Tuesday, 19 April 2022

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.


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. 

Saturday, 26 March 2022

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. 

Monday, 14 March 2022

Fiery Sky


A Fiery Sky at sunset yesterday evening.

view taken from my bay window.

Saturday, 26 February 2022

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. 

Tuesday, 18 January 2022

Red Sky

 A weirdly mottled red sky over Crewe this morning. 

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

Thursday, 13 January 2022

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.

Thursday, 23 December 2021

Fibonacci and Combinations

 I've been looking through my books on number theory and combinatorics, but none of them seem to mention anywhere the following simple relationship of Fibonacci sequence to combinations. In the following I use the notation nCr = n!/(n-r)!r! for the number of ways of choosing r from n. 

An explicit sum for F(n) can be found by summing the upward diagonals of the combination table that lists n against r and is sometimes presented in the form of Pascal's Triangle. 

It can be expressed in the general form:

F(n) = (n-1)C0 + (n-2)C1 + (n-3)C2 + (n-4)C3 + ... + (n-k)C(k-1) 

                = Summation (r=1 to k) (n-r)C(r-1).

Where k =  n/2 or (n+1)/ when n is even or odd respectively. 

In the symmetric form of Pascal's Triangle the upward diagonals are transformed into knight-lines.

Instead of the above elementary formula for F(n) that uses only addition, subtraction, multiplication and division of whole numbers, the mathematical texts focus on the exotic Binet formula that expresses F(n) in terms of root five, or root five and the golden ratios, which are irrational numbers that all cancel each other out.

A First Course of Combinatorial Mathematics by Ian Anderson (Oxford University Clarendon Press (1979) page 43 derives from the Binet formula a more complicated relationship involving the summation of expressions 5^r.(n+1)Cr divided by 2^n, but does not simplify it to the above form. This is the nearest I have found in my limited sources.

Of course there is also a simple relationship of 2^n to the combinations:

  2^n = nC0 + nC1 + nC2 + ... + nCn.

Does anyone have other sources they could direct me to? (see my Knight's Tour pages for my email)

Does anyone have a concrete combinatorial explanation for the formula?

CORRECTION: I have since noticed that Anderson does mention the above expression briefly, without further explanation, in Exercises 4.2 (4) on page 44.

FURTHER: I have found another reference in Principles of Combinatorics by Claude Berge (Academic Press 1971) page 31, though it has n+1 instead of n. It implies that (n-k)Ck counts the number of ways of choosing k items from a row of n, but with no two items being adjacent.

FURTHER: I have modified the equation, since there seems to be some divergence in the way that the Fibonacci numbers are labelled. I take F(0) = 0 and F(1) = 1, so that F(2) = 1 and F(3) = 2 and so on, whereas other sources start from F(0) = 1, F(1) = 1, F(2) = 2, etc. I hope it is now correct!