Updating Website

 New results sent to me on enumeration of the smallest board tours by Andrew Usher in the USA and of queen tours by Walter Trump in Germany have led me to upgrade the relevant Knight's Tour Notes pages on my website. The trouble is that each of these updates has led to the realisation that all the other related pages also need updating.  First I did the Shaped Boards section. Then the Walker Tours section (on pieces with lateral and diagonal moves. 

A result about the impossibility of magic queen tours with rotary symmetry on the 6x6 board has led me to put new results into the Magic Theory page, and that led me to reinstate the page on 4x4 Magic Squares, which had become mixed up with the page on Multimover tours. Now I am recasting the whole section on Other Pieces. A lot of the pages are out of date compared with the PDF versions I issued in 2019.

The whole of the History pages also need to be updated, due to newer information, some of which was put into the 2019 PDF versions. So it's proving a big task. After that I will have to try to put everything into book form again. Or maybe I need to start a dedicated website for Knight's Tour Notes alone. All this leaves little time for other original work.

The Onitiu Problem Published

 I've now completed my account of the Onitiu Problem, which has featured in these diary pages over the past two years. It forms a third page in the Figured Tours section on my Knight's Tour Notes pages.

https://www.mayhematics.com/t/f3_Onitiu%20Problem.htm

Quite a lot of work there that I'm pleased with, though still a number of gaps, and constructing solutions on larger boards is beyond the limits of my drawing program.

Giraffe Move Pattern

 This is a differently coloured and enlarged version of the Giraffe Move Pattern that I published here in 2021. It shows the successive cells reached from the centre cell in successive numbers of Giraffe {1,4} moves. The colouring is still based on the rainbow sequence, but achieves more contrast between areas, with the orange and yellow areas coming between the darker areas. Is it too fanciful to imagine it as an architectural plan of some ancient temple?

It would be interesting to expand the pattern outwards further, repeating the colour sequence, so black cells wold represent 9, red cells 10 and so on, but this is close to the limits of my drawing programme..   

More Demolition at the Bus Station

 I seem to have missed a month. I've been busy updating my website pages dealing with Knight's Tours on Shaped and Holey Boards. This was stimulated by new enumerations of 11-cell and 14-cell tours sent to me by Andrew Usher from the US. I've found it rather laborious, partly because I'm experiencing eye strain, and have to ration the time I spend online, to avoid getting a headache. 

There have also been developments concerning magic queen tours on odd-sided boards. This is in work by Walter Trump in Germany. I suppose I will have to update the magic tour pages next.

Meanwhile in Crewe town centre work on clearing the site for the Bus Station is under way very slowly.

Here is another photo

Latest photo of the demolition work, taken 14 July, below: 

Beautiful blue sky today and cool in the shade, 

but I'm not looking forward to the heat wave forecast for next week.

Demolition work now complete (4 September 2022). Fortunately three trees along Delamere Street have survived the devastation so far.

Numbers and a Tour

 I have a new publication! Directory of Numbers. It is a 24-page listing of the numbers up to 9999 with prime factorisations, except that four-digit numbers divisible by 2 or 5 are omitted. These are easily divided by 2, 5 or 10 and the factors of the resulting smaller number can then be looked up.

https://www.mayhematics.com/p/p.htm

I have also found a solution to the 18x18 Onitiu Problem of a knight tour with 180 degree rotational symmetry. This just fills a gap in my work, since I have not been able to find a solution to the 90 degree rotation on this board.  Here is the new result:

The square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324 are shown on the red line. The blue line is a rotation of the red. There are no intersections between the red and blue lines, but single links 63-64 and 225-226, shown as heavy black moves. 

Tree Cutting in Crewe

 

There has been much cutting down of trees around the bus station in Crewe. 

This is presumably to clear the way for new entrances and exits to the new bus station that is planned. 

Though I have not seen any publicity of the design of the new station.

I took these three photos.

Some of the trees cut down were substantial, so I hope others can be planted to replace them.

There was also substantial tree-felling of a row of trees on Godard Street. 

This is probably to clear the way for a new housing development. 

Fiery Sky

 

A Fiery Sky at sunset yesterday evening.

view taken from my bay window.

Work in Slow Progress

Despite not posting anything here I have been busy this month, but not making much progress. 

Partly this is because I've had trouble with blisters on a leg and losing a couple of front teeth, so have been kept occupied visiting dentist and GP surgeries and hospital for checkups.

I'm still trying to complete my study of the Onitiu Problem but am stuck on the 14x14 case. 

I'm also trying to edit my 12 PDFs on Knight's Tours into a shorter book. 

Another project is a book on Numbers, which I may call Numerology or maybe Arithmology, with the subtitle The Wisdom and Folly of Numbers, .since it includes chapters on Arithmosophy and Numeromancy. It also includes my notes on Figurate Numbers as well as basic Arithmetic. 

Red Sky

 A weirdly mottled red sky over Crewe this morning. 

I went out specially to catch the image before it faded.

On Sacred Geometry

 On Sacred Geometry

I've been looking at various YouTube sites that have videos on "Sacred Geometry". Some are completely vacuous waffle to me. However a few do contain genuine arithmetic and geometric results that seem of interest from a mathematical viewpoint. In particular there is a series from the "Jain Academy" fronted by an affable Australian lecturer who calls himself "Jain108". His mathematics is mostly correct, except where he obsesses about the "true" value of pi being 3.144... The Jain Academy deals in what I can only call "New Age Eclecticism". In other words it takes bits from everywhere, Hinduism, Kabbala, Christianity, Islam, Buddhism, Astrology, and so on, regardless of dogma. 

The number 108 is apparently ubiquitous in Hindu mysticism. It is said in several sources to be the ratio of distance to diameter for both the Sun and the Moon. For any celestial body this ratio would be near enough the cotangent or cosecant of its apparent angular diameter. The angle whose cotangent or cosecant is 108 turns out to be 31 minutes and 50 seconds to the nearest second. In Patrick Moore's "Atlas of the Universe" (1994), which I happen to have to hand, the mean apparent diameter of the Sun is given as 32' 1" and the mean apparent diameter of the Moon as 31' 6". So the number 108 seems to be a reasonably good estimate. The closeness of the apparent diameters of Moon and Sun is of course why Solar Eclipses can be so spectacular.

Another obsession of the Sacred Geometers is, as might be expected, the dimensions of the Great Pyramid of Khufu at Giza. The Jain Academy makes much of the triangle formed by the pyramid as seen at a distance from a side. Taking its base width to be 2 units it is claimed that its slope length is the golden ratio 1.618 and its height is the square root of the golden ratio. According to Wikipedia the pyramid's original height was 280 cubits and its base 440 cubits. The slope length is thus the square root of (280^2 + 220^2) = root 126800 = 356.09 cubits. The ratio of slope to half base is thus 356.09/220 = 1.61859, and the golden ratio is 1.6180339. So again a plausible approximation. The angle of slope is 51 degrees 5 minutes so the visible angle at the summit would be 77 degrees 50 minutes.

When people make a model of a pyramid they nearly always take the faces to be exact equilateral triangles, but this gives a pyramid whose slope height is half of root three and whose height is half of root two. If the slope height of the pyramid is the golden ratio then this must be the altitude of the triangular face shape. The base angle then works out at 58 degrees 17 minutes and the apex angle as 63 degrees 23 minutes Differing from the equilateral by 3.6 degrees at the apex. 

The symbol used for the golden ratio in Sacred Geometry is phi (though mathematical texts often use tau). The claim about the new value for pi is that is should be 4/(root phi) = 3.144605...as opposed to 3.14159... Or equivalently that 4/pi = 1.2732395... should be root phi = 1.2720196... Whether this means that the Laws of Nature themselves are going to change when the New Age dawns, or only human consciousness of them is not at all clear.