CTK Exchange
Front Page
Movie shortcuts
Personal info
Awards
Reciprocal links
Terms of use
Privacy Policy

Interactive Activities

Cut The Knot!
MSET99 Talk
Games & Puzzles
Arithmetic/Algebra
Geometry
Probability
Eye Opener
Analog Gadgets
Inventor's Paradox
Did you know?...
Proofs
Math as Language
Things Impossible
My Logo
Math Poll
Other Math sit's
Guest book
News sit's

Recommend this site

Manifesto: what CTK is about Search CTK Buying a book is a commitment to learning Table of content Products to download and subscription Things you can find on CTK Chronology of updates Email to Cut The Knot Recommend this page

CTK Exchange

Subject: "Two paradoxes: perfect spheres, and one is less than one"     Previous Topic | Next Topic
Printer-friendly copy     Email this topic to a friend    
Conferences The CTK Exchange This and that Topic #799
Reading Topic #799
panda24
Member since Dec-11-07
Dec-11-07, 03:13 PM (EST)
Click to EMail panda24 Click to send private message to panda24 Click to view user profileClick to add this user to your buddy list  
"Two paradoxes: perfect spheres, and one is less than one"
 
   Two (apparent) paradoxes were given to me by a maths teacher 25 years ago. I'd like help disproving them.

A
Perfect spheres don’t touch.
He claimed that, on the surface of a perfect sphere, the most prominent part is a point. As a result, two perfect spheres cannot touch, because then a point would be touching a point, and points have no area, so nothing would be touching nothing. His solution was only a practical one: perfect spheres don’t exist, so any real, roughly spherical object is actually a bit flat at the molecular level. But this doesn’t really help.

My best guess is that a point is not a measurement of area but a coordinate. If the surface really were made up of nil-area points, then the whole sphere would have no area, because n times 0 is 0. So a point is just a location, a place where something happens, which may as well be touching as anything else.

B
1 = 0.9 recurring
Apologies for my lack of a “recurring” dot. My teacher argued that since 1 is also 3/3, and 1/3 is 0.3 recurring, then 1 must be 3 x 0.3 recurring, which is 0.9 recurring. Thus 1 = 0.9 recurring.

My best answer is that 1/3 isn’t really 0.3 recurring. In other words, 1/3 describes, effectively, a problem (“divide one by three”), whereas the decimal version is a solution. But because each of the point 3s, or point 9s, leaves another one to come, they cannot really be regarded as a completed series. So there is a bit more left at the end of 0.9 recurring.

As you can tell, I’m not a mathematician!!!

But any help/thoughts gratefully received, the problems may seem obvious to you, but they've perplexed me all my adult life.


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
alexb
Charter Member
2150 posts
Dec-12-07, 02:59 PM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
1. "RE: Two paradoxes: perfect spheres, and one is less than one"
In response to message #0
 
   I must confess that little of what you wrote made any sense to me. Please do not take an offense.

I would suggest you take a look at this:

https://www.cut-the-knot.org/ctk/NatureOfProof.shtml

The main point there is the importance of definitions.

We may discuss your quandary ad nauseum without getting anywhere. So before we start you should first tell us what does it mean for spheres to touch. Not that this is impossible but simply when would you or your teacher say that the two spheres touch.

A point is a geometric object that has area of zero, true enough. However, a sphere does consists of points. Area is a function on sets such that for a finite number of nonintersecting sets the area of their union equals to the sum of their areas. This is also true for countably many sets provided the series of their areas converges. However, this is not true for a continuum of sets - points on a sphere being one such case.

Lastly, for the recurring quandary check

https://www.cut-the-knot.org/arithmetic/999999.shtml

>
>A
>Perfect spheres don’t touch.
>He claimed that, on the surface of a perfect sphere, the
>most prominent part is a point. As a result, two perfect
>spheres cannot touch, because then a point would be touching
>a point, and points have no area, so nothing would be
>touching nothing. His solution was only a practical one:
>perfect spheres don’t exist, so any real, roughly spherical
>object is actually a bit flat at the molecular level. But
>this doesn’t really help.
>
>My best guess is that a point is not a measurement of area
>but a coordinate. If the surface really were made up of
>nil-area points, then the whole sphere would have no area,
>because n times 0 is 0. So a point is just a location, a
>place where something happens, which may as well be touching
>as anything else.
>
>B
>1 = 0.9 recurring
>Apologies for my lack of a “recurring” dot. My teacher
>argued that since 1 is also 3/3, and 1/3 is 0.3 recurring,
>then 1 must be 3 x 0.3 recurring, which is 0.9 recurring.
>Thus 1 = 0.9 recurring.
>
>My best answer is that 1/3 isn’t really 0.3 recurring. In
>other words, 1/3 describes, effectively, a problem (“divide
>one by three”), whereas the decimal version is a solution.
>But because each of the point 3s, or point 9s, leaves
>another one to come, they cannot really be regarded as a
>completed series. So there is a bit more left at the end of
>0.9 recurring.
>
>As you can tell, I’m not a mathematician!!!
>
>But any help/thoughts gratefully received, the problems may
>seem obvious to you, but they've perplexed me all my adult
>life.


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
panda24
Member since Dec-11-07
Dec-19-07, 00:49 AM (EST)
Click to EMail panda24 Click to send private message to panda24 Click to view user profileClick to add this user to your buddy list  
2. "RE: Two paradoxes: perfect spheres, and one is less than one"
In response to message #1
 
   Thank you very much for the hyperlink on the recurring quandary, I understand. In my non-specialist language, I was getting at the same basic point. Many thanks.

On the spheres, what I mean is this: let's suppose we have two pool balls, which are perfectly spherical. We know that we can hit one ball into the other. We can take a cue, hit the first ball, and it will bump into the other ball. So, the surfaces of the two balls will come into contact when they bump. This is what I mean by "touch" - the surface of one makes contact with the surface of the other.

Yet, how can they touch if their most prominent parts (those which touch) are only points, with no area? Is it that touching doesn't have to involve area? My assumption (and what my teacher was suggesting) is that to touch, an area on one surface has to be in contact with an area on the other.

I have no mastery of mathematical terms, but I think this is a "real" situation, not just an accident of poor definition.


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
alexb
Charter Member
2150 posts
Dec-20-07, 10:13 AM (EST)
Click to EMail alexb Click to send private message to alexb Click to view user profileClick to add this user to your buddy list  
3. "RE: Two paradoxes: perfect spheres, and one is less than one"
In response to message #2
 
   >On the spheres, what I mean is this: let's suppose we have
>two pool balls, which are perfectly spherical. We know that
>we can hit one ball into the other. We can take a cue, hit
>the first ball, and it will bump into the other ball. So,
>the surfaces of the two balls will come into contact when
>they bump. This is what I mean by "touch" - the surface of
>one makes contact with the surface of the other.

Excellent. Let me add that in mathematics "touch" means "having a common point" (assuming of course there is no intersection like for the graphs y = x2 and y = x3. This is also impossible as long as you have in mind "physical" bodies.)

>Yet, how can they touch if their most prominent parts (those
>which touch) are only points, with no area?

What's wrong about that? They are solids, not elastic bodies. Abstractedly, they do have a common point.

<p>Is it that
>touching doesn't have to involve area?

Yes, it does not.

>My assumption (and
>what my teacher was suggesting) is that to touch, an area on
>one surface has to be in contact with an area on the other.

So you see this is indeed a matter of defition. If you, along with your teacher, insist that touching involves non-zero area (which I and other mathematicians disagree with) then of course there is no touching for the pool balls. I just do not see what is the problem.

If touching means sharing a point, then they touch. If touching needs non-zero areas, they do not. There is nothing else to it.

>I have no mastery of mathematical terms, but I think this is
>a "real" situation, not just an accident of poor definition.

Of course it's a real situation. I believe you exist, I know for sure that I also exist, and, in addition, some billiard balls are known to exist (I have a pool table in my basement.)


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top
panda24
Member since Dec-11-07
Dec-22-07, 08:28 AM (EST)
Click to EMail panda24 Click to send private message to panda24 Click to view user profileClick to add this user to your buddy list  
4. "RE: Two paradoxes: perfect spheres, and one is less than one"
In response to message #3
 
   Excellent, thank you very much for that.

If I knew where my old schoolteacher lived now, I would let him know the solution.

Happy Christmas to you and all at CTK!


  Alert | IP Printer-friendly page | Reply | Reply With Quote | Top

Conferences | Forums | Topics | Previous Topic | Next Topic

You may be curious to have a look at the old CTK Exchange archive.
Please do not post there.

Copyright © 1996-2018 Alexander Bogomolny

Search:
Keywords:

Google
Web CTK