The 80s called. They want their store back

tracecom

Joined Apr 16, 2010
3,944
For those of you who pride yourselves in your 3d visualization skills, try this.

Imagine a paper Mobius strip that has had a pencil line drawn along its length until the pencil line connects with its beginning. Then imagine taking a pair of scissors and cutting the strip along the pencil line. What will that produce?

Think it through and then try it to see if you visualized it correctly.

There is a next step, but I can't reveal that without spoiling the answer to the first step.
 

Thread Starter

spinnaker

Joined Oct 29, 2009
7,830
For those of you who pride yourselves in your 3d visualization skills, try this.

Imagine a paper Mobius strip that has had a pencil line drawn along its length until the pencil line connects with its beginning. Then imagine taking a pair of scissors and cutting the strip along the pencil line. What will that produce?

Think it through and then try it to see if you visualized it correctly.

There is a next step, but I can't reveal that without spoiling the answer to the first step.
Two loops of paper hooked together.
 

strantor

Joined Oct 3, 2010
6,782
For those of you who pride yourselves in your 3d visualization skills, try this.

Imagine a paper Mobius strip that has had a pencil line drawn along its length until the pencil line connects with its beginning. Then imagine taking a pair of scissors and cutting the strip along the pencil line. What will that produce?

Think it through and then try it to see if you visualized it correctly.

There is a next step, but I can't reveal that without spoiling the answer to the first step.
Im thinking, you double the circumference of the mobius strip, and introduce one more half-twist.
 

tracecom

Joined Apr 16, 2010
3,944
Im thinking, you double the circumference of the mobius strip, and introduce one more half-twist.
Well, everyone has had a chance to try this, so I'll announce that strantor is almost correct. It doubles the circumference and actually adds three more half-twists.

Now, if only wbahn would come along and show the mathematical explanation for this.

The next question is what will you get if you perform the same operation again on the same strip?
 
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THE_RB

Joined Feb 11, 2008
5,438
Ah yes! :) Well technically it still has 6 sides as do all piece of paper. At least in this dimension. :)
No actually if you want to be pedantic it would have 2 sides. One "face" side, and one edge. :)

3D objects can have all numbers of sides. An egg is a real object and like a sphere has only one side. A UFO shaped disc with a sharp edge has only two sides, etc.
 

DerStrom8

Joined Feb 20, 2011
2,390
No actually if you want to be pedantic it would have 2 sides. One "face" side, and one edge. :)

3D objects can have all numbers of sides. An egg is a real object and like a sphere has only one side. A UFO shaped disc with a sharp edge has only two sides, etc.
That point of view has all but been phased out. It is more realistic, and more generally accepted now, that spheres and ovoids have an infinite number of sides. Same with 2-D objects such as ovals and circles.

A piece of paper still has 6 sides. It's just a squashed rectangular prism, but it has not lost any of its dimensions.
 

Thread Starter

spinnaker

Joined Oct 29, 2009
7,830
No actually if you want to be pedantic it would have 2 sides. One "face" side, and one edge. :)

3D objects can have all numbers of sides. An egg is a real object and like a sphere has only one side. A UFO shaped disc with a sharp edge has only two sides, etc.
Actually I was speaking about a piece of paper. It has four sides. You could say it has 3. The outside edge and each face. You could not count each face since it is interrupted by the edge. Though the airlines does set their dimension limits in linear inches.
 

Thread Starter

spinnaker

Joined Oct 29, 2009
7,830
That point of view has all but been phased out. It is more realistic, and more generally accepted now, that spheres and ovoids have an infinite number of sides. Same with 2-D objects such as ovals and circles.

A piece of paper still has 6 sides. It's just a squashed rectangular prism,
but it has not lost any of its dimensions.
And to add to your and my argument. All rectangles have 4 sides. So why would not a piece of paper?

Would a sphere have more infinite sides than a circle? :)
 

DerStrom8

Joined Feb 20, 2011
2,390
And to add to your and my argument. All rectangles have 4 sides. So why would not a piece of paper?
A piece of paper is not a rectangle. A rectangle is 2-D, a piece of paper is 3-D. So now you have 6 sides--the front face, the back face, the right face, the left face, the top face, and the bottom face (though the last 4 are very narrow, they are still faces). You could say it has 4 sides, a front, and a back though. It would depend on your definition of "sides". :D

Would a sphere have more infinite sides than a circle? :)
Never thought of that before, it's a very interesting thought....

Infinity is mind-boggling.
 

DerStrom8

Joined Feb 20, 2011
2,390
Add onto that, that some physics needs bigger and smaller infinities to work. :p

Just an infinity is so mind bogglingly incomprehensible....
Obviously there isn't really a "bigger infinity" or "smaller infinity", but I get what you're saying :p

Another example of why it is believed a circle has an infinite number of sides:

Draw a 3-sided figure with sides of equal length. Do the same with 4 sides, then 5, then 6, then 7, then 8, and so on. Notice that the more sides you add, the more it looks like a circle. So with a very high number of sides, it will look very much like a circle. Therefore it is generally assumed that an infinite number of sides creates the circle.
 

Sparky49

Joined Jul 16, 2011
833
There really is bigger and smaller infinities. :D

The Casimir force (dealing with negative energies) relies upon the infinite sum of frequencies from virtual particles between two metal plates of half-wavelength being smaller than the sum of infinite frequencies of any wavelength outside two metal plates.

I believe the analogy is:

Consider the series: 1+2+3+4+5+.... for infinity.

Then consider: 1.1+1.2+1.3+1.4+1.5+.... for infinity.

Even though infinity itself is not an absolute value, there can be relative differences between them.
 

DerStrom8

Joined Feb 20, 2011
2,390
There really is bigger and smaller infinities. :D

The Casimir force (dealing with negative energies) relies upon the infinite sum of frequencies from virtual particles between two metal plates of half-wavelength being smaller than the sum of infinite frequencies of any wavelength outside two metal plates.

I believe the analogy is:

Consider the series: 1+2+3+4+5+.... for infinity.

Then consider: 1.1+1.2+1.3+1.4+1.5+.... for infinity.

Even though infinity itself is not an absolute value, there can be relative differences between them.
Hmm, that does sound vaguely familiar. I guess it depends on the point of view. I may have read an article that looked at infinity in a much different way than I usually looked at it. Wish I could find that article....
 

Georacer

Joined Nov 25, 2009
5,182
Hmm, that does sound vaguely familiar. I guess it depends on the point of view. I may have read an article that looked at infinity in a much different way than I usually looked at it. Wish I could find that article....
You can start looking up countable infinites and uncountable infinities.

The cardinality of the set of Natural numbers is infinity. Starting from any natural number you can count towards infinity, without missing any number inbetween.

However, the set of Real numbers is both infinite and uncountable. There is always a number between any two real numbers, thus you can't count up from any real number without missing any.

These two sorts of infinity are considered separate in mathematics.
 

Thread Starter

spinnaker

Joined Oct 29, 2009
7,830
A piece of paper is not a rectangle. A rectangle is 2-D, a piece of paper is 3-D. So now you have 6 sides--the front face, the back face, the right face, the left face, the top face, and the bottom face (though the last 4 are very narrow, they are still faces). You could say it has 4 sides, a front, and a back though. It would depend on your definition of "sides". :D


I did not say a piece of paper was a rectangle. I said a rectangle has 4 sides. If a rectangle has 4 sides then a piece of paper would have to have an additional 2. Front and back making it 6 sides.

Never thought of that before, it's a very interesting thought....

Infinity is mind-boggling.
Actually the circle idea makes it a lot easier to comprehend.
 
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