The Games and Puzzles Journal — Issue 23, May-August 2002

This issue, wanders in the realms of direction-finding, astronomy, astrology and science fiction.

Back to: GPJ Index Page
Sections on this page: (21) Equidirectional Knight's Tour. (22) Thirteenth Month and Zodiac Sign. (23) A Space Compass for the Constellations. (24) Bode's Law Revisited. (25) Approximation to Ratios by Continued Fractions. End

## é(21) An Equidirectional Knight's Tour.

A solution is presented here to a problem that has remained unsolved for 60 years: to construct a knight's tour that has “eight moves in each direction”. It is a lot more tricky than it seems at first sight (unless I've overlooked something that may be obvious to the more clear-headed).
First it needs to be said that English has a difficulty with the “sense” (in more ways than one) of the word “direction”, and I'm not sure of the best way to resolve it. In most everyday uses one points in a given direction, or indicates it by an arrow. However in geometrical applications “direction” is also used to mean any one of a system of parallel lines, regardless of “sense”, i.e. not taking account of motion or “direction” along the lines. It would be helpful if there were another word for this “senseless” direction.
The first diagram below, by E. Lange, Sphinx 1931, solved the problem of constructing a tour with 16 moves in each of the four (unarrowed) directions in which the knight can move. The tour is not quite symmetric, as the four darker moves indicate. In fact a symmetric solution of this problem is impossible. Here is a proof I have found of this fact.

Theorem: A symmetric 8×8 knight's tour with 16 moves in each direction is impossible.

Proof: A symmetric tour can be split into two equal halves of 32 moves joining a corner to an opposite corner, say a8-h1. This journey is equivalent, over all, to a move of type {7, 7}. If this half-tour contains m moves (1, 2) then to meet the required equality in the whole tour, the half-tour must have 8 – m opposite but parallel moves (–1, –2) and these combine to give a resultant move [m – (8 – m)](1, 2) = (2m – 8, 4m – 16) in which both coordinates are even. Four such even moves cannot combine to give a move with odd coordinates.

It follows that it is impossible to construct a symmetric tour with 8 moves of each of the eight directed types: (1, 2), (1, –2), (–1, 2), (–1, –2), (2, 1), (2, –1), (–2, 1), (–2, –1), since ignoring the direction of movement (leaving out the arrows) reduces it to the above problem.
This long-standing problem is mentioned by H. J. R. Murray in his 1942 manuscript on The Knight's Problem. After quoting the Lange diagram he writes: “A more difficult task, for which we know no solution, is to draw a tour in diametral symmetry with 8 vectors of each kind.” The above theorem shows that no symmetric solution is possible. He makes no mention of knowing any asymmetric solution.

On 5 June 2002 I found the asymmetric solution shown in the second diagram. It was constructed from Roget's four nets (formed by the ‘straight’ knight moves that cross only one grid line) using the linkage polygon shown on the right, which consists of alternate straights and ‘slants’ (moves that cross two grid lines). The deleted straights and the inserted slants (shown darker) occur in similarly directed pairs of moves. The reason for using the Roget nets is that any circuit of a net (connecting 16 cells by 16 moves) has the moves in oppositely directed pairs. After finding the above answer I spent a while looking for a solution using only four slants, but the only suitable octagon I found failed to engage all four nets.

çHere is a more colourful presentation of the tour to show the equipartition more clearly. 8 moves in each direction. It's a pity I wasn't able to get the arrows to interweave alternately over and under.

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## é(22) A Thirteenth Month and Zodiac Sign.

At the end of 2001, annoyed by fact that Christmas and New Year's Day fell on a Tuesday, thus disrupting normal activities for an unnecessary length of time, I sent a letter to the editor of the Leicester Mercury, published on 5th January, which won the ‘letter of the week prize’, proposing a simple calendar reform. The proposal, which is not new of course, was simply that Christmas and New Year should be on fixed days, at the weekend, and not in the middle of the week. Some people seemed to think this required changing the date of Christmas, so I wrote again on 14th January, to clarify my proposal as follows: “Since a year of 365 days consists of 52 weeks and one day, all that is necessary is to make one day per year a non-weekday. It could be called ‘Year Day’ and inserted between a Saturday and Sunday to make a long weekend at some suitable point in the year. In compensation one of the months of 31 days would be reduced to 30. Similarly the ‘Leap Day’ inserted every four years would be a non-weekday.”
Alan Pendragon, who also wrote on the subject (concerning its effect on religious festivals), expressed the view, in a later discussion I had with him, that he did not like the resulting effect of having his birthday on the same day of the week every year. There is no pleasing some people! I feel there is enough disorder about life anyway and that the calendar should not be designed to introduce more.
Anyone interested in exploring the various ways that have been proposed for reforming the calendar will find a whole range of proposals at: http://www.calendarzone.com/Reform.

An alternative to the above calendar scheme, since 364 = 13 × 28, would be to have 13 months each of exactly four 7-day weeks. The division into 12 months is of course more practical from the organisational point of view since it easily divides the year into halves, thirds, quarters, and sixths. The thirteen month scheme is sometimes argued to take better account of integrating the Moon's phases with the solar calendar. However the Moon's phase period (also called a lunation or synodic month) is 29.53059 days, and the length of a tropical year, between two spring equinoxes, is 365.24219 days, so the actual number of months in a year is 12.36827, which is nearer to 12 than to 13.
To reconcile the phase month and the year we need to express the fractional part 0.36827 as a ratio of whole numbers p/q, then after q years a shift of p months will accrue. By the process of calculating the reciprocal and noting and removing the integral part we express the decimal as a continued fraction: 0 + 1/(2 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + 1/(17 ..., which is calculated from right to left, and the successive values (convergents) are: 0, 1/2, 1/3, 3/8, 4/11, 7/19, 123/334, these are alternately less than and greater than the decimal value but gradually get closer to it. [For more on this method go to item (25) below.] The ratio 7/19 is correct to three places of decimals and gives the Metonic cycle of 19 years (or 19 ×12 + 7 = 235 months), which is named after Meton, a 5th century BC Athenian astronomer.

Also at the beginning of the year I received a programme from the Leicester Astrological Society in which a lecture is announced to be given in November by their former chairman Edward Crabtree with the title “Is There a Thirteenth Sign?”. By chance I also happened to come across, in a second-hand book shop, a copy of The Thirteenth Zodiac; The Sign of Arachne by James Vogh (Mayflower Books 1979). The American edition of this was published under the title Arachne Rising: The Thirteenth Sign of the Zodiac. It gives a quite well reasoned case for a thirteenth sign (which he identifies roughly with Auriga, between Taurus and Gemini), to be used in conjunction with the usual twelve, and presumably taking more account of the supposed influence of the Moon.
Unfortunately the author, who was the science fiction writer John Sladek (born 15 December 1937, died 10 March 2000) writing under a pseudonym, is said to have stated, in an interview by David Langford that: “The James Vogh books, Arachne Rising and The Cosmic Factor, were conceived as jokes, but very quickly turned into moneymaking enterprises. Only they didn't make a lot of money, either. So finally they turn out to have been a gigantic waste of time. Except that I can say that I invented or discovered the lost 13th sign of the zodiac.”
See Guardian Obituary and Sladek Interview [found on a new site].
Some subsequent commentators have been more forceful, calling the book a ‘spoof’ or even a ‘hoax’. Sladek however says only that it was “conceived as a joke”, which then, in the course of his research for the book, seems to have turned into a genuine discovery. Skeptics should however bear in mind that David Langford has also turned his hand to spoofing, since he was involved in bringing the Necronomicon to reality. [For those not into fantasy literature the Necronomicon is a supposed book of dangerous occult lore that features in the horror stories by H. P. Lovecraft.]

When I read the book by James Vogh, before learning of the above interview, his reference on page 3 to a Babylonian astronomer called ‘Kidinnu’ (Kidding You?) indeed made me wonder if he was joking, but on further enquiry it seems that Kidinnu is a genuine alternative version of the name of an astronomer cited by the Roman authors Livy and Pliny as Cidenas, who is credited with the discovery of the precession of the equinoxes. Presumably this phenomenon, which involves a 26,000 year cycle, could only be discovered by a serious astronomer with a considerable tradition of accurate observations over many years, possibly centuries, on which to rely.

An alternative for the thirteenth sign, espoused in particular by author Walter Berg, is the constellation Ophiuchus. This sign however seems to be favoured more by astronomers who are out to discredit astrology rather than by astrologers themselves. An examination of a star-map will show that Ophiuchus, Auriga and perhaps Orion are the primary contenders for the thirteenth sign, since they are the only prominent signs near enough to the ecliptic. Orion and Auriga form together with Gemini and Taurus a very prominent group located at the summer solstice with Auriga above and Orion below the ecliptic (from the point of view of the Northern hemisphere). The advocates of Ophiuchus consider that the zodiac signs should be the same as the actual constellations that appear along the ecliptic (the apparent path of the sun against the background of the stars). But this depends on just where the lines dividing one constellation from another are drawn, and they seem to be pretty arbitrary. [This led to item (23) that follows.]

## é(23) A Space Compass for the Constellations.

According to Patrick Moore [Atlas of the Universe (Philips 1994) p.197] “The 19th century astronomer Sir John Herschel said that the patterns of the constellations had been drawn up to be as inconvenient as possible. In 1933, modified constellation boundaries were laid down by the International Astronomical Union. There have been occasional attempts to revise the entire nomenclature, but it is unlikely any radical change will now be made. The present-day constellations have been accepted for too long to be altered.”

What puzzles me is why the astronomers still bother with constellations at all. They are after all merely visual patterns of stars as viewed from Earth, not in most cases actual systems of stars in close proximity. For example Castor and Pollux, the main stars in Gemini are 46 and 36 light years away respectively. The seven main stars in the Plough (Ursa Major) are from 59 (Mizar) to 108 (Alkaid) light years away. In Orion some distances are: Betelgeux 260 ly, Rigel 900 ly and Saiph 2200 ly (figures from Moore and from the Collins Dictionary).

It occurred to me that for a division of the night sky to be truly scientific the areas (better called sectors than constellations) should all be of the same shape and size. So what methods are available? First I should mention that the celestial sphere is an imaginary sphere of large radius centred on the Earth; its poles and equator are found by extending the Earth's axis and equator to meet the sphere. The ecliptic is the apparent path of the sun across the celestial sphere. The ecliptic is inclined at 23.5 degrees to the equator (due to the tilt of the Earth's axis) and crosses the equator at points that mark the spring and autumn equinoxes. The highest and lowest points of the ecliptic mark the summer and winter solstices.

Obviously the Platonic solids, projected onto the celestial sphere, provide five solutions. The octahedron is equivalent to three great circles: such as the equator, and the circles connecting the poles with the equinoxes and the solstices, thus dividing the celestial sphere into 8 parts (octants). The cube (hexahedron) can similarly be oriented so that the centres of its faces correspond to the poles, equinoxes and solstices, dividing the sphere into 6 parts. The tetrahedron, dodecahedron and icosahedron cannot be placed so as to mark these six significant points, so we disregard these solutions.

By combining the octahedron and cube, and extending the great circles that form the edges of the projected cube, we get the pattern of 6×8 = 48 triangular sectors shown below. This is my proposal for dividing the heavens into equal sectors. Each of the 6 faces of the cube is divided by its diagonals and medians into 8 sectors. Each of the 8 faces of the octahedron is divided by its medians (from vertex to middle of opposite side) into 6 sectors. Each sector is a 45, 60, 90 triangle (bear in mind that on a circle the angles of a triangle always add to more than 180 degrees). The two circular areas in the diagram should be thought of as hollow hemispheres with the points R and L depressed into the page. They can be imagined as folding together at W to form a complete sphere with the observer (and the Earth) inside.

It so happens that the ecliptic, shown by the red line, still passes through twelve sectors, so the twelve-fold Zodiac can be maintained. However, the times the sun spends in the 12 sectors are not completely equal, so the dates of start and finish of the signs should be revised. Since the 48 sectors are defined in terms of the position of the equinoxes they will precess with the equinoxes. Precession is due to a wobble in the Earth's axis occurring over a period of 26,000 years, so that since the Classical period the spring equinox, often called the ‘First Point of Aries’ has moved and is now in fact in Pisces and moving towards Aquarius — presumably it is when it arrives there that the ‘Age of Aquarius’ properly begins.

The following diagrams show a method of naming the 48 sectors in terms of the six cardinal points N, S, E, W, R, L. These letters can be thought of as North, South, East, West, Right, Left. Alternatively W = Winter, E = Estival (i.e. summer), and R = Rising, L = Lowering (since the sun rises above the equator at the spring equinox in the northern hemisphere and falls below it at the autumn equinox), or R = Ram (Aries), L = Libra.

The following table gives the constellations that occur in each sector. The most prominent (printed bold) can be regarded as giving their names to the sectors, but it needs to be borne in mind that over the centuries the constellations will shift into different sectors due to precession. Aries (in its present astronomical definition) seems to be quite a small constellation whereas Taurus is very large. I have assumed, to fit astrological custom, that part of Taurus should be subsumed to Aries. Also, some constellations span two or more sectors. Orion, in particular, sits over the cardinal direction E, marking the summer solstice (though of course it cannot then be seen because the sun is nearby — it becomes most prominent in the winter sky). Ophiuchus similarly (but less recognisably) sits over the winter solstice. Hydra, Eridanus and Draco are long thin wriggly constellations that extend over three or four sectors. Auriga, Carina, Scutum, Ursa Minor, Ursa Major have distinct parts in two sectors. No doubt several of my assignments may be found questionable if more accurate maps are used.

 NER perseus, camelopardis, auriga NRE cassiopeia NRW cepheus, lacerta NWR cygnus NWL draco, ursa minor NLW ursa minor/major, draco NLE ursa major NEL lynx SER reticulum, dorado, horologium SRE phoenix, hydrus SRW grus, tucana, indus SWR octans, pavo, telescopium, corona australis SWL triangulum australis, apus, ara, norma, lupus SLW crux, musca, centaurus SLE chameleon, carina, vela SEL carina, volans, pictor, puppis ENR taurus, orion, auriga ERN aries, taurus, orion ERS eridanus, orion ESR lepus, caelium, orion ESL canis major, columba, puppis ELS monoceros ELN cancer, canis minor ENL gemini WNR lyra, vulpecula WRN aquila, sagitta, delphinus WRS capricorn, scutum WSR sagittarius, scutum WSL scorpius , ophiuchus WLS libra, ophiuchus WLN serpens (caput), corona borealis, ophiuchus WNL hercules, ophiuchus REN pisces, aries, cetus RNE andromeda, triangulum RNW pegasus, pisces RWN equuleus RWS aquarius, capricorn RSW piscis austrinus, microscop. RSE sculptor, cetus RES cetus LEN leo, hydra LNE leo minor, leo LNW coma berenices, canes venatici LWN bootes LWS virgo LSW centaurus, corvus, hydra LSE crater, antlia, hydra LES sextans, pyxis, hydra

The ecliptic, i.e. the Zodiac, runs through the sectors
ENR, ERN, REN, RWS, WRS, WSR, WSL, WLS, LWS, LEN, ELN, ENL

The Milky Way, i.e. the plane of the Galaxy, runs through the sectors
ENR, NER, NRE, NRW, NWR, WNR, WSL, SWL, SLW, SLE, SEL, ESL

Other schemes of labelling the sectors are of course possible. For instance it so happens that there are 26 vertices, one for each letter of the alphabet; though there is no one method of lettering that suggests itself. Since each node is even (of degree 4, 6 or 8) a single sequence of lettering will cover the whole, beginning from any node, and can be made reentrant.

## é(24) Bode's Law Revisted.

I went to a meeting of the Leicester Astronomical Society, held at the new National Space Centre on January 8th 2002. The entertaining talk, by David Connor, had the title: “Surely, you must be joking; or myths and old wives tales in modern astronomy.” One of the topics that came up was “Bode's Law” which is an empirical formula that purports to approximate the relative distances of the planets from the sun, though as far as I am aware no theoretical explanation has ever been given as to why the distances should be so related. The formula is usually attributed to Johann Elert Bode (1747-1826) who wrote about it in 1772, though apparently a form of it was proposed in 1766 by Johann Daniel Titius (1729-1796), whose name is also spelt Tietz.

The speaker presented the Law in the following form, which is the form given, under Bode, in a copy I happen to have of The New Standard Encyclopedia 1933, which reads: “The number 3 is doubled and then redoubled thus: 0, 3, 6, 12, 24, 48, 96, and so on, and 4 is added to each. The resulting numbers, 4, 7, 10, 16, etc., show the proportionate distances of the planets from the sun.” Dividing throughout by 10 gives the distances in astronomical units (i.e. in terms of the distance of the Earth from the Sun). In the discussion at the end of the talk I pointed out that the first term in the “doubling” sequence should be 1.5 and not 0, and that this would result in an anomalous value of 0.55 instead of 0.4 for the orbit of Mercury. However the speaker declined to agree with my point.

Later in the month I was reading Scientific Method; An historical and philosophical introduction by Barry Gower (Routledge 1997) in which, on page 175, in passing, the Titius–Bode Law is presented (in slightly different notation) in the form: d(n) = 0.4 + [0.3 × 2^(n–2)], where d(n) is the distance, in AU, of the nth planet from the Sun (the asteroids counting as the fifth planet). This mathematical expression of the formula thus agrees with my contention that the series should begin with d(1) = 0.55. The following table gives the true and calculated values for the distances. I calculated the true values from the mean distances expressed in millions of kilometres as given in Patrick Moore's Atlas of the Universe.

 Planet Mercury Venus Earth Mars Ceres* Jupiter Saturn Uranus Neptune Pluto M 0.387 0.723 1.000 1.523 2.666 5.200 9.539 19.18 30.06 39.44 n 1 2 3 4 5 6 7 8 9 10 B 0.55 0.70 1.00 1.60 2.80 5.20 10.0 19.6 38.8 77.2
M = Mean distance in AU. B = Bode's Law value. *The distance of Ceres, the largest asteroid, is given by Moore as varying from 2.55 to 2.77 AU. Other asteroid orbits vary widely, thus 2.666 is just a crude average.

Looking again at the above figures after a break it seemed to me that the distances of the inner planets obey an arithmetical progression and that the geometrical doubling only sets in beyond Mars. Accordingly I tried the following formula:
d(n) = (n + 2^(n-3))/4. This gives the series:

 n 0 1 2 3 4 5 6 7 8 9 10 d(n) 0.03 0.31 0.62 1 1.5 2.25 3.5 5.75 10 18.25 34.5

This gets reasonably close (±20%) to the planetary distances provided we count Jupiter as the seventh planet so that the asteroids are regarded as taking the place of two planets, 5 and 6. The distance 34.5 AU falls between the mean distances of Neptune and Pluto, suggesting that they should be regarded as in the same slot. [Could Pluto be an escaped moon of Neptune? — it is only a very small body, 0.002 Earth mass, smaller than our Moon which is 0.012 Earth mass.] If there is another planet beyond Neptune and Pluto it should be at distance 66.75 AU according to my formula. If the 0 value represents the surface of the Sun (rather than the mythical planet Vulcan!) d(0) should be about 0.005.

There are many features of the planets that are anomalous and seem to require some sort of catastrophe as explanation. For example, their inclinations to the ecliptic vary enormously; they are, in degrees: Mercury 2, Venus –2, Earth 23.5, Mars 24, Jupiter 3, Saturn 26.5, Uranus –82, Neptune 29, Pluto –58. (All figures from Moore, p.110, except that I have replaced figures greater than 90 by negative values obtained by subtraction from 180. The negative sign indicates counter-rotation relative to the Earth.) Their periods of rotation are also very variable; in days: Mercury 58.646, Venus 243.16 (longer than its year!), Earth 1.000, Mars 1.026, Jupiter 0.413, Saturn 0.426, Uranus 0.717, Neptune 0.672, Pluto 6.39. (Figures derived from those given by Moore in terms of days, hours, minutes and seconds.)
In terms of inclination Uranus and Venus are odd, while in terms of period of rotation Mercury and Venus are odd. Patrick Moore writes on Uranus (p.110): “The reason for this exceptional tilt is not known. It is often thought that at an early stage in its evolution Uranus was hit by a massive body, and literally knocked sideways. This does not sound very likely, but it is hard to think of anything better.” And on Venus (p.66) he writes almost identically: “The reason for this retrograde motion is not known. According to one theory, Venus was hit by a massive body early in its evolution and literally knocked over. This does not sound very plausible, but it is not easy to think of anything better.” Concerning the Moon (p.40) he has: “It may be that the Earth and Moon were formed together from the solar nebula, but there is increasing support for the idea that the origin of the Moon was due to a collision between the Earth and a large wandering body, ... ”
If one collision is implausible are two or three more implausible or more likely? Could it be that there were indeed originally two planets where the asteroids now are? Their catastrophic collision perhaps creating the asteroids, some of the satellites, and sending out fragments in all directions, some perhaps quite large, to account for all the other anomalies? This theory, which is probably just science fiction, reminds one of the theory of Immanuel Velikovsky (1895-1979), expounded in Worlds in Collision (1950) and other books, which envisaged a realignment of the inner planets (he seems to have had Venus originating from Jupiter somehow). However, he put this realignment in historical rather than geological or cosmological time which is implausible.

Speaking of cosmological time. Back in issue 15 p.268 Professor Cranium queried the accuracy of the figure of 15,000 million years cited as the age of the Universe since the Big Bang by various authorities. He pointed out that for any quantity to have scientific credibility there must be an indication of the degree of accuracy claimed for it. Recently I heard a more accurate statement for this age, on a radio programme. The speaker was Martin Gorst and his figure was 13.4 ±1.6 billion years. This gives a range of possible values from 11.8 to 15.0 billion years.

## é(25) Approximation to Ratios by Continued Fractions

In the section on the Thirteenth Month I make use of a procedure involving continued fractions, which I believe should be much better known. The following account is a shortened version of an article that I wrote for the last issue of the Dozenal Journal (1992), published by the Dozenal Society of Great Britain; the first section also appeared in the Games and Puzzles Journal, issue 4, p.57 (1988).

The ratio s = a/b of the shorter to the longer side of a non-square rectangle can be regarded as expressing its 'shape' or 'squareness' or 'proportions'. The rational number s also expresses the ratio of the area of the rectangle to the area of the smallest square containing it, also the ratio of the area of a largest square contained in it to the area of the rectangle. In other terms: s = a/b = (a×a)/(a×b) = (a×b)/(b×b).

Any non-square rectangle can be divided up into squares by the process of removing from one end of it a square of the largest possible size, then applying the same procedure to the smaller rectangle that results. If the number of squares removed in this way, of the first, second, third etc. sizes are m(1), m(2), m(3), ... what is the ratio s expressed in terms of these numbers? The answer is the continued fraction: 1/(m(1) + 1/(m(2) + 1/(m(3) + ...))). This can be proved by repeated application of the relation s = 1/[m(1) + s(1)] where s(1) is the shape of the rectangle left when m(1) squares a×a are removed.

In practice the dissection process always terminates after a finite number of steps, because the squares get smaller and smaller until they go beyond the accuracy of our measuring capabilities (or beyond the physical limits of observability).

A ratio (rational number) s = a/b can always be expressed uniquely in the form p/q where p and q are whole numbers with no common factor. If we calculate the continued fraction for p/q up to the rth term we get a series of ratios p(1)/q(1), p(2)/q(2), ..., p(r)/q(r) which are convergents to p/q. These are alternately larger and smaller than p/q and get closer and closer to p/q. Moreover p(r)/q(r) is the best approximation possible to p/q with denominator q(r) or less. Any convergent can be calculated from the previous two convergents and the next term m(r+1) by the relations: p(r+1) = m(r+1)p(r) + p(r1) and q(r+1) = m(r+1)q(r) + q(r1).

If we have a ratio s expressed as a decimal fraction, say e = 2.7182818, that we wish to express as a fraction we follow the repeated procedure, on an electronic calculator, of noting and removing its whole-number part and then finding the reciprocal of the remainder. Thus we get, in this example, the whole number parts 2, 1, 2, 1, 1, 4, 1, 1, 6, ... which indicate the continued fraction 2 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + 1/(4 + 1/(1 + 1/(1 + 1/(6 + ... which gives the convergents: 2, 3, 8/3, 11/8, 19/7, 87/32, 106/39, 193/71, 1264/465, ...Of course the further we go the more inaccuracies will mount up, due to working within a limited number of decimal places and inverting small numbers.

Practical applications of this procedure occur where we wish to find compatible units for measuring a quantity according to two disparate standards. The Month and Year discussion leading to the Metonic cycle was one application. The following are further examples.

Years and Days. As noted earlier, the mean tropical year = 365.24219 days. As a continued fraction we find 365 + 1/(4 + 1/(7 + 1/(1 + ... with the convergents 365, 1461/4, 10592/29, 12053/33, 46751/128. The last denominator 128 = 2^7. And 46751/128 = 365 + 31/128. Thus in an accurate calendar we should insert an extra 31 days every 128 years, i.e. 1 leap day every 4 years except on one occasion. This also suggests that we should count historical time in periods of 2^7 years rather than centuries.

The Hammond Temperature Scale. The freezing and boiling points of water are at 273.16 and 373.16 on the Kelvin temperature scale, which uses the centigrade degree (i.e. 100 degrees between the two fixed points). Consider the convergents to 273.16/373.16 = 0.7320184; the whole number parts are: 0, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, ... giving convergents: 0, 1, 2/3, 3/4, 8/11, 11/15, 30/41, 41/56, 71/97, 112/153, 519/709, ... The value 41/56, correct to three decimal places, is attractive since 56 41 = 15 and multiplying by 12 gives a freezing point of 492 and a boiling point of 672 with a difference of 180, which of course is the Fahrenheit degree! This is the scale proposed by Donald Hammond (Dozenal Journal issue 1 p.39; except of course that he expressed the numbers in base 12, in which 492 and 672 become the rounder figures 350 and 480).

Approximating Pi. We find: p = 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ... with convergents: 3, 22/7, 333/106, 355/113, 103993/33102, ... The value 22/7 was known to Archimedes; the value 355/113 was obtained by a Chinese astronomer Tsu Ch'ung-chih c.450AD.

Degrees and Radians. These convergents to pi enable us to work out the best units in which the degree and radian can both be expressed in whole numbers. An answer is radian/13751 = degree/240. This unit is simply the angle of rotation of the Earth in one second of time, equal to 15 seconds of arc.

On the [new] DSGB website there is an interesting article by Nigel Corrigan on The 10,000 Year Old Inch in which he defines a nautical mile as a minute-of-arc = 1/(360 × 60) = 1/21600 of the equatorial circumference, and a cubit as the rotation of the equatorial circumference in a millisecond-of-time = 1/(24 × 60 × 60 × 1000) = 1/86400000 of the equatorial circumference, so that nautical mile = 4000 cubits. The above unit equal to 15 seconds of arc is thus equivalent at the equator to 1000 cubits or a quarter nautical mile.

Corrigan argues from other evidence that cubit = 18.24 inch so that nautical mile = 6080 feet (assuming 12 inches to a foot). According to the value for the equatorial circumference, 24902.4 (statute) miles, cited by the CRC Standard Mathematical Tables 17th edition 1969 (the most up-to-date reference to hand!) one minute of arc is 6087.2533 (modern) feet, i.e. (24902.4 × 5280)/21600. This would give: cubit (as defined above) = 18.26176 (modern) inch. Does the fact that 6080 of today's feet makes a "UK Admiralty nautical mile" derive from the above considerations, or is it a numerical coincidence, since presumably this unit is based on a minute of arc along a geodesic in UK regional waters, not along the equator?

Another relationship mentioned at the end of the article, accurately expressed is: 180/(p^2) = 18.237813 which is close to 18.24, but this seems to be merely a numerical coincidence, and no reason why a unit of length should be based on this particular constant.

Sections on this page: (21) Equidirectional Knight's Tour. (22) Thirteenth Month and Zodiac Sign. (23) A Space Compass for the Constellations. (24) Bode's Law Revisited. (25) Approximation to Ratios by Continued Fractions. End
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