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davidk 
Posted: Oct 30 2017, 10:08 AM

Casual Member Group: Members Posts: 132 Member No.: 1,157 Joined: 25October 15 
The design of the first phase of Stonehenge circa 3200 BCE.
It appears to be based on 32 periods of 91 days or 8 years of 364 days and can be represented by the number 261381120. This number suggests that base 2 and other primes from 317 are included in the design as follows. http://numbermatics.com/n/261381120/ Interestingly the system incorporates the bases 99, 100, 101, and 102. The evidence is included in my latest book which can be reached via a recent publication by the archaeologist Bill Wilkinson, the book is linked via other purchases as well as Bill’s book. https://www.amazon.co.uk/DartmoorsStoneCi...=bill+wilkinson 
Double sharp 
Posted: Oct 30 2017, 10:53 AM

Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
Surely this is more relevant to a thread about Stonehenge than a thread about number bases in general, not just the ones you speculate to have been used there?

davidk 
Posted: Oct 30 2017, 04:12 PM

Casual Member Group: Members Posts: 132 Member No.: 1,157 Joined: 25October 15 
It depends on how persuasive the evidence supporting the speculation is possibly.

Shaun 
Posted: Oct 30 2017, 04:45 PM

Dozens Disciple Group: Admin Posts: 1,115 Member No.: 3 Joined: 2August 05 
This did not belong in a topic devoted to basic Number Theory, and I have therefore created this new subforum.

davidk 
Posted: Oct 30 2017, 04:46 PM

Casual Member Group: Members Posts: 132 Member No.: 1,157 Joined: 25October 15 
wow
thanks shaun so impressed by the clarity of thinking on this website. Stonehenge is possibly about Pi and the manipulation of it. Fibonacci Pi possibly Jim posted on this but i have lost it. What was it Jim? 
davidk 
Posted: Oct 30 2017, 05:20 PM

Casual Member Group: Members Posts: 132 Member No.: 1,157 Joined: 25October 15 
The message form our ancestors is that they were possibly being playful and laughing at us when they designed the imperial system.
the imperial system is a base 99 system with a base 100 unit lying on top of it. The imperial foot. It is out of place 11.88 inches defines the imperial system not 12 inches. 5280/0.99 feet = 5333.3333r recurring infinite this is the link https://drive.google.com/file/d/0Bwx4SWhy8p...3RuLXUzRkE/view 
icarus 
Posted: Oct 30 2017, 11:23 PM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
Look I think it's fair. You may speculate here to your heart's content.

davidk 
Posted: Oct 31 2017, 03:53 PM

Casual Member Group: Members Posts: 132 Member No.: 1,157 Joined: 25October 15 
This is the design of the Stonehenge Aubrey circle
285.60 x 22/7 = 897.60. 22/7 is not Pi because it is a repeating infinite number that can be expressed as a fraction, it is a larger number than Pi. Pi cannot be expressed in this way because it is irrational. So how does the creator of this circumference 897.60 show that it contains real pi as any circle must do? The solution appears to have been to create another version/approximation of Pi of a lower value than actual Pi, and I have already posted on the origin of this version as 1632/520. 897.6 / 1632/520 = 286. So using two diameters of 285.60 feet and 286 feet a difference of 0.4 of a foot or 4.8 inches in the diameter produces two circles of exactly the same circumference 897.60. Real Pi lies within the 0.4 scale. Real pi is 3.1415927 to seven significant figures. Dividing by Thom’s megalithic inch of 0.816 imperial inches gives 3.849991 So taking 3.85 gives 0.816 x 3.85 = 3.1416 exactly a very fair approximation of Pi. So the versions of Pi are 22/7(1760/560), 1632/520 and 3.1416. 22/7 – 3.1416 = 0.0012571 22/7 / 0.0012571 = 2500 exactly and 3.1416 divided by 0.0012571 gives 2499. Speculation? This is a very odd and unexpected result. Try it with the other version of Pi. 3.1416  1632/520 = 0.003138 1632/520 / 0.003138 Is 1000 exactly and into 3.1416 gives 1001 exactly. Speculation possibly but once is a shock and twice needs explanation Bill has produced Fibonacci Pi as 3.1418181818r 3.1418181818r – 3.1416 = 0.000218 3.1418181818r / 0.000218 = 14400 This is 12 x 12 x 10 x 10 3 times and the question has to be Why is this happening? What kind of speculation is this? Can anyone explain it please? It could be because as Thom sugessted they wanted everything in whole numbers 897.60 / 22/7 / 1632/520 = 91 exactly. ( one season in days?) http://calendars.wikia.com/wiki/World_Season_Calendar 
davidk 
Posted: Nov 1 2017, 03:44 PM

Casual Member Group: Members Posts: 132 Member No.: 1,157 Joined: 25October 15 
1001 is a really important number at SH
as icarus says Sorry, what I was referring to was merely the 210 > 2310 pattern. The more visible 30030 > 510510 > 9699690 pattern is created by the threeplace arrangement, i.e. 7 * 11 * 13 = 1001. First though the basic circle of 897.60 feet can be broken down into the two versions of pi that is designed against 40.8 x 22 = 897.60 13 x 7 = 91 the two versions illustate just why number 91 results from division by the two versions of Pi multiplied together. 13 and 7 are not included in 897.60. 897.60 / 17 = 52.80 897.60 is derived from two systems one base 17 and the other base 11 and the pi numerators reflect this. So the imperial system is missing base 7 5280 x 7 = 36960 this is made up as follows http://numbermatics.com/n/36960/ Where does 1001 fit in? It is 11 x 91 (base 7 11 and 13) 13 is not yet in the mix so the calculation is missing primes 13 and 17 in the 40.8 / 13 version of Pi. 36960 x 13 = 480480 x 17 = 8168160. Divide by the base unit in Thom's system 816 and get 1001. reconciled as 89760 x 13 x 17 = 8168160 http://numbermatics.com/n/8168160/ 
icarus 
Posted: Nov 2 2017, 09:31 PM

Dozens Demigod Group: Admin Posts: 1,913 Member No.: 50 Joined: 11April 06 
I have marveled at the "missing" sevenness in the Imperial system, and the puzzling presence of larger primes, particularly 11 (as in the mile). It is for this puzzlement that I do value your speculations and thoughts. I don't know the exact thoughtlines on how our predecessors arrived at the units (you all seem to have a much deeper grasp of these) but it is interesting for example that the current definition of "mile" = 2^5 * 3 * 5 * 11 = 3080 (dozenal). Note missing 7. It might've been 4620 feet (decimal) thus 2^2 * 3 * 5 * 7 * 11 = 2810 (dozenal). But our predecessors' considerations suited their needs, and not an abstract want to include the 5 smallest primes.
Not sure many have thought of decimal 1001 = prime(4) prime(5) prime(6), but obviously our forefathers were no less intelligent than we are today and had a lot more time on their hands despite maybe not being as secure (sudden Viking attacks, plagues, famines, etc.) What would you think were the reasons for our predecessors' seeming desire to incorporate these relatively large primes? (7, 11, 13, etc.) 
Double sharp 
Posted: Nov 3 2017, 04:04 AM

Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
There were many units between a foot and a mile which multiplied together to create this 5280, that have since fallen out of use. (I think: the only time I've ever encountered many of these units is when I visited the US. Do people actually use rods, chains, and furlongs anymore?)
As John Baez's exposition notes, the mile was defined to be 80 Gunter's chains, and each of those was thus 66 feet long. Why did Gunter make them 66 feet long? For backward compatibility with an older British unit, the rod; four rods made 66 feet. So what happens is that the long units make sense among themselves: 4 rods per chain, 10 chains per furlong, 8 furlongs per mile. The short ones also make sense: 12 inches per foot, 3 feet per yard. But the clash between the two is horrible, perhaps because distances in different orders of magnitude were being measured by different people, and conversion was rarely needed: there are 5 1/2 yards in a rod. This 5 1/2 = 11/2 is where the factor of 11 comes from. This comes from a very old English standardisation of the year 1270 of 16 1/2 feet in a rod, a rather inconvenient number. (But this may have originally been the much nicer 16; the English rod is very close to 16 Danish or Prussian feet!) So this is a demonstration of what happens when different units accumulate gradually with different orders of magnitude being different occupations' specialties. But there may be some method in this madness: at a certain point, it seems that higher and higher primes become an expected part of factorisation. It is the Achilles heel of purebase thinking: sure, 12 works very well at its scale, but take several powers of it, and we get 12^6 = 2985984, which only resolves {2, 3} (admittedly at great depth) at a scale where you can have numbers like 4324320 resolving {2, 3, 5, 7, 11, 13}. Although one might argue that with so many digits there is ample room to zero in to the 5ths, 7ths, 11ths, and 13ths to more precision than you could ever want, even with a pure power of a human scale base. At this stage, one should also consider that the improvement each successive prime gives to the totient ratio is getting a little negligible. 7ths don't really improve it much more than 11ths or 13ths, whereas the need for 2 is vital. So I get the feeling that the idea was to have flexibility by incorporating many factors into the system and see what came out of people's needs. (I still think metric is much better, mostly for the "so many digits" reason, and because pure bases are just so much easier than mixed bases.) 
Dan 
Posted: Nov 3 2017, 05:15 AM


Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
As far as I'm aware, furlongs are used only for horseracing distances, and rods and chains aren't used at all. Yards are in common use (especially in football), and so are miles, but nothing in between. 

Dan 
Posted: Nov 3 2017, 05:25 AM

Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
Anyhow, I agree that the "weird" conversion factors are trying to fit two incompatible systems together.
Another example of this is the international nautical mile, standardized since 1929 as 1852 meters. Why? Well, conceptually, a nautical mile is meant to be one minute (1/21600) of the earth's circumference, whereas a kilometer is meant to be a centigrad (1/40000). If the units had been standardized at the same time, based on the same survey of the earth's size, then a nautical mile would be exactly 1851.851 851... meters. Rounding this to 1852 meters is close enough for practical purposes. 
Oschkar 
Posted: Nov 3 2017, 06:17 AM

Dozens Disciple Group: Members Posts: 575 Member No.: 623 Joined: 19November 11 
I think I’ve read that the factor of 5.5 has to do with the discrepancy between the Roman units of measurement used in Roman Britain, and the German units used by the AngloSaxon invaders. The rod and larger units were taken from the Saxon system, because cadastral measurements had already been standardized around them, but the yard and smaller units were taken from the Roman system as a compromise, and set at 10/11 of their previous values.

jim 
Posted: Nov 3 2017, 11:19 AM


Regular Group: Members Posts: 165 Member No.: 641 Joined: 20April 12 
'As far as I'm aware, furlongs are used only for horseracing distances, and rods and chains aren't used at all.' My god that is heresy. The cricket pitch 22 yards. 1 chain or 4 rods. The Ashes begin next month fork out and buy fox sport. Jim 

davidk 
Posted: Nov 5 2017, 10:31 AM

Casual Member Group: Members Posts: 132 Member No.: 1,157 Joined: 25October 15 
i always think of Don Bradman when the chain is mentioned because i feel for sure that when he was scored in test matches natural human errors were made and snicks for 4 were recorded as byes. So to me i know he recorded an average of 100 in tests even though the scorebook does not.
i played and umpired cricket matches into my sixties and laid out many a good and bad wicket with my teammates on Saturday mornings for Sunday afternoon games. It has to be exactly 22 yards nothing else will do or ever replace it, or the sound of leather on willow, and nothing was more satisfying than watching Dennis Lillee throw his aluminium cricket bat away in anger. 22 does not have the missing number 7, the imperial system does. Number 10 or 100 is built into the imperial system and the system Thom discovered and the Scotish and Welsh systems of ancient measure. The objective of my books is to reraise the profile of Alexander Thom's work in the 1960's and 1970's and also change our understanding of how the imperial system works. Thom concluded that the circle builders wanted to predict eclipses and used a unit of 2.72 feet to help them do this. He also said they wanted units of 2.5 in their circumferences but this is actually 2.5 x 4. If Thom is correct our understanding of pre history in the UK is woefully wrong and i have written a book called 'Proving the Megalithic Yard' about this. The book is more a series of working papers but the conclusion that the megalithic yard exists is founded on a very simple proof. The imperial system x 34/33 or x 102/99 is the system discovered by Thom. So the imperial was around when the circles were built, circa 5000 tears ago. I know what it is to live in a dozenal society and 12 pennies in the shilling and 12 inches in the foot were standard. This was ripped from our grasp by political sleight of hand amid absolute uproar at the time. Bookmakers in my home town were still illegally using pennies six years after decimalisation. To me one penny is still 2.4 real pennies and always will be. Thom said they predicted eclipses and base 10 or 100 or 1000 etc were universally sacred units. Thom's work is so complex that i have only been able to study eclipses and the numerical aspect of his discoveries. But i feel that this has been enough to prove his work beyond any doubt because the criteia is much higher than beyond reasonable doubt because of the nature of the investigation. Furlongs come into the proof. 8 furlongs is 5280 feet one mile 10 furlongs is 6600 feet a base 10 system so according to Thom universally sacred. This can be converted using 34/33 to give the equivalent Thom unit of 6800 feet. The nodal cycle of the moon is 6800 days. Thom's second book was named ' Megalithic Lunar Observatories' which just as a title is quite mind blowing and that is obviously the intention. So the chain is 66 feet and this is 100 links and the chain as 22 yards is a circle based on Pi as 22/7 it could not be more simple. 100 links is the giveaway and the post on this thread identifying that Gunter nicked his system. So what about 68 feet in Thom's system ? You guessed it it represents 100 units of 8.16 inches and this is 10 of Thom's megalithic inches of 0.816 imperial inches. So as an imperial system Thom's represents the nodal cycle of the Moon and as a megalithic system represents 1000 units of a circumference. So the proof is not difficult to understand but requires a 'quantum leap' in our understanding of our ancestors capabilities. The real 'proof' however is in restating the imperial system as a system that predicts eclipses and to do this the familiar number of 360 is required to restate the mile circle circumference 5280 / 360 = 14.666r and this i have named the eclipse unit. Using pi as 22/7 it is a base 10 unit as follows: 14.666r / 22 x 7 = 4.666r The circumference minus the diameter is base 10 and this only works using the imperial system. Then we need to go back to 10 furlongs 6600 feet and subtract this base 10 number of 14.666r to give 6585.3333r. This is a NASA accurate calculation of the Saros cycle. This is the only number that needs to be known to predict eclipses, but the nodal cycle can predict warning signs. At Stonehenge these two systems are married together in the Aubrey circle circumference and the base unit is 0.8976 feet. This gives a circle circumference of 100000 units. The mile is 5280 feet and Thom's equivalent is 5440 feet and divided by 2.72 gives 2000 units so obviously 5440 changes to 2720. So the two units working together at the aubrey are 5280 and 2720 added together to give 8000 feet exactly. this is where the 8 to 10 Furlong relationship comes into play The units become 6600 and 3400 making 10000. Everything resolves to base 10 as Thom told us. Finally where does the unit 0.8976 come from? It is derived from base 7. Pi as 3.1416 is 0.8976 x 3.5 = 3.1416. Both versions of Pi as 22/7 and as 3.1416 are in Thom's first book on page 47 identifying 3.1416 being used at Avebury and Brodgar and 22/7 at the Aubrey circle. 22/7  3.1416 = 0.001257143 This gives an exact ratio between the two versions of Pi as 996 to 1000. So the system being used at the Aubrey is 1760/560 x 1632/520 x 91 = 89760. The numerator is the saros cycle x the nodal cycle and the denominator is lunar 56 times solar 52. This represents one day and is multiplied by 91 to give one season. What i have been looking for is evidence they knew 29.53 and Jim Wakefield has produced this by using the imperial system from Saxon times. The ratio between Pi as 1632 / 520 to 3.1416 is 1000 to 1001. If this is not enough proof the eclipse line that runs from the centre of Old Sarum to Brodgar where Jim has recently visited cuts through the Stonehenge Greater Cursus, the West Kennet Long Barrow the centre of the Avebury Ring and the centre of the Brodgar Ring.An incredible five key megalithic sites in an exact line only visible by using Google earth cannot be by chance. The burden of proof is very heavy but the evidence is rock solid. It is important to understand that number 52 is soli  lunar, 52 weeks is solar but 520 days is lunar as 520/3 x 2 = 346.6666r days a lunar year. 56 is 8 x 7 solar weeks and the same number of lunar weeks. The numbers 89760, 52 and 56 are all evidenced at the Aubrey circle. 89760 x 56 x 52 = 261381120 http://numbermatics.com/n/261381120/ Double sharp said 'There were many units between a foot and a mile which multiplied together to create this 5280, that have since fallen out of use. (I think: the only time I've ever encountered many of these units is when I visited the US. Do people actually use rods, chains, and furlongs anymore?)' The US sent men to the moon using the imperial system. NASA refuse to use anything else. Can we have it back please? 
Double sharp 
Posted: Nov 5 2017, 11:48 AM


Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
Well, the computers used for Apollo used metric internally (source). It was just that they converted everything, so the input and output were displayed in US customary units. (Also, doesn't NASA use SI now? I remember hearing about that a while ago, but may be misremembering. It would certainly be interesting trying to collaborate on projects like Cassini with the ESA without using metric...) I do not think you will ever get them back, mostly because the younger generation in most countries that have metricated have not heard all that much of the old units. The only time I think in feet and inches is when converting for someone who is not about to understand metres. And the same for pounds and ounces versus kilograms, and for Fahrenheit versus Celsius. And my mental estimates for length, mass, and temperature are all rounded to metric, because I never grew up having to use anything else. So I think the resistance on both sides comes more from a resistance to change than anything else. If you are just thinking of lengths to measure heights and lengths to drive, I don't think there's much of a difference between the systems. The difference comes mostly in science, which are mostly the advantages of a mostlycoherent system using the customary number base. Something like Primel succeeds in being a "better metric" than metric, and I may even dare to say that the majority of it comes from its fuller coherence than even SI (everything is related without even factors of the base's powers), rather than it being dozenal (although that is a factor too). 

davidk 
Posted: Nov 5 2017, 01:53 PM

Casual Member Group: Members Posts: 132 Member No.: 1,157 Joined: 25October 15 
I am inclined to agree with you but perhaps the imperial system could be re adopted but not used, just taught in schools perhaps as maths/history. The USA is a massive exception to the rule.
the UK systems appear to have evolved from a root base 100 system that is very simple. 1x2x2x2x2x2x 1.25 x 2 x 1.25 = 100 Imperial uses the rod as 16.5 to give 16.5, 33, 66, 132, 264, 528,660.1320, 1650 Thom uses 17 to give 17,34,68,136,272,544,680,1360. 1700. These are the 2 systems locked together at the Aubrey Circle Stonehenge 17 x 5280 = 89760. 
Double sharp 
Posted: Nov 5 2017, 02:12 PM

Dozens Disciple Group: Members Posts: 1,401 Member No.: 1,150 Joined: 19September 15 
I did learn of some premetric measures as mathematical history, but they were not covered in much detail, since we would never have to use them; just a brief mention of their existence and a few of the most common ones that we might need to understand when dealing with the USA and some historical texts. All I really remember was feet and inches, pounds and ounces, and degrees Fahrenheit, and a few traditional Chinese and Japanese units (mostly namedropping)  many are shared anyway due to the immense Chinese influence on Japan. The idea that the US customary inch was not the only inch historically was unknown to me for a good while, actually.
It might be of use to cover them officially, though, since the conversion is a practice of arithmetic and teaches the need to avoid false precision. Regarding the Chinese units, I should note that they already went a significant distance even traditionally towards metric thinking: traditional Chinese units were already decimal, with ten cùn (寸) to one chǐ (尺), and ten chǐ to one zhàng (丈). Only, when the Nationalist Government standardised the length units to give metric equivalents in 1930, they set one chǐ equal to that awkward fraction: one third of a metre! 
davidk 
Posted: Nov 5 2017, 07:58 PM

Casual Member Group: Members Posts: 132 Member No.: 1,157 Joined: 25October 15 
That is marvellous
i have speculated that to dozenalise the metre 39.60 units has to represent the metre. And the Chinese go for 1/3 of this 1320 10x 11 x 12 this is Jim Wakefield's unit 
Dan 
Posted: Nov 5 2017, 11:49 PM


Dozens Disciple Group: Members Posts: 1,463 Member No.: 19 Joined: 8August 05 
Well, cricket isn't very wellknown here in Texas. Do people actually think of the length of a cricket pitch as being "1 chain", or do they call it "22 yards" or "66 feet" without knowing why this figure was chosen? 

davidk 
Posted: Nov 6 2017, 08:57 AM

Casual Member Group: Members Posts: 132 Member No.: 1,157 Joined: 25October 15 
Hi Dan
depends how old you are, born in the fifties chains were used to mark out sports pitches. At school we ran the 110 yards the 220 yards the quarter mile 440 yards the half mile 880 yards and famously Rodger Bannister was the first man to break the 4 minute mile 1760 yards 5280 feet. Multiply this by 17 to get 89760 the unit used at Stonehenge. Everytime i go to the USA two things hit me. One when you go for gas ( petrol in UK) it is gallons only. Two no kilometers It is great to be in the USA. 
Shaun 
Posted: Nov 6 2017, 03:40 PM


Dozens Disciple Group: Admin Posts: 1,115 Member No.: 3 Joined: 2August 05 
I've always thought of it as "22 yards" rather than a chain. 

Shaun 
Posted: Nov 6 2017, 03:42 PM


Dozens Disciple Group: Admin Posts: 1,115 Member No.: 3 Joined: 2August 05 
Being slightly older than most of you I can remember running the 100 yards at school ... 

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