Mathematics and Number Wisdom

I've at last got round to updating the mathematics section of my website. The "Rational Mathematics", "Alternative Mathematics" and "Geometry" sections are now all under one index page. I've also transferred the pages on "Altairian Arithmetic", "Numerology" and "Numeromancy" from Esoterica to the Alternative Mathematics section, together with a new page on "Arithmosophy".

I'm not sure that Numeromancy and Arithmosophy can really be called Mathematics but they use numbers and require a knowledge of some arithmetic. I base my interpretation of "Wisdom of Numbers" on multiplicative relationships rather than additive properties. This makes it a more disciplined realm.

I have masses of notes on Geometry, but lack of satisfactory software for producing diagrams has held it back. There is nothing particularly new, just improved arrangement and presentation, making use of CSS.

A Bright Morning's Walk

I seem to have been neglecting this Diary. The last message I intended to post here, about seeing a rainbow at London Bridge, went to the Hastings Humanists blog by mistake.

It was a beautiful sunny day this morning, so I took a walk along the front as far as Glyne Gap to see how the new cycle path is progressing, and there is stil quite a lot of work to be done, but when finished it will be possible to cycle to Bexhill without going on a main road at all.

I went to the Ravenside retail park to have a look at the computers on display in PCWorld and ended up buying one. It's a large Samsung laptop, which I'm hoping will be able to replace the old desk-top machine on which I'm typing this message.

I left about 10 am and got back about 2 pm. There was quite a cold wind, whipping up surf on the sea, which was near to high tide, but the sky was blue like a spring day, and the birds were singing. But soon after I got back the sky darkened and the rain came down. It's just started again now as I write.

Last month, after my visits to Leicester and Bournemouth for the chess, I had some trouble with my right leg and had to rest it, but it seems to have held up today, but maybe a bit stiff in the morning, well see. I needed the exercise anyway.

Words

A day or two ago I finally got round to updating the Words section of my website. There is not much difference really except in the presentation, which now makes use of CSS (Cascading Style Sheets) to ensure the pages are in a consistent style throughout. There is a lot more to CSS that I need to learn, I've only used the basics to determine the background colours, typefaces, margin settings and suchlike.

In the simplified spelling I've gone back closer to the orignal 1997 article, though with j used for the th sound in 'then' and q for the indefinite vowel. The only new feature is the use of th for the tch sound in 'match' and dh for the dge sound in 'hedge', which I consider an analogous use of h as for sh in 'fish' and in the combination zh often used for the sound in 'leisure'. It always seems to me that there must be another little tweak somewhere that will make the system better, but when I try it there is usually some unforeseen problem. I'll probably come back to the question in another three years.

The z-section, which used to contain the verse anthologies, is now devoted to lists of links, and the s-section whch contained the stories is no more. Instead they are all in the w for Words section.

Bournemouth Weekend

Around the 7th October I got the idea into my head that I should play some more chess and enter the Bournemouth Congress which was on the next weekend 14 to 16 October. Thanks to the internet and email I managed to send an entry form, book a bed and breakfast for two nights and work out a railway journey, along the south coast, changing at Brighton and Southampton Central.

The travelling went very well apart from getting a slow train from Southampton and having to change again at Brockenhurst, to catch the Cross-Country train from Manchester that I should probably have caught in the first place.

This was the first time I'd been in Bournemouth since the memorable BCPS weekend there in 1989. The weather was excellent and I did a lot of walking around the central areas of the town; probably too much as by the end I was struggling to get up the hill to the venue with a stiff right knee.

My results were as poor as at Leicester, winning only the last game, and that only because my opponent blundered away a rook, and refused to agree a draw, which I thought would have been fairer, since he had played well and had the advantage up to that point. There are photos of the event here.

Since I got back on the Sunday night I've been feeling pretty tired and have endeavoured to take it easy and rest my brain and my leg. The photo is of a tethered balloon on which people were booking rides in the Winter Gardens.

Weekend in Leicester

I spent the weekend from Thursday 29 September to Monday 2 October in Leicester, playing in the Chess Congress and attending meetings at Secular Hall. The weather was very hot and sunny, not what I had expected for this time of year, and not ideal for playing chess. However I suppose I can't really blame the weather for my poor results, just two draws (in the first and fifth games) and one of those was against a junior.

At Secular Hall there was a lecture on 29 September about the situation in Libya, though the speaker was a supported of the Gaddaffi regime, who described his 40-year rule as a type of socialist utopia! What is going to happen there in future is of course difficult to predict. On Sunday there was a meeting of a new History Group in the afternoon, and in the evening a most interesting talk on Thomas Babbington and his friends and family and their work for the abolition of the Slave Trade. They did a lot of the work in preparing evidence for William Wilberforce to use in his speeches in Parliament (from which Quakers and nonconformists were excluded).

I travelled to London by train, and to Leicester by coach. On the return journey, also by coach from Leicester I thought I would try the coach to Hastings. This proved to be a mistake, as the so-called 'Express' took nearly 4 hours! It began by going south-west to Mitcham, and so on to Coulsdon and East Grinstead. Then it went on a grand round tour of East Sussex, stopping at Uckfield, Hailsham, Eastbourne, Pevensey, Bexhill and other places.

Allotting Computer Time

Sorry I've not found time to post anything here for several weeks. Most of my time on the computer has been taken up either with Twitter or trying to improve my understanding of CSS (Cascading Style Sheets) so that I can improve my website and also help to improve the Hastings Chess Club's website. If you click on the "Events" link you will be taken to a page I've designed, and you will find another if you click on the link to the page abour Jude Lenier's simultaneous chess display. I would like feedback to know how this style is perceived, whether the rest of the site should be updated similarly, and if anyone can offer suggestions for improvements. I've also done a lot of work on sections of my own website, but the pages can only be posted to the site when they are all ready, because they all link together, so there is nothing new to see there yet.

The Birds!

This photo is taken from the window that faces the opposite direction to the previous photo. The local pigeons are usualy flying about in disordered groups. However on this occasion I noticed that they all lined up with miltary precision along the roof ridges of the Holy Child college (formerly a convent), and stayed there quite still for a longish time. I wondered which is the general: the one on the point of the gable, or the one on the head of the christ child? (You may need to click on the photo and look at the larger version to see it clearly).

The local pigeons are of very varied colours, a few white, a few black, and quite a number an attractive brown. On the old RDFRS forum I queried why this might be, when the common pigeons seen in Warrior Square are more uniformly the traditional blue-grey, and seagulls have no noticeable variation. I was informed that this must be due to human intervention. Perhaps by the nuns who used to live there breeding and selecting them.

The Eyes in the Sky!

I've been intending to post this photo since the day after I posted the last message. I was half-dozing in my chair in the afternoon and suddenly woke up to see what appeared to be two demonic eyes glaring at me through the narrow window opposite. By the time I had realised what it was and got my camera ready the cloud formation, with the sun behind it, had moved on slightly, but the two "eyes" are still visible, but not so dramatic as they looked a minute or so before. Also the view shown is much wider than the one I saw framed by the window. One can appreciate from such an experience how people can attribute such phenomena to the actions of supernatural beings, and perhaps interpret them as personal warnings, since they depend on the location of the viewer and would not be seen much more widely.

The Two Ritas

This morning around mid-day I felt in need of some exercise, and it was a sunny day, possibly the last for a while, so I decided on a walk along the prom towards West St Leonards and perhaps a meal out. On the promenade near the Azur restaurant I was stopped by two ladies, who turned out to be Jehovah's Witnesses. I'd met at least one of them before, having argued with her at the door of my home, and she remembered me as being a Humanist. We exchanged names and some banter. She was Rita, and so was her companion.

I carried on with my constitutional, which Rita had called "power-walking", as far as the end of the cycle-path. No sign yet of it being extended to Bexhill. Turning back there I noticed that the "Bridge Cafe" on the corner was open, so went in for a late breakfast and cup of tea. I didn't remember it being open on other times I've walked past, but apparently the people who run it have been there for ten years. While I was there a group of eight businessmen came in, though there's not much sign of business around there now, since the wood-store closed down.

Maybe I ate my sausage, egg, bacon, beans and tomatoes too quickly, or didn't rest long enough to let them digest, because as soon as I got back home I began to have a bout of hiccups. Once it starts I find it difficult to stop. After a rest it seems to stop, but as soon as I make some strenuous move it starts up again. I think I will call it a bout of "The Two Ritas" in future!

Digging, Hacking and Tweeting

On Tuesday morning I finally got round to weeding and digging over the little strip of front garden I have. The next day I felt very stiff, no doubt due to using unaccustomed muscles. There was a forecast of rain for the afternoon, and I was thinking of sowing some grass seed, but the rain never came, and it has remained dry since. I suspect if I sowed the grass seed in the dry ground the pigeons would soon have it, so I'll wait for some rain.

Yesterday I had an email fron Vodafone saying they were sending me a SIM card. This was odd since I've never had any dealings with them. I deleted it as spam. Today I received the SIM card. It says "We're delighted to enclose your free SIM card and details of the Vodafone TopUp and Get International Freebee you selected." But I've never "selected" any such thing. I have an old pay-as-you-go mobile phone that I got from Virgin but have not used for over a year, and have no intention of reviving. I did think of emailing Vodafone, but their website wasn't very helpful and came with lots of warnings and popups, so I think it best to just ignore it.

My twitter account was apparently hacked a few days ago, and a message under my name was sent to a number of my "followers". Originally the message came from someone I follow and said "is it you in this photo" I clicked on this thinking it might refer to the photo of me with Ken Ham taken in Leicester in 2006 that was published on the Answers in Genesis site. However the link just led back to the twitter log-in site. I changed the password on my account, and the problem has not recurred, but it seems this sort of annoyance must be very easy to do.

I've now sent over a hundred tweets and have over 30 followers, though a number of them just seem to be local businesses in St Leonards that are no doubt angling for my trade. The most useful contacts I have made are with mathematicians who have provided a lot of interesting links, to websites and PDFs. I also had an amusing exchange with Bob Churchill, formerly of the BHA, about the type face "Comic Sans" which I use for the navigation links on the Mayhematics site, but he doesn't like. He wrote that we must be "aesthetically incommensurable", to which I replied that that must mean I'm rational while he is irrational (a mathematical joke for those not in the know).

Light Music Festival

I've been following the series of programmes on Radio 3 in the BBC's "Light Fantastic" music festival over the past week, since I enjoy this type of music, at least when it is well done.

An article by Stephen Hough in the Telegraph on line praises light music and has some links to several classics.

On the Sunday however I went to the concert put on by the Cory Band at the Queen Elizabeth Hall and, although I stood it all through to the end, found it extremely disagreeable simply because of the sheer loudness. Their only interpretation of forte was as fortissisimo. Maybe on a bandstand in the open air the sound dissipates, but at the back of the hall it was unpleasant.

Some pieces I've not heard for years, and have not so far heard on the programmes are "Parade of the Tin Soldiers" by Leon Jessel and "Do Not Forsake Me" (from the film "High Noon") by Dmitri Tiomkin; but I suppose these don't count as British Light Music. Another is the Oxford Street march by Eric Coates, although I have a CD with an old recording of this; I found the tune running through my head earlier today, and had to check that's what it was.

Website Upgrade

Yesterday my mayhematics website was unavailable due to the bandwidth limit being exceeded. This appears to be due to more people looking at it and downloading pages from it. There was a big spike at 4pm on Saturday for some reason. So I have had to upgrade to the next level, which about doubles the price to be paid each year to £60, which still seems very reasonable. At any rate I hope this resolves the problem. Perhaps I should look into ways of charging for donwloads.

I'm also thinking of registering some other domain names, since the subjects covered on the site do not all fall logically under the "mayhematics" heading, they have all accumulated in one place due to previous service providers closing down or not continuing to provide support, and due to changes from dial-up, to cable broadband, to telephone-line broadband connections.

Probably it is about time to upgrade my computer as well, but I dread all the problems that that could throw up so will probably leave it to the last minute as usual when it will be unavoidable.

Coughs and Wheezes

I've had a touch of flu for several days, probably exacerbated by standing around in the cold on London Bridge station platforms last Monday. It's been making it difficult to get any sleep, since for some reason I start coughing whenever I lie down and try to relax. When it gets bad, the cough seems to go through my vocal chords so that it makes a mad laughing noise rather than a wheeze. I tried a packet of lemsip but it didn't seem to do any good. A mint toffee is effective for a while. Fortunately it now seems to be clearing up. I must get back to taking more regular exercise, haven't been swimming so far this year.

Rationalism Advances

I took this photo at the AGM of the Rationalist Association, held on Monday evening in the Great Hall of the Art Workers Guild in Queens Square, London. This is a place I've not been to before. The bust above the big chair is of William Morris. In the middle Jim Herrick is talking to Caspar Melville (CEO of the RA, and Editor of New Humanist) and Laurie Taylor (Chairman). It seems the RA and New Humanist is making a determined move to be more active in promoting debates and vastly increasing circulation of the magazine. I asked whether this might mean some "dumbing down" of the content - not that I'm against this, I might be able to understand more of it!

Ceremonials

On Wednesday 18th May I went along to the White Rock Theatre to attend the Mayor-Making ceremony (the present Mayor was re-elected) and presentation of the Freedom of Hastings to former MP Michael Foster. Having to keep on "be upstanding" every time the mayor entered or exited was a bit wearing. It was curious to note that the conservative councillors still observe the ceremonial aspects by wearing reddish-coloured robes, while the labour side were in normal dress. The "processing" in and out could well also be done away with. The best part of the show were the performances by children from local schools, though one of the songs was mildly religious. The dance by the older girls was apparently to celebrate the suffragettes.

On Saturday 21st May I accepted an invitation to attend a meeting of the Freemasons Lodge formed by Old Olavians from my former school. This was held at the Freemasons Hall in Great Queen Sreet in London. This is a very large building, in art deco style, built as a memorial after the first world war. We were admitted to the so-called "Indian Temple" which is apparently only one of the smaller temples in the building, and treated to a talk by one of the Masters. The temple was a large rectangular room with domed roof. The masons in their ceremonial aprons sat on thrones, three at each end, and one in the middle of each side, with lighted candles in front of them.

According to the talk the masons are now purely a philanthropic organisation, raising money for work done by other charities like the red cross, or to supply equipment to local hospitals, although they also provide relief for old masons fallen on hard times. However they require applicants to declare belief in a "supreme being" to become a member, and swear their oaths on a bible, which rules me out. Though doesn't this now contravene discrimination laws, if they are not an explicitly religious organisation?

I'm afraid I found their costumes and their "processing" only rather comical. They made me think of Tweedledum and Tweedledee, especially as most of the members were of portly proportions. This may be a consequence of their dinners, there was one following the meeting which declined to attend, since it cost £35, a price that represents a whole week of dinners for me.

On Monday I'm due to attend the AGM of the Rationalist Association. I wonder if they go in for robes and ceremonials? I hope not.

Break Building

I spent most of the Easter and May holiday week-ends watching the snooker world championship from Sheffield, in the hope that newcomer Judd Trump might win over the old guard John Higgins, and he came quite close.

Rather than abolish the May Day holiday break, surely it is time to fix the date of Easter closer to the Spring equinox, in March, so that the two holidays don't come so close. It's getting like Christmas and the New Year.

Much of the rest of my time was spent revising and checking the section on King tours on the Knight's Tour Notes pages, particularly the enumeration of the tours on boards of two ranks. The totals are given by recursion relations and by formulae involving the Fibonacci numbers. This proved quite troublesome, but maybe that's because my brain isn't working a smoothly as it did even a few years ago.

On the 2x8 board there are 128 closed tours, which is easy to verify, but the number of open tour diagrams works out to the surprisingly large, and surprisingly round number T = 32000. I still wonder whether I've got this right, but the other figures seem to be consistent with this. On the same board there are 584 reentrant tours, and G = 8176 geometrically distinct open tours, of which S = 352 are symmetric. These figures are related by G = (T + 2S)/4.

Magic Rook Tours

The image shows three diagonally magic rook tours that I constructed on 20-21 May 1986 but then seem to have forgotten about. (It was a busy year for me!) The tours have biaxial symmetry and each quarter tour is a bisatin (i.e. uses two cells in each rank and file). If the numbering is shifted by one quarter, so 17 becomes 1, 18 becomes 2 and so on, the rank and file magic property is automatically retained. However for the diagonals to remain magic there must be two numbers less than or equal to 16 in each (i.e. the bisatin has to be diagonal as well). The work was inspired by a less structured diagonally magic rook tour by J. Brugge that appeared in the German chess problem magazine Die Schwalbe in August 1985. Anyone care to enumerate all the diagonal bisatins that could be used?

Tours and Twitters

I've just been trying to remove two blogs from the list of those that I "follow", but there doesn't seem to be any way to do it. The instructions explain how to do it, and they are supposed to lead to a page where you can click a "stop following this blog" button. But they don't!

A day or so ago I uploaded a new page on King and Queen tours to my Knight's Tour Notes pages on the Mayhematics site. This is a page that has been waiting to be uploaded to the old KTN site for several years. There's stil a lot of other material, and updates to existing material, to be added. It's a continuous process.

I've been getting more involved with Twitter. I now have about 15 followers, have sent about 30 tweets and follow about 70 people and organisations. Mostly these are to do with mathematical recreations and humanism. One message led me to this site on mathematics by John D. Cook where the old subject of the misattribution of Beverley's magic tour to Euler came up, and I've contributed to the discussion.

Honeycomb Tours

I have added a new page about tours of Honeycomb Boards to the Knight's Tour Notes pages. This collage shows a selection of results. Some date back to 1974 when I sent an example tour to W. Glinski whose Hexagonal Chess on a 91-cell board was popular. Others are new results. I've included wazir, king and knight tours that show various different types of symmetry.

A Disorienting Day

There was a concert of light music, of which I am a fan, due to be held at St George's Church in Beckenham this evening. So I thought I would look into the possibility of travelling there by public transport. According to the National Rail Enquiries website, which I've found to be reliable previously there was a train at 4:55 that would get me to Beckenham Junction by 17:15 with one change at East Croydon.

However, when I tried to buy such a ticket at Hastings station I was told such a trip was "impossible"! Also that the National Rail Enquiries were now run by a bus company who didn't understand the railways. Apparently the link from East Croydon to Beckenham Junction is via a Tram line. This appears to be correct: it is on something called the London Tramlink, which I've never heard of before, though it has been around for ten years.

So, lacking a clear route, I aborted the trip and decided to listen to a concert on Classic FM. This was a Prokofiev concert introduced by Howard Goodall. The first item announced was his Classical Symphony, number 1. However the music played bore little resemblance to previous performances of that work I had heard. I suspect it was some other of his symphonies, in far more modern style. The second item was announced as the music from Romeo and Juliet, including the "Dance of the Knights". However, unless I fell asleep, this was not the music played.

Tomorrow apparently it's time to put the clocks back, or is it forward? I'm not sure if I'm coming or going.

A Magic Knight Rectangle

Back in 2003 I was able to prove that magic knight's tours were not possible on boards 4n+2 by 4m+2, but a proof for the 4n by 4m+2 case eluded me. I now see that that is because there is no such proof! Thanks to a suggestion by John Beasley, that since there is a simple magic knight+wazir tour on the 2x4 board, a magic knight tour should be possible on a sufficiently large 4n by 4m+2 board, I looked at the subject again and found two 12x14 examples last night, of which this is the first:

141 122 143 038 139 124 127 042 045 030 131 026 047 028
144 037 140 123 128 039 044 125 130 041 046 029 132 025
121 142 035 138 119 126 129 040 043 050 031 134 027 048
036 145 120 063 034 137 014 155 032 135 106 049 024 133
011 064 061 118 013 154 033 136 015 156 051 108 105 158
146 117 012 151 062 059 016 153 110 107 018 157 052 023
065 010 115 060 149 152 111 058 017 020 109 054 159 104
116 147 150 009 114 057 094 075 112 055 160 019 022 053
091 066 007 148 093 074 113 056 095 076 021 162 103 078
006 069 092 073 008 003 082 085 168 161 096 077 100 163
067 090 071 004 083 088 167 002 081 086 165 098 079 102
070 005 068 089 072 001 084 087 166 097 080 101 164 099

It is constructed by the "rolling pin" method that I devised for 12x12 magic tours. It's surprising I hadn't thought of trying this before. It's just a matter of widening the board. The files add to 1014 = 169x6 and the ranks add to 1183 = 169x7. Each file consists of three pairs adding to 127 and three pairs adding to 211. The ranks are made up of pairs of complements adding to 169.

Knight's Tour Notes website gone

I've just noticed that my old Knight's Tour Notes website has vanished into the ether. This is not unexpected, since it was housed on a dial-up site which I have been unable to access for several years. I will now have to relaunch it on my Mayhematics site, or start up a new URL. This may take some time. I will probably want to put it into a revised format and improve some of the diagrams.

Outing to London and Leicester

On Saturday morning I decided on something of an impulse to combine a trip to London to attend the Conway Hall talk by Jim Moore on "Darwin and the Sin of Slavery", with a trip to Leicester to attend the lecture by Ken MacLeod on "Darwin Dawkins and the Left". This was just about possible in the time by using booking.com to locate a Hotel and the National Rail Enquiries to book the trains, using a debit card and collecting the tickets from the machine at Hastings station. The machine was a bit recalcitrant but finally came up with the tickets.

One of the interesting points from the Moore lecture that I'd not been aware of was the role of Louis Agassiz (now best known for his work on ice ages) in diverting Darwin's efforts into countering Agassiz's strange theory of multiple 'creations' of separate human races known as polygenism.

Ken McLeod's thesis was that many people on the political Left have deliberately misunderstood Richard Dawkins, or the implications of his "Selfish Gene" idea, though it seems to me that many of other persuasions have been equally free in criticising Dawkins without having apparently read his books.

I was hoping also to fit in a Gresham Lecture by Christopher Hogwood, given at the Museum of London at 1pm today (Tuesday). But this was scuppered since my train was over half an hour late arriving at St Pancras, due apparently to signaling problems. It was also raining, so I used my bus pass to get the No.17 to London Bridge, but managed to get there just in time to miss the first Hastings train.

Twittering

Well the beard didn't last very long. I'll have to change my photo again. It felt too uncomfortable, and was just as much trouble to maintain as shaving every day or two, as well as making me feel old. Srangely three people complimented me on the beard, but unfortunately after I'd already got rid of it.

I joined Twitter a few days ago. As if I need more ways to waste my time. Somehow I have already acquired five followers and sixteen subjects to follow, and posted two tweets, all this mostly related to humanism.

Nostalgia Apologia

I thought I'd let my beard grow during the cold spell over the Winter Solstice, thinking that I would probably shave it off when the weather changed, but it seems I have now become attached to it, at least until any really warm weather should come along. I've also updated my photo here and on facebook accordingly.

This evening, having seen it previewed in the local paper, I went to the White Rock Theatre for an evening of nostalgia, where the Glenn Miller Orchestra was performing. The tunes were good, and some of the individual performers, but it was just far too loud. After the interval I tried moving further back in the Gallery but couldn't escape the sheer blast of sound. It was getting almost painful. So when they started playing something from The Sound of Music, which I certainly don't associate with the Glenn Miller era, I decided I'd had enough.

Queen Victoria in Warrior Square

There was a spectacular sunset this evening around 4:30. I went out with my camera to try to capture an image, but by then the view had changed, first the sun came out from behind the clouds, then it quickly set. I got some images, but they were similar to others obtained before. I ended up on the sea-front by Warrior Square and noticed that the statue of Queen Victoria was illuminated with a spotlight and that the plinth seems to have been cleaned. Hence the photo shown here.

I did very poorly in the Weekend Chess Congress, playing in the Minor tournament, since I managed only a draw, and that was against the youngest competitor, where I ended up with a knight and pawn against two knights and pawn. The middle game of the five, played in the evening of the first day, was weird. I made a muddle of the opening and was well down, then somehow revived to fight back to a winning position, and then went to sleep again in the end game and threw it away. Sheer tiredness I think. There are another five games to go in the New Year tournament, which I suspect will be against stronger opposition.