# Mathematical question



## Thread man (Jul 23, 2022)

I've always regarded myself as being good at geometry (and trigonometry) and I suppose you have to be working with threads. However, and to my surprise, I got the wrong answer to this and I even regarded it as simple!

Use your head or pencil and paper but don't cut anything out. That'd be cheating ☺

How many revolutions of the small circle (R1) will it take to circle the circumference of the large circle (R3) to arrive back at the starting point?

Any answers?


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## TonyM (Jul 23, 2022)

3. Circumference is Pi x D. As Pi is constant then it's simply Diameter 6 divided by Diameter 2. Same rules for cam gears on a four stroke which are 2:1 ratio on pitch diameter and number of teeth.


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## mu38&Bg# (Jul 23, 2022)

enter epicycloid


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## Thread man (Jul 23, 2022)

TonyM said:


> 3. Circumference is Pi x D. As Pi is constant then it's simply Diameter 6 divided by Diameter 2. Same rules for cam gears on a four stroke which are 2:1 ratio on pitch diameter and number of teeth.


I though so too but that's wrong.


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## Thread man (Jul 23, 2022)

dieselpilot said:


> enter epicycloid


You gave the answer but didn't answer the question. Why didn't you also just add the number to your reply?


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## kvom (Jul 23, 2022)

Answer is 4, or ratio + 1 for integral ratios.


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## stevehuckss396 (Jul 23, 2022)

Why is that wrong? In the example r1 has a Dia of 2 and r3 has a Dia of 6 so 6/2 = 3. 3 revs and your back. Tony's answer sounds correct to me. What am I missing?


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## Thread man (Jul 23, 2022)

stevehuckss396 said:


> Why is that wrong? In the example r1 has a Dia of 2 and r3 has a Dia of 6 so 6/2 = 3. 3 revs and your back. Tony's answer sounds correct to me. What am I missing?


You're missing the same as I missed. Take two identical coins and have them touch. Turn one of them around the circumference until back to the same point as you started. You'll find what should have taken one turn takes two. Regardless of the two circle diameters always add a 1.

To be honest I'm still trying to figure out why. 

Dieselpilot and kvom got it right but didn't do as asked. If it were an exam I'd have failed them both for their "non answers"


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## stevehuckss396 (Jul 23, 2022)

If both coins stay in there position and are both rotated they will both rotate once. The trick is that only one coin is allowed to move in video so that one coin has to rotate twice to achieve the same as both coins rotating once.

We as "engine guys" will probably never look at the problem and not imagine both circles rotating and maintaining there position.


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## Thread man (Jul 23, 2022)

stevehuckss396 said:


> If both coins stay in there position and are both rotated they will both rotate once. The trick is that only one coin is allowed to move in video so that one coin has to rotate twice to achieve the same as both coins rotating once.
> 
> We as "engine guys" will probably never look at the problem and not imagine both circles rotating and maintaining there position.


When I saw it on Youtube I didn't get the right answer and I am an engineer that's pretty good at geometry and trigonometry. I'm guessing I shouldn't post any more tricky questions (at the time I regarded is as interesting rather than insulting to"engine guys" ) if there are other "touchy feely" types like yourself.

Re "engine guys" then among the various jobs I had before I started my own company, I was Quality Engineer for a Danish company making parts for the F-16. The contract at the time stated that the Danish company would make half the number. Ended up that the Danish company made them all.


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## Thread man (Jul 23, 2022)

stevehuckss396 said:


> If both coins stay in there position and are both rotated they will both rotate once. The trick is that only one coin is allowed to move in video so that one coin has to rotate twice to achieve the same as both coins rotating once.
> 
> My first reaction was "That's the answer" until, after reading again, it could be understood that the correct answer would be the "calculated" rotations multiplied by 2.  It is in fact always the "calculated" result plus 1.


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## Thread man (Jul 23, 2022)

stevehuckss396 said:


> If both coins stay in there position and are both rotated they will both rotate once. The trick is that only one coin is allowed to move in video so that one coin has to rotate twice to achieve the same as both coins rotating once.
> 
> We as "engine guys" will probably never look at the problem and not imagine both circles rotating and maintaining there position.


When I first read that I thought "That's the reason". I then thought that it could be read as the correct answer for the number of rotations necessary was twice that of the "calculated".  It is in fact always the "calculated" (calculatted from the circumference distance) plus 1 regardless of the two circle diameters.

I don't remember at any time referring to the two circles as gears and neither does the Youtube video. It was to me a simple mathematical question that turned out to be harder than I thought LOL

I showed it to my wife and she said "It should be 3 but I feel it is more than 3". I've got a clever wife  At least I think so.


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## mcostello (Jul 23, 2022)

It's what She thinks that matters.


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## stevehuckss396 (Jul 23, 2022)

I think what is tripping me up is that the smaller circle makes a full rotation when compared to horizontal but not along the surface of the circle. Make a small line where the two circles meet and then turn the small circle until the line touches the surface of the large circle and it will be 3 turns to make the trip. It's the fact that the small circle is displaced from the starting point that throws the brain off. Kinda like a rod in a radial engine has to be compensated so the pistons hit top dead center at the right time.


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## scottyp (Jul 23, 2022)

Interesting...Like steve h said, when circle A is one quarter of the way around circle B, it has rotated 360 degrees around it's own center but it's outer edge has not yet traveled 1 circumference distance around Circle B, that doesn't happen for another quarter turn in this case.


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## willray (Jul 23, 2022)

stevehuckss396 said:


> If both coins stay in there position and are both rotated they will both rotate once. The trick is that only one coin is allowed to move in video so that one coin has to rotate twice to achieve the same as both coins rotating once.



It's not really "that one needs to rotate twice to achieve the same as both rotating once".

The reason there's 1 additional rotation, is because the one that's moving is _rolling_around_a_circle_.   If you linearize the circumference of the bottom circle, you get exactly the ratio of "diameters" that you would expect.  When you roll it around a circle though, the path _itself_ contains one rotation, so you get the rolling rotations, plus the rotation of the path.


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## Ken I (Jul 24, 2022)

I haven't looked at anything and my answer is 4 - the smaller outer turns 3 revs against the inner but in so doing rotates an additional revolution.

Kind of like the plot kicker in "Around The World In 80 Days" - Fogg thinks he has lost the bet as he has seen 80 sunsets - but that was in 79 days (if my memory serves me correctly).

If you like geometric puzzles, here's one called Curry's Triangle :-




Curry originally presented this to his lecturer as a cardboard cut-out - painted red one side and blue the other - presenting the red side first and then turning it over to the revised pattern which has a 2 unit "hole" in it - he told his lecturer that this proved a triangle had a different area when viewed from the front versus the back.

The triangle is 12 units tall and 10 units wide and therefore has an area of 60 square units - but the "blue" version is only 58 as two have gone missing.

As a curiosity arranging the bits into rectangle (R.H. image) occupies only 59 units.

Attached a printable version if you want to make the cardboard cut-outs and drive yourself nuts.

P.S. If you know the answer keep it to yourself for a while at least (it drove me nuts when I first came across it).

Regards, Ken


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## mu38&Bg# (Jul 24, 2022)

Thread man said:


> You gave the answer but didn't answer the question. Why didn't you also just add the number to your reply?


To lead the horse to water. What fun is a thread that's over in 3 posts?


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## Thread man (Jul 24, 2022)

dieselpilot said:


> To lead the horse to water. What fun is a thread that's over in 3 posts?


I'm assuming you know the rest of your quote? I didn't realise that replying to threads was supposed to be "just" for fun. Up until now I had the impression this forum was about asking for help and helping when possible. One of us is in the wrong forum and if it's me then I'll just quit.

Here's a joke with realism.


*A man goes into a pet shop to buy a parrot.
There are 4, each on a perch and with prices on each perch. First one 1,000, second one 2,000, third one 5,000 and the fourth 25,000.
The man asks the pet shop owner what the parrot at 1,000 can do.
“Speaks one language fluently” is the reply.
“Impressive”, say the man. “What can the one at 2,000 do?”
“Speaks two languages fluently” is the reply by the pet shop owner.
“Gosh” says the man. “And the one at 5,000?”
“Not only can it speak three languages fluently but it can operate a computer”, replies the pet shop owner.
“I hardly dare ask” says the man “but what about the one at 25,000?”
The pet shop owner leans forward and whispers in the man’s ear, “I’ve never seen it do anything or heard it say anything but the other three call it “BOSS”.*


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## Steamchick (Jul 25, 2022)

Kvom: A good thread! 
A friend answered thus...
"There are a number of answers :

1)  Mathematical
Assming nothing slips you will only get back to the starting point after N rotations of circle R1 ( assuming the larger cicle stays still ) where
Circumfrence of Circle R3 =2 pi R3
Circumfrence of circle R1 = 2 pi R1
N = 2 pi R3 divided by 2 pi R1 = R3 divided by R1

2) Philosophical
When both circles have reached a mutually beneficial understanding

3)  Fatalistic
Who cares ?

4)  Quantum
They have already reached the starting point but we cannot percieve it."

So I sent him the explanation for "4"... (having got the wrong answer myself).
Thanks for the fun!
K2


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## mu38&Bg# (Jul 25, 2022)

Thread man said:


> I'm assuming you know the rest of your quote?



Of course I do. Some will investigate, some won't. You can stay, I'll close my account.


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## awake (Jul 25, 2022)

Thread man said:


> When I saw it on Youtube I didn't get the right answer and I am an engineer that's pretty good at geometry and trigonometry. I'm guessing I shouldn't post any more tricky questions (at the time I regarded is as interesting rather than insulting to"engine guys" ) if there are other "touchy feely" types like yourself.
> 
> Re "engine guys" then among the various jobs I had before I started my own company, I was Quality Engineer for a Danish company making parts for the F-16. The contract at the time stated that the Danish company would make half the number. Ended up that the Danish company made them all.


??? What in Steve's reply suggested anything touchy-feely or that he felt insulted?


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## Thread man (Jul 25, 2022)

awake said:


> ??? What in Steve's reply suggested anything touchy-feely or that he felt insulted?


I didn't write that he felt insulted. I wrote that I did. I suggest you read some of the posts he writes to others and either you get what I mean or you don't.


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## awake (Jul 25, 2022)

Hmm, sorry that I misunderstood. I have to say, though, I have been reading Steve's posts for years, and have not felt any "sharp edges" to them. YMMV.


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## Thread man (Jul 25, 2022)

awake said:


> Hmm, sorry that I misunderstood. I have to say, though, I have been reading Steve's posts for years, and have not felt any "sharp edges" to them. YMMV.


Depends on how you read his replies. I've only read a few but to me, and obviously not to you, I read what he writes as condescending rather han helpful although I suppose to some (most?) they appear helpful.

How would you read and understand the reply I got when I asked why he didn't just give an answer? If that isn't enough look at the recent replies he's been giving posts.

I've tried but can't find a thread he's started. I'm sure he has so if you let me know I'll read it.  Who knows, I might change my opinion


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## awake (Jul 25, 2022)

Thread man said:


> How would you read and understand the reply I got when I asked why he didn't just give an answer?


I did not hear that reply as condescending, but rather as good humored. Always hard to tell "tone of voice" in an internet forum, so I generally assume the best. YMMV.


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## Thread man (Jul 25, 2022)

awake said:


> I did not hear that reply as condescending, but rather as good humored. Always hard to tell "tone of voice" in an internet forum, so I generally assume the best. YMMV.


I know you didn't "hear" it. You read it. That's one of the biggest problem with the internet. You can't see the expression on people's face. A very good reason to think before posting. 

Interesting too is that you seem to dislike more me objecting than why I objected.

*There are 2 rules in life:
1. NEVER GIVE OUT ALL THE INFORMATION*


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## Thread man (Jul 26, 2022)

I don't regard myself as stupid but it did take me a while to figure out why I originally got the answer wrong. There have been many explanations but in my own head I've got it explained thus   

The distance the smaller circle is travelling around the larger circle isn't the circumference of the larger. It's the larger circumference that the smaller circle has from its radius. I know others have more or less written the same but if I was explaining it to anyone (in this case myself) that's how  I'd write or say it.

*"Statistics is the art of never having to say you're wrong."*


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## mklotz (Jul 26, 2022)

Thread man said:


> I don't regard myself as stupid but it did take me a while to figure out why I originally got the answer wrong. There have been many explanations but in my own head I've got it explained thus
> 
> The distance the smaller circle is travelling around the larger circle isn't the circumference of the larger. It's the larger circumference that the smaller circle has from its radius. I know others have more or less written the same but if I was explaining it to anyone (in this case myself) that's how  I'd write or say it.




Easy to say after reading what I posted last Sunday to your other thread (Reading, understanding and following instructions)...

A coin of diameter 2R is held stationary while a smaller coin of diameter 2r is rolled around its circumference until it returns to its starting position.  How many revolutions does the smaller coin make ?

The number of revolutions (N) can be found by dividing the distance the center of the coin has traveled (T) by its circumference...

N = T/(2*pi*r)

The center of the smaller coin travels around a circle with a radius of (R+r); thus T = 2*pi*(R+r) and...

N = (R+r)/r = R/r +1

In the case of R = r (two identical coins), N = 2.


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## stevehuckss396 (Jul 26, 2022)

Wow. I have been on this forum sence 2006? And have never found Mr klotz ( Mr used loosely) to be anything but helpful.  Math skills way above normal. Just an all around helpful when he can guy. I have been staying away from this thread as not to exacerbate some negative feelings but I can't help to defend Mr klotz. In my opinion he has been nothing out side of helpful.


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## kuhncw (Jul 26, 2022)

Marv's posts are always helpful and worth reading.  

Chuck


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## awake (Jul 26, 2022)

All right, I've had it. Thread Man, you have a grand total of 50 posts, and you've been a member for just over 3 weeks. In that time, you have repeatedly posted that various members are condescending, being a smart ass, not following directions, etc. - and these members are folks who have been around this forum for many years, with many hundreds or thousands of posts and many hundreds or thousands of likes - in other words, long valued members who are widely regarded as some of the most helpful posters on this forum.

All this to say, I heartily endorse your statement that it is time for you to leave the forum. Sorry to be blunt, but there it is.


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## willray (Jul 26, 2022)

Thread man said:


> I don't regard myself as stupid but it did take me a while to figure out why I originally got the answer wrong. There have been many explanations but in my own head I've got it explained thus
> 
> The distance the smaller circle is travelling around the larger circle isn't the circumference of the larger. It's the larger circumference that the smaller circle has from its radius. I know others have more or less written the same but if I was explaining it to anyone (in this case myself) that's how  I'd write or say it.
> 
> *"Statistics is the art of never having to say you're wrong."*



Sorry, but no.

Take the larger circle, split it in the bottom center and turn the two tails out so that it's shaped like an Omega symbol.  Roll your smaller circle along the Omega-shaped path.   This will have almost exactly the same "larger circumference that the smaller circle has from its radius", but the smaller circle will turn exactly the expected "ratio of the diameters" number of turns.

The reason there's an extra rotation when the path is a circle, is because the path contains a rotation.  You only get as much extra rotation as the difference in rotation of the beginning and ending of the path, so if the beginning and ending are co-linear (and there are no rotations between them), there is zero extra rotation in the rolling circle, nomatter how far its center travels.


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## dnalot (Jul 27, 2022)

Thread man said:


> You're right. What I wrote was easier to say than what you wrote. There's smart and there's smartass. You seem to be the latter. It's also why I choose to quit this forum as I don't want to be the same place as folk like you.
> 
> Bye all!



I don't think you will be missed. 
Happy trails

Mark T


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## clockworkcheval (Jul 27, 2022)

Wow! We didn't search for it, but found and lost the Big Bang. Confusing experience.


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## Ken I (Jul 27, 2022)

Well I was wrong and of course Marv was right but I had to draw it out to convince myself





By rotating a 10T planet gear about a 60T sun gear (3:1) ratio in 180° (10 tooth) increments you can see that after being turned though 3 half turns the datum dot is now in its starting position and thus after travelling all the way around it has only traveled through 2 revolutions.
It loses a rev not gained a rev as I had (wrongly) guessed.

Regards, Ken


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## Mike Ginn (Jul 27, 2022)

We have a very helpful and wonderful engineering community and we should be proud of that and not be offensive to each other.  Mathematics is full of interesting and non-intuitive puzzles (which should really reside on YouTube) which often don't have obvious answers or even answers at all.  That's how it is!

I did like Ken's approach to the problem - thank you

Although ThreadMan used a cartoon which depicts terms which remain in use in todays Politically Correct world, I personally thought the cartoon unnecessarily vulgar and was surprised that it was allowed by the moderators.  I made no comment at the time.

I do however endorse the criticism aimed at ThreadMan - he was out of order - completely out of order and he is no loss to the community.

As a mathematician I would also take issue with his statement "*Statistics is the art of never having to say you're wrong".  *He should only make comments he understands but fortunately we will never hear his response!

Mike


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## clockworkcheval (Jul 27, 2022)

The cartoon reminds me of my first visit to Copenhagen Airport about 1981. We were welcomed by a large poster of a ferocious Viking complete with helmet and sword and the words 'We have already been trading for more than a thousand years with the rest of Europe', clearly to support the then Danish campaign for acces to the EC.


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## minh-thanh (Jul 27, 2022)

Thread man said:


> You're right. What I wrote was easier to say than what you wrote. There's smart and there's smartass. You seem to be the latter. It's also why I choose to quit this forum as I don't want to be the same place as folk like you.
> 
> Bye all!


I don't know if you read it but I hope you will
   As I said : " Wellcome !
  Please share your knowledge ... . "
  You may be good at math , geometry ... but there is always someone better than you ( definitely not me ) in this forum .
   So sit down, calm down, have a cup of coffee or beer and think.


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## Mike Ginn (Jul 27, 2022)

Clockworkcheval, Yes it is interesting that different countries have completely different views about humour - some can't even spell humor!  Japan is an interesting example of adverts which would never be allowed in the UK.  I was once told by my Head Teacher at school to never discuss/joke about politics or religion with someone from Ireland.  Having worked in the US I would change that to "never discuss/joke about politics, religion, guns or abortion" with anyone from the outside the upper East/West coast area of the US ie the middle and South.

Its a strange world but fortunately modelling is common to many countries but interestingly not many EU countries.  

Mike


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## willray (Jul 27, 2022)

Ken I said:


> By rotating a 10T planet gear about a 60T sun gear (3:1) ratio in 180° (10 tooth) increments you can see that after being turned though 3 half turns the datum dot is now in its starting position and thus after travelling all the way around it has only traveled through 2 revolutions.
> It loses a rev not gained a rev as I had (wrongly) guessed.



Umm... I think you've miscounted... 

The datum dot also returns to its starting point midway between your "Rotate 180 (10T)" and "Rotate 360 (20T)" positions...


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## Ken I (Jul 27, 2022)

Willray - I see your point - and got confused all over again so I drew in the other positions (rotation degrees indicated are for the 20T planet wheel so rotating it 180° equates to 10 Teeth) :-





As you can see at the North South East & West positions the datum tooth (now highlighted in red) is indeed at the south position 4 times.
However if you look at the East position the gear has only turned 3/4 of a revolution but the datum dot appears to have turned through a full revolution because it was carried on the Sun gear the additional 90°.

So the planetary gear only rotated three times relevant to the sun gear but 4 times relevant to its "compass" orientation. Hoo boy!

So who's right - It depends on your point of view ???

Regards, Ken


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## Charles Lamont (Jul 27, 2022)

On the matter of Ken's triangle area conundrum, a hint for anyone still puzzled: similarity.


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## Mike Ginn (Jul 28, 2022)

Guys
What we are talking about is the Coin Rotation Paradox which has similarities with the famous Asistotles Wheel which mathematician puzzled about for centuries.  Looking at the coin Paradox and simplifying it and making use of Stan's tooth idea we can have a linear gear of 60 teeth and a cog of 20 teeth.  I think we can all agree the the cog rotates 3 times.  Stan's last diagram is also correct and shows that the cog rotated 4 times.  How can this be?  It's because the cog is rolling around a circle as can be seen in Stan's diagram.  The result is that if you work out the number of times the coin rotates based on the circumference ratio you then need to add 1 to take account of having to go around the centre of the inner gear.  This is true for any size wheel/gear/disc system.

If you really want to blow your mind then consider that the sum of all natural numbers :- 1+2+3+4....to infinity = -1/12

Its controversial but backed up by very eminent mathemations.  The proof is not easy but well documented!

Maybe for a wet afternoon!

Have fun

Mike


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## Foozer (Jul 29, 2022)

This old man is not to bright with math - But would the pattern be Circum A / Circum B ?


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## Ken I (Jul 30, 2022)

Foozer,If you are looking for the locus of motion - here it is - red dot on the pitch circle diameter so ignore the teeth and consider it two wheels running on each other (at their PCDs). A red dot indicates its position per tooth of rotation.




It traces three epicyclic paths - demonstrating that the gear indeed rotated three revolutions against the sun wheel.

I have drawn the planet gear at the North, South, East & West positions, the reference red dot (timing mark) is at the South position indicating the outer wheel has rotated four revolutions relative to its compass orientation.

Regards, Ken


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## mklotz (Jul 30, 2022)

Foozer said:


> This old man is not to bright with math - But would the pattern be Circum A / Circum B ?View attachment 138892


What you suggest is exactly what I used in my earlier post (post number 29 in this thread).  However, one must do the math to understand why the number of rotations has that '1' added to the ratio of the radii of the two circles.

Go back and read #29.  The math is trivial, involving nothing more than the formula for the circumference of a circle and simple division.


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## Foozer (Jul 30, 2022)

Ken I said:


> View attachment 138902
> 
> It traces three epicyclic paths - demonstrating that the gear indeed rotated three revolutions against the sun wheel.
> 
> ...



Where Grandas head is at with this circle thing .
I have grandkids 7-15 years. I like to use and encourage them to find the pattern in things - Simple Pattern - So in circle query what is actually moving around the center point of the fixed circle is the center point of the moving circle. If the unwrap distance was from the point of contact the circle center point would have traveled farther - be great for gas mileage as the wheel is just once circle rotating around another, but it don't work that way does it. [mayhaps it does but time differentials? No Thanks] 

So the pattern I try to get the grandkids to see is that the number of rotations of X is the circumference of a circle r1+r2/ circumference of the circle r2 [ Older one can do Circumference].
So a 4 in fixed and a 2 in mover gives 18.849/6.283 - - 3 times.

Finding a pattern one thing, trying to explain it is another.







Asked the grandkids once to add up 1+2+3...+10.  Their facial expressions were cute.  Grandpa is crazy . Told them to hold up their hands fingers spread - Do you see the pattern I ask them?
10 fingers numbered. 
1-10 - First [1] and Last [10] add up to 11. 10 fingers divided by 2 is 5, 5x11 is 55 so 1+2+3...+10=55 - Oh Look, you were born with the answer, Open hands before you says 55 ... 1+2+3+4...+100 same pattern -5050

If I went off into the weeds, hey I have grandkids to entertain
Robert


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## Foozer (Jul 30, 2022)

mklotz said:


> However, one must do the math to understand why the number of rotations has that '1' added to the ratio of the radii of the two circles.


Have always liked your post - Missed you while you were away. So a plus 1 when the mover is outside the fixed and a -1 for when the mover is inside the fixed?


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## Mike Ginn (Jul 30, 2022)

Guys
The coin rotation paradox has caused many minds to melt!  It can easily be shown physically that with 2 coins of equal size - (I always favor 2 silver dollars pieces!) then the moving coin rotates twice.  David Ding has produced an elegant proof for those  those who like to tidy up loose ends.  

His proof is found here:- www.davidyding.com/navPages/coinRotation 

In any discussion it is important to define the initial conditions and this is what Ding says:-

"_How many revolutions does a circular coin make while rolling around another circular coin of the exact same size without slipping?_"

"At first glance, the intuitive answer might be, well, one. After all, the rolling coin has a circumference, say c, and it is rolled, without slipping, against another coin with circumference c. This is basically saying how many revolutions does a circle with circumference c make while rolled against a path with length c. Surely, the answer is one?

Well, the actual answer is *two*. And this seemingly strange result is called the “coin rotation paradox” and that Wikipedia has a short article describing it. Here, I will use math to solve this paradox in the general case and dish out some insights for the general result."

Ding's final equation is as follows:-





Where rf and rr are the radii of the coins so if the coins are identical then N=2.  To fully understand this you will need to work through the other 4 equations.

Note that Ding's proof covers coins/disks of any size provided the ratio of the disk radii is an integer.

Mike


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