In reply to Robin Downie (
25 January 2023), space is three-dimensional because we need three numbers to say where something is. For example, a pilot might quote latitude, longitude and altitude to say where the plane is. Miss one out and the position isn't fully defined. Any four-number set-up will contain redundancy, in that one can be deduced from the other three.
Space-time is four-dimensional because defining an event, as opposed to a position, requires a fourth number, namely time. In the plane example, it's where it was and when it was there.
Abstract maths allows for structures with any number of dimensions, finite and infinite. Observation says that three-dimensional space, within four-dimensional space-time, is where we live.
Dave Dempsey
I have been enjoying Manfredi La Manna's series of articles but I fear he may have stumbled in his description of the billiard table to illustrate Bayes' Theorem (
31 January 2023). The probability of a randomly thrown ball coming to rest x feet from the left-hand side of the table is given as x/L, where L is the length of the table. If so, then the probability of the ball resting on the right-hand edge is 1 (L/L) and therefore the probability of it being anywhere else is 0. This is obviously wrong.
I think that the probability is actually 1/L (assuming x is an integer), so that the probability of the ball being two feet from the left-hand side of a five-foot table is 1/5 = 0.2 and the probability of the same event on a 10-foot table is 1/10 = 0.1. (More precisely, the probability is 1/(L+1) to include the possibility of the ball being 0 feet from the left hand edge.)
The important conclusion, that the probability of the ball being two feet from the left-hand side is larger if the table is five-feet long than if the table is 10-feet long, still holds.
Douglas Carson
One of the delights of the internet is surprises. This may have something to do with my personal lack of keyboard skills and general ignorance of where my stuttering fingers are taking me. Nevertheless, most unexpected developments have been fun. Last week, for example, I stumbled across a friend who had been lost for over 30 years.
I think we found each other on LinkedIn, a career building app I inadvertently downloaded several moons ago. Although my personal condition is firmly retired, it still works away with encouraging messages about how my career is being 'noticed' and 'regularly checked'. Last week, it mentioned that so-and-so had been searching for me. I clicked eagerly and jumped back 30 years. A few more clicks and a telephone reunion was set for later in the day.
When the phone rang, we were in the middle of a discussion about the appalling potholes gracing our local roads and how they were threatening to damage our precious car. The deep warm voice of the old friend was a welcome relief. He was calling from Tel Aviv where he and his wife have a house. Asked whether that was the safest place to stay, he replied it was, 'but we don't live here all the year. Most of the time, we live on a boat'.
Living on the Ayrshire coast, our minds immediately imaged cabin cruisers and yachts. 'No. A big boat called The World,' said our friend. It was at this stage that 30 years became an aeon. We had our potholes. He had become seriously rich. Not a millionaire but a multi-millionaire. To live on MV The World, you must have a net worth of $10m or an after tax annual income of $1m.
His 'boat' cruises the world continually, transporting around 150 passengers from exotic location to location. On 12 decks, it offers 165 homes – from studios at £3m to £20m penthouses. It has the only full-sized at-sea tennis court, a 7,000 sq. ft. spa and fitness centre, and a 12,000 bottle wine cellar. Other facilities include a swimming pool, billiard room, golf simulator, jogging track, six restaurants, cocktail lounges, movie theatre, library and space for musical performances. Since each residence has a kitchen, there is a grocery store and a delicatessen.
The World is now owned by its residents who plan the itinerary two years ahead. They also vet carefully who comes aboard – as a visitor or as a potential buyer. Paparazzi are definitely not welcome nor are celebrities. As their online brochure explains, there is a code of confidentiality and privacy. There is even a committee to vet and approve any interior design you may wish to execute on your multi-million pound pad.
We did not quiz our old friend on his interior design afloat. We remembered days together in Amsterdam, New York, New Orleans and Glasgow, where he began as an important client and later became a pal. We ended the call with vague suggestions of meeting. He disappeared into Tel Aviv. We came back down to our potholes and energy bills. Envy has never been one my follies but it would be nice if our next internet surprise could be from the National Lottery. We have a shrewd idea of where that might lead.
David Donald
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