have heard regularly. The logic of this is the reason why most Americans pronounce quarks KWORKS to rhyme with forks.
But physicists in Europe would also have been familiar with German soft cheese called Quark (pronounced in Germany KVAHK). More particularly, physicists in England would have known "The Hunting of the Snark" published in 1876 by by Lewis Carroll. They would also be very aware of the terrible and fatal consequence that the Snark may turn out to be a Boo … (look it up for yourself!). Anyway, James Joyce was clearly mixing the word Snark with the word "quart" to get the invented word "quark." And as the other lines in his poem end with a word rhyming with Mark, so does the European pronunciation of quark as KWAHK to rhyme with park.
[November 2007]
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“Property for sale, unusual prospects, suit small quark”

There are more than four dimensions (X, Y, Z and time) according to string theory. In fact there are seven further dimensions which are needed to make sense of the universe, according to the latest theories. And guess what? Someone is already buying up property rights in the new dimensions. Jonathon Keats, a conceptual artist exhibiting in the Modernism Gallery, San Francisco, says, "The important thing is that real estate in cities from San Francisco to New York is selling at a premium, unaffordable to many, and here are half a dozen or more extra dimensions of space, just going to waste."
Mr. Keats, working in consultation with leading researchers including Shaw Prize-winning cosmologist Saul Perlmutter, realized that rights to develop in these extra dimensions could be bought very inexpensively. "The legal framework was already in place," he says. ""People like Donald Trump buy and sell air rights over city buildings all the time. If the third dimension is negotiable, the higher dimensions must be as well." Accordingly, the artist/developer bought extra-dimensional rights to his first property, a flat in San Francisco’s exclusive North Beach district, in August 2006. While the lower-dimensional space is valued at approximately $1,027,000, Mr. Keats purchased rights to the extra dimensions, with a legally-binding contract, for a mere $5.
He has since bought higher-dimensional rights to five other properties for between $1.80 and $15. "Nobody really wanted the rights," Mr. Keats recalls, "and I guess that I can understand why." The extra dimensions, like the strings vibrating in them, are very small, many orders of magnitude smaller than an atom. "They’re a bit inaccessible by conventional means, but they’re everywhere."
For the record the maximum length of a "string" is 1.6E-33 cm. The diameter of a proton is about 1E-13 cm.
[December 2006]
Posted in Art | Leave a comment

How are lenses made?

Updated Feb 2008: There is a new pictorial presentation on the web, showing how a lens (such as a TV camera lens) is made. It shows all the processes used to make a single glass element, and to cement it to another one. Then it goes on to show how the metalwork is built up around it to form the complete camera lens. Finally there are some ideas for alternative methods by which lenses can be mounted. This presentation is on the website of Ancient and Modern Optics at: Please post any comments below.
Posted in Optics | Leave a comment