Prime Numbers

Because 7 is just a good friendly number (despite all those nasty rumors).

This response was brought to you today by the letters "R", "V", and "Z" and the numbers "1" and "7".

Japanese bloke I worked with not long after the introduction of the 50p coin in the UK asked similar questions. "How do they get the angles correct - there's no geometric construction ...?" and "Why ...?"

There's a weird math story about that coin. Now if I could only remember... it's something like those weird triangular manhole covers (in San Francisco, I think). I'll search for my memory neuron (I think I saw it at the bass fishing tournament being used as bait, so..) and will get back to you, provided I remember to do that.

That's all very well, though most other coins, apart from the 20p piece, are circular.

It must waste a lot of metal at the stamping operation.

I once asked someone about the recurring numbers of 7 and 12 in the bible. He gave me an answer that really made since (to me at least).

When you think about the first few numbers, 1 and 2 cannot support a structure. Numbers 3 and 4 have long been considered the first two stable numbers (a triangle or rectangle base). These numbers can support structures. 3+4=7. 3*4=12. So 7 and 12 were considered perfect numbers. If I remember right, mankind was typically represented with a number 6 for imperfection.

I'm not trying to bring religion (I'm sure there's similar reasoning for all religions) into this thread but if this is true then it would mankind's fixation on the number 7.

This phenomenon is espontaneously reproduced by bees while constructing honeycombs, which lead "scientists" to believe that bees had some type of "sense of geometry"

L.O.L... I mean, Yahlasit

"Heck, I have problems understanding how ... all the ... Gothic masterpieces were built in the 12th Century without cranes!" - Should've asked Tricky Dicky - sure he'd've pointed you in the wrong direction!

OK, so why do egg boxes contain 6 eggs?

Because that's half a dozen, of course

Oh, alright, it's because [traditionally] one doesn't eat breakfast on Sundays, so there was a need for 6*e eggs per person each week (where e = number of eggs eaten in each breakfast).

Either that or it makes a convenient sized box. Five (5) would be a silly number of a box...as I said earlier, it's a nasty number! Seven would mean one left over to go off.

Re: "...why do egg boxes contain 6 eggs?" ...

'round these parts, it would ONLY be because somebody already consumed either 6 or 12 of them. All our grocery shelves are loaded with boxes of either a dozen or 18!

there is some more to relate this problem to other prime numbers and even to the periodic table and the structure of condensed matter .. try to google the title 'Primes, Geometry and Condensed Matter'.

yup ... with 6 or 8 circles you can't make the circles touching each other as you make with 7 circles

You forgot to put the basic condition that all circles should be of equal diameter.

(yep... I copy and pasted that right from Wiki.)

You forgot to copy from somewhere else that there are only 7 basic crystaline configurations.

Yahlasit

Seven days?

Sorry, I disagree with all those who have given you GA.

Your long list is not an answer to original question.

be surprise where the answer is hidden.......it may not be where you think.

The original question as I read it was about the prime number 7, and the gear thing was just an example and therefore all things relating to the prime number 7 is an answer.. although the OP was not very specific in his inquiry. Perhaps he could be asked to clarify?

Chris

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you brought up a point, I would like to add. iwhen I was in college we had a class in spc and was cover a bit about Demming.

In which it was brought up when you are document processes say on a chart, at the time it was said the which things were going up rarely did it go up 7 times in a row, or down 7 times in a row. and I believe it was the number 7 anyways. We were discussing this and looking at ways to apply it. Flipping a coin, rolling of dice.

An example, if you were to flip a coin, it would not get 7 heads in a row.

I suggested 21 (blackjack) later I found out that this was a proceedure called progressive gambling.

I did a quick google search and looked for with this using number 7, but did not com up with a definitive answer but this one was close on what we talked about called the Bernoulli Process.

http://books.google.com/books?id=IuS3o5Eq5rgC&pg=PA253&lpg=PA253&dq=statistic+process+control+and+the+number+7&source=bl&ots=ONwclNpDcE&sig=kvq1CZlgOIACmtm_xOb_AMro-rI&hl=en&ei=YWKASqqDGMOGtgfii_jkAQ&sa=X&oi=book_result&ct=result&resnum=1#v=onepage&q=&f=false

phoenix911

just noticed.....black jack (21) 7 goes into 21,,,3 times.

Could this just be simuliar to 6 degrees of separation.....wait 6 degrees that one less than 7.

get my point....

Could this just be similiar to 6 degrees of separation.....wait 6 degrees that one less than 7.

Would that be Six Degrees of Bergerac ???

What if you start counting from zero?

But you do! It's the links that counted, so if you're first person is counted as zero, the the link between him/her and the next person is link one. boing!

This of course has everything to do with the Illuminati and the Masons.

Don't forget Peter, Janet, Jack, Colin, George, Pam and Barbara.

The sheriff's badge can be 5, 6, or 7 points. But 7 is magic; it comes from the old Celtic: the Sun, the Forest, the Sea, Magick, the Moon, the Wind and Spirit. See, Shire Reifs used to ride around in the dark forests where there were all manners of goblins and witches and they used Celtic magic against the haints.

Actually, you can sorta get gears to work out if you use an epicycle planetary drive. You can get all kinds of weird ratios.

Otherwise, I'd bet on the Book of Revelations for the popularity of the number 7.

Seven days to the week, 7 branches on the menorah. Seven is a very symbolic number in Judeo-Christian tradition.

1 + 6 = 7

One represents God, 6 represents man.

2 + 5 = 7

Two represents witness, 5 represents grace.

3 + 4 = 7

Three is another number associated with God, 4 is associated with the Gospel.

Seven is also the largest single digit prime. Pi is roughly 22/7.

Any explanation why 2 represents witness and 5; grace or indeed one for God and six for man. How about 4 for elephant and three for diode?

Not sure if the ancient Hebrews knew about elephants...pretty sure they didn't have diodes

1 = God: presumably because there's one one (Abrahamic) God

6 = Mankind: someone else said imperfection, perhaps because we surround God (back to the 7 circles), most probably just to make the sums work! Also the day on which Mankind was created according to Genesis.

As to the rest, the number associations in themselves aren't important, the list as a whole acts as a form of mnemonic - vital in an illiterate culture.

Have you never listened to Deck of Cards?

You quoted the examples, but did not really gave the reason behind odd gear ratios. Anti resonance... I am not aware or sure.

But simple reason to have odd gear ratios is to minimise the contact of same gear teeth again and again. This protects gears from damage of particular gears due to any possible hard spots on some singular gear.

I am sure that in the catalog literature you can find many examples of sprockets, gears, screw compressor rotors, and the like with prime numbers of teeth or lobes. One advantage is that on successive revolutions, the same teeth or lobes do not always mesh with the same teeth or lobes. This spreads out wear and can reduce vibration and drive-chain whip. Also, such gear ratios are useful in lathe cutting of screw threads, for instance. (Or if not prime numbers, pairs of numbers that are relatively prime to each other.)

It is true that for most primes, there is no (and certainly no easy) geometric construction. However, CAD programs can accurately do these subdivisions. This certainly beats using dividers to lay out 97 pins in a circle to engage with a 1" pitch 16-tooth sprocket, say.

"CAD programs can accurately do these subdivisions." I tried a layout on Autocad and using the polar array command and 7 divisions within a 360° circle, came up with an even numbered angular value as long as the number of decimal points accuracy was set at 1. When I set the accuracy to 8 decimal points, I got 51.42857143°.

" ... came up with an even numbered angular value as long as the number of decimal points accuracy was set at 1."

Don't understand. Are you talking about the dimensioning accuracy?

Are you talking about the dimensioning accuracy?

The accuracy is not the point. How for example would you layout 51.4285143°? You could round down to 51.43 or even 51.4, but it would not be exact.

How do you know when you layout 10.0 deg its exact?

Ah! - I see what you're getting at. That was the sort of thing that my Japanese friend was fretting about (see #2). [When you said "even numbered", I assumed you meant divisible by 2 without remainder].

You can choose almost any angle you like between two lines - whether or not the numeric representation is an integral number of degrees - and it can be (theoretically) made exactly provided that there's a geometric construction that will give that angle.

If you try to make something with two lines at an angle which cannot be geometrically constructed, then (theoretically, again) it can never be made exactly - there will always be a more precise result.

(I've generalized here a bit to any initial angle to be subdivided - not just 360°. If the angle between the two lines was 360 ÷ (4 x √2), which is "about" 63.639610306789277196075992589436°, it can still be bisected exactly - never mind how many decimal places are used).

_{^{<keeping my head down now coz it's late and I don't want to wait for the ricochet>}}

Ah! Now I see what you meant in the early post - and apologise for not getting it then. WRT coins and machines, I think the level of accuracy required is low enough that it doesn't become an issue.

_{^{to wait for the ricochet>}}

I can't see the site, but I presume you're dodging flying lead and tryong to lead us to considering why time is divided into base 60 divisions (that's them Aztecs) Sweet!

"...why time is divided into base 60 divisions (that's them Aztecs)..."

The Babylonians & Sumerians used a base-60 number system. The Aztecs, Mayans, Celts and Basques used a base-20 system.

Did you google 7 ?

The properties of 7 looks to be unique as a prime number. It is a member of too many groups of prime numbers

http://en.wikipedia.org/wiki/7_(number)

• Seven is the fourth prime number. It is not only a Mersenne prime (since 2³ − 1 = 7) but also a double Mersenne prime since it is itself the exponent for another Mersenne prime (127). It is also a Newman-Shanks-Williams prime, a Woodall prime, a factorial prime, a lucky prime, a happy number and a safe prime. http://en.wikipedia.org/wiki/Mersenne_prime Quite a few of these look to be very very interesting. I am not sure when they took up seven, then they thought about these or simply by trial and error they 7 did fit? BTW - there are a few numbers in the common subset, but 7 is the first one. So may be as you start trial and error, the 7 being the smallest stuck. Another interesting part from wiki
• A seven-sided shape is a heptagon. The regular n-gons for n ≤ 6 can be constructed by compass and straightedge alone, but the regular heptagon cannot. Figurate numbers representing heptagons (including seven) are called heptagonal numbers. Seven is also a centered hexagonal number.

Reducing gear wear is the best answer given.

More generally, prime numbers are regarded as the building blocks of all numbers. Elemental freaks that pop up without pattern for ever and ever and ever.

Sorry, Ron, but I don't see anything extra special about 7.

5 - now there's a number to be reckoned with !

Hey Kris. Now your 2nd pic is what I would call a PRIME #.

5 is a very nasty number. 4 is much more cuddly.

2 and 3 are too close together and too similar for my liking.

Synesthesia anyone?

Oooooooooh, yes!

Cuisenaire rods!

Nope, never encountered them...but was taught lots of maths before I started school, so would have ignored them anyway, in that I would have found they took longer to use than I would to work it out. I did that a lot in maths...

I did love using the SRA boxes at junior (7* - 11 yo) school - yes even in the UK. Stupid Education Departments for dropping them

.

*See still on-topic!

I think many of you have missed the point. Let me present it in an example. Most lug arrangements on automobile wheels are a 4, 5, 6 or 8 lug pattern. I have never seen a 7 lug pattern nor has anyone else. There must be a reason why you would not choose 7. Truck wheels have 10 and 12 lug patterns, but never 11. Lug patterns are always a number divided into 360 that comes up even.

Maybe it's even numbers so you can tighten opposite pairs without forgetting any. 5 is OK, because it's easy enough not to forget, but a larger odd number might lead a person to miss some. 5 is readily seen, but 7 is not as readily noted by the brain ? On the other hand, it might be superstition.

Actually some Ford vans and light duty F250's use a 7 lug pattern

Not so for the 1980 VW Passat, which had 7 lugs to hold its hub caps on.

Hmm... I will have to find my 7 lug wheel hub I found in the scrap yard then and post a picture! Once I figure out how to do that.

I kept it just because its a unusual odd numbered lug system. Good bet winner too!

I checked with a wheel company and a 7-lug pattern was used on heavy duty Ford F-150's and F-250's around 2000-2002. I still don't understand why they didn't go to an 8-lug pattern like the more recent trucks use. Was it cost consideration (1 less stud/lug per wheel)? Granted, a 7-lug pattern can be lay-ed out pretty accurately, but never perfectly (as an even divisor into 360), but why use 7 which complicates a normally simple layout? If there is a perfectly simple answer why, I'd like to know. Sorry for kicking this dead dog around for so long.

Its surprising the savings that could happen with just one lug per wheel.

They must have taken the que from the airline industry.

http://www.nytimes.com/2005/06/10/business/worldbusiness/10iht-air.html

Sorry for kicking this dead dog around for so long.

That dog isn't dead...its resting....you see it moved. (a skit from Monty Python dead parrot skit) http://www.youtube.com/watch?v=4vuW6tQ0218

phoenix911

could it be a legacy from indexed dividing heads / rotary table when these things were (initially at least) done by hand in the toolroom?

(I remember being introduced to the intricacies of the dividing head by Dennis Ingram at ATG Aylesbury when producing a graduated dial for my lathe).

Perhaps this has already been suggested and I've missed it in the thread.

Have you guys noticed that any whole number divided by 7 and which result is not a whole number, will always have a mantissa repeating 428571 apparently endlessly (depending on what number you divide the mantissa will start from 4 or 2 or 8 etc.)?.

Just throw the chalk to my head, not good at math anyway.

Yahlasit

you may have had a boring day, but you were not alone.........I was doing something simuiar trying to find a pattern. It was on a tangent to a problem but for a reason the escapes me at the moment.

Apparently you can only fold a sheet of paper in half not more than seven times.

Hmmmm...... well... does that hold true for.. say a "D" size? or "E" size sheet of paper? we'll need some clarification.. or better yet some experimentation...

I just tried with an 11x17 sheet... definitely could not fold more than seven.. (in fact the bast i got is 6 folds in half.

Would that hold true even If the sheet was the size of a football field??

Yup - try it if you think you're up to it .

^{_{<if you think you're 'ard enough - come an' 'ave a go ... >}}

The original topic was about dividing a circle (360 degrees) into a prime number of equal sectors. As I pointed out before, there are several good reasons to do this. For a further example, check out cycloid gear speed reducers, in which prime numbers of lobes figure prominently (Sumitomo SM-Cyclo, for example).

The post about six wires in a cable surrounding a seventh central wire is interesting in its own right, but it isn't strictly relevant, because the circle is divided into six parts. Add 12 more wires and you get 19, another prime number and a common wire stranding pattern. Add 18 more wires for a total of 37, again prime and common. If you keep going, you get mostly prime numbers with a few composites from time to time. (These are all called "hexagonal numbers," by the way, and they are useful to know in laying out tube sheets for heat exchangers.)

The paper-folding exercise is completely off-topic, though it too has an unrelated component of interest. And I don't care about the Seven Samurai, or 7-Up, either.

"The original topic was about..." the prime number 7.

I found it!

This problem was solved in 2002 by a high school student, who derived the limiting equations and as a consequence ceated a unique series of numbers that is now registered.

The limit is actually 12 for same direction folding - oh look! we're back to that other "magic" and "holy" number!!

I remember "Record Breakers" trying this when I was a kid - can't remember what the result was!

For a picture of some 7-tooth gears, see this link:

http://cgi.ebay.com/Pinion-gear-7-tooth-48-pitch-brass-press-on-new-NR-BRP_W0QQitemZ370229584368QQcmdZViewItemQQimsxZ20090715?IMSfp=TL090715214005r6590

And there are lots of gears in the world with other prime numbers of teeth.