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Number Theory - Calculating e and The Moebius Strip

02/08/2008 11:35 PM

Calculating e

The "magic" number e is fundamentally defined by the formula:

which means the same as:

In other words, as m approaches infinity, the result approaches e. The value of e can be calculated from this formula by a computer program like for pi, but convergence is very slow. We can also write:

Now if we expand

by the Binomial Theorem and take n to the limit we get:

which is the series expansion for ex. If x is made equal to 1, the series will give the value of e. The series converges rapidly, so not too many terms are needed for reasonable accuracy.

The Moebius Strip

A strip of paper bent into a circle and stapled has 2 surfaces, the outside and the inside. Such a hoop has 2 edges also, with a measurable distance between them. When one end of the strip is twisted a half turn before stapling, it is called a moebius strip. How many surfaces does it have? How many edges? The books say one each. Some people say two each like the hoop. Clearly there is still a measurable width as before. Is this not the distance between 2 edges? One can look at a section of the paper and turn it over to look on the 'other' side. If it has 2 sides, then it has 2 surfaces they say. Logic demands it.

You can draw a center line around the outside of the hoop until it joins itself at the starting point. The inside of the hoop has no line. If you do this with the moebius strip, the line ends up on 'both' sides of any section of the paper. This proves that it has only one surface, some say. The same thing happens if you color 'one' edge. 'Both' edges get the color. It must be the same edge, but how can there be a distance between 1 edge? The human mind is certainly twisted along with the paper!

Maybe mathematics can solve the debate. When you cut the hoop on the center line, 2 pieces result. These 2 pieces have 4 surfaces. With the moebius strip you get only 1 piece. The piece that you get has 2 full twists! The piece has 2 surfaces. Let's take this to stage two. We cut each of the two hoop pieces down the center as before. Now we have 4 pieces with 8 surfaces. After cutting the moebius piece we find that we have 2 distinct pieces that are entangled. Each piece has 1 full twist, so there are 4 surfaces. When the pieces are laid down on a table, they can be arranged into two figure 8's that are woven like a lawn chair.

Now let's write some formulas for the hoop, and apply them to the moebius strip. Let the number of surfaces before the first cut be designated Sb. Let the number of surfaces after the cut be designated Sa. Let the number of pieces after the cut be designated Pa. For stage one we have the two formulas: Sb = Pa, and Sb = Sa/2. For stage two we have the two formulas: Sb = Pa/2, and Sb = Sa/4. Applying these formulas to the moebius strip, we have:

Sb = Pa = 1

Sb = Sa/2 = 2/2 = 1

Sb = Pa/2 = 2/2 = 1

Sb = Sa/4 = 4/4 = 1

All four formulas agree that the moebius strip had 1 surface before the cut! Algebra has come to the rescue.

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#1

Re: Number Theory - Calculating e and The Moebius Strip

02/09/2008 6:19 AM

A mathematician confided,
That a Moebius strip is one sided,
You get quite a laugh,
when you cut it in half
'cos it stays in one piece when divided....

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#2
In reply to #1

Re: Number Theory - Calculating e and The Moebius Strip

02/09/2008 12:26 PM

Very good.

There once was a cat named Dell,

who fell into a well.

The water was freezin',

so he started sneezin',

and said "I'd be warmer in hell."

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#3
In reply to #2

Re: Number Theory - Calculating e and The Moebius Strip

02/09/2008 12:31 PM

Doctor Bell fell down a well
and broke his collar bone.
A doctor should attend the sick,
and leave the well alone....

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#17
In reply to #3

Re: Number Theory - Calculating e and The Moebius Strip

02/11/2008 7:11 AM

There once was a woman from Bude

Who went for a swim in the pond

A man in a punt

Stuck his pole in the water

And said, "You can't swim here, it's private".

<Splutter>

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#6
In reply to #2

Re: Number Theory - Calculating e and The Moebius Strip

02/10/2008 1:34 AM

A flea and a fly in a flue

Were imprisoned, so what could they do?

Said the flea, let us fly

Said the fly, let us flee

So they flew through a flaw in the flue.

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#4

Re: Number Theory - Calculating e and The Moebius Strip

02/10/2008 12:13 AM

The sister or cousin to the Moebius strips in the Klein Manifold, usually referred to as a Klein Bottle. It has no inside nor outside, just one side.

Glass Klein Bottles

Whatever you do, don't get trapped in one!

There are a few other interesting glass 'containers' on this site.

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#7
In reply to #4

Re: Number Theory - Calculating e and The Moebius Strip

02/10/2008 2:18 AM

I damn near got trapped on the linked site! Wow!

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#5

Re: Number Theory - Calculating e and The Moebius Strip

02/10/2008 12:42 AM

Here are some mathematical versions, sorry I didn't keep the formulas or the source but I think it was science mag. within the last year.

Brad

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#10
In reply to #5

Re: Number Theory - Calculating e and The Moebius Strip

02/10/2008 7:03 PM

Hi U_V,

An interesting picture. Thanks. The one on the lower left looks like a variation to the Moebius strip.

S

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#8

Re: Number Theory - Calculating e and The Moebius Strip

02/10/2008 11:07 AM

Suppose for a moment that we assume the hoop and the mobius strip to be hollow (very thin, but hollow), and we have a blow up valve positioned on each of them. Now, neglecting the fact that there is an inner surface, if we blow each of them up they both become- a toroid!

How many surfaces does a toroid have (again neglecting the inner surface)? One of course.

What happens if we cut the toroid along a line drawn circumferentially around its surface? We have two hoops. What happens if we cut the toroid axially, then twist it 1/2 turn, then sew it back together? We have the same toroid as before.

Just some food for thought.

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#12
In reply to #8

Re: Number Theory - Calculating e and The Moebius Strip

02/10/2008 7:20 PM

Hi JJ,

"Just some food for thought"

You are making me hungry. Do you have a toroidal donuts?

S

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#15
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Re: Number Theory - Calculating e and The Moebius Strip

02/10/2008 8:12 PM

Krispy Creme™ has been manufacturing toroids for quite a few years now.

Anyway S, don't you agree that a mobius strip is just a flattened toroid?

What would you get if you cut & paste a toroid that has a triangular cross section?

Hmmmm, I think I'll try it and see.

-John

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#16
In reply to #15

Re: Number Theory - Calculating e and The Moebius Strip

02/11/2008 3:50 AM

Hi..I just tried to 'cut and paste' the triangular section toroid.

The computer asked 'WTF have you done to 'clipboard' it now has no back or front'...the diplay on the monitor shrank to a small dot and the PC imploded.

I am sending you the bill for a new PC...

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#18
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Re: Number Theory - Calculating e and The Moebius Strip

02/11/2008 12:06 PM

Hahahahahahaha......!!!

I'm afraid all I have is Confederate currency to pay you with Del.

I have not yet tried to cut the triangular toroid but I will. For best visual effects I wonder if I should twist the cut section 1/3 turn or 2/3 turn before re-attaching? Maybe I should try both just for good measure.

The moebus strip is an aberation of the laws of physics! It's a ploy created by aliens to bend our feeble minds.

-John

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#25
In reply to #18

Re: Number Theory - Calculating e and The Moebius Strip

03/24/2008 4:18 PM

John-John, upon re-reading your commentary in #8, it shows that you were the one to first posit a hollow strip that could be twisted, connected, and then inflated, to make a hollow toroid, and yet (SS) "vents" on me, instead. Do you think, maybe, he should cut back a little bit on (the salt and the red meat)... You seem much more ready to (put the shoe on the other foot), and I appreciate the rigor of your mental flexibility. Thank you... Do you think that maybe "surface" and "edge" are mathmatically inadequate terms to describe the phenomena we have been trying to discuss... Maybe we should distinguish between Topological Space and Euclidean Space?... or even Mobius Space?...

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#27
In reply to #25

Re: Number Theory - Calculating e and The Moebius Strip

03/24/2008 5:57 PM

"...and yet (SS) "vents" on me, instead."

Who is me? As vermin says, "Horton hears a who?"

Be much easier Guest if you logged in and we knew your name.

-John

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#30
In reply to #27

Re: Number Theory - Calculating e and The Moebius Strip

04/03/2008 9:21 PM

Ah, I now see that I was wrong. I mistook rigid-mindedness for mental rigor. I now stand (enlightened?). Thankyou...

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#28
In reply to #25

Re: Number Theory - Calculating e and The Moebius Strip

03/24/2008 6:43 PM

Hi Guest,

As I said in my opening post "How can you have a distance between 1 edge?" But you can say that it has the property of "width". This is the same kind of problem that physicists have with sub-atomic particles, and they may have to do with more dimensions.

S

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#31
In reply to #28

Re: Number Theory - Calculating e and The Moebius Strip

04/09/2008 9:55 PM

Hmmmm, but wait a minute. If you had a non-zero, circular portion of a 2-dimensional plane, and you scribe a line through the center of said circle, from one side, clear across to the other side, through the circle's center, you would have a line that represents the diameter of said circle. Said line would be the finite length of the circles' diameter, by definition. However, said disk has only one edge, (ie: along it's circumference) and said diameter would necessarily go from one point on the disks' edge to the point which is directly opposite on the very same circumference. Is that not just one of the several ways that you can have a finite "distance between 1 edge"?...

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#32
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Re: Number Theory - Calculating e and The Moebius Strip

04/12/2008 3:36 PM

Hi guest,

I don't think I agree with your assessment. A 2-dimensional circle would only have 1 side by definition, but a circle has an infinite number of edges by my reasoning. A moebius strip can have thickness and still only have 1 edge because of the half twist.

regards,

S

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#33
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Re: Number Theory - Calculating e and The Moebius Strip

04/12/2008 6:05 PM

Hi S,

I repeat my earlier assertion: the mobius strip is simply a flattened toroid.

Let me rephrase, how many edges does a toroid have?

If I place a sphere in a press and compress it to flatness, does it then have an edge, or is it just a flattened sphere, or is it a two dimensional disk?

More generally, can we (three dimensional beings) create one less dimension than we start with? (I don't think so, we only create the illusion of doing so).

Conversely, can the Flat-landers of Two-dimensional world "inflate" anything in their world to become three dimensions? I suspect not!

-John

P.S. I'm still looking for a triangular tube that I can cut, reattach, and see how many surfaces it has. Any guesses? (a tri-mobius thing).

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#34
In reply to #33

Re: Number Theory - Calculating e and The Moebius Strip

04/12/2008 6:58 PM

Erratum to last post:

Concerning flattening a sphere into a disk, I'm a little puzzled here. Maybe you can shed some light on this, S.

Using standard formulas for area, i.e., a sphere: A = 4∏r2 and a disk: A = ∏r².

Now, is I take a sphere of, say, 2" diameter, A would be 4∏ or 12.56636".

If I compress the sphere into a disk, what would be the area? Seems to me it would be 12.56636" / 2, or 6.28318". Am I right? But what about the area of the "edge"? Does it just get thrown away?

Consider the flattened sphere:

Using A = ∏r², 6.28318 = 3.14159 (r²), then

√6.28318 = √3.14.159 (r) or

2.50663 = 1.7724 (r) or r = 1.41422

Therefore 0.41422 is the area of the edge! We can't just throw it away.

So we can't create a two dimensional object! QED (I was wrong once though and I stand to be corrected).

-John

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#35
In reply to #34

Re: Number Theory - Calculating e and The Moebius Strip

04/12/2008 11:49 PM

Hi JJ,

A toroid would have 1 edge, or infinite edges, depending on your viewpoint.

On the sphere flattened to a disk, it would be 2-dimensional by definition, so the edge would have no area. In practicality it could not be done.

Examine your math a little closer. If you cut the sphere in two you would have a hemisphere. It would have more area than a circle of the same diameter. A 2" circle would have a radius of 1, so A = ∏(12) = ∏.

We can't create a 2-dimensional object, but are there some in this universe? It seems unlikely. What if you had a sheet of material that was 1 Planck-length thick - but that is thinner than an atom, right? So the material wouldn't exist!

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#36
In reply to #33

Re: Number Theory - Calculating e and The Moebius Strip

04/13/2008 3:13 PM

Hello JJ,

You do not need a triangular tube. Take a flexible tube such as the one shown at the right. To represent the three sides of a triangular tube, paint it three different colors, say red, white and blue. Each color covers exactly one third of the circumference for the entire length.

Now rotate and reattach. How much do you plan to rotate? If you rotate one half turn, as in a moebius strip, the vertices will not align. If you attach one side as in a moebius strip, there will be five surfaces, one for the moebius strip and two for each of the others. If you attach so that the round hose meets correctly, you will have six surfaces, two for each of the three sides.

If you rotate 120 degrees before attaching, red aligns with white, white with blue and blue with red and all vertices meet correctly. You will then have two surfaces, one inside and one outside. The outside surfaces will have three different colors, red white and blue.

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#37
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Re: Number Theory - Calculating e and The Moebius Strip

04/13/2008 3:26 PM

Very interesting ba/ael!

But why do you want to inflict such pain on this simple mind?

I shall now go curl up in my soft toroid and sleep a while.

-John

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#38
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Re: Number Theory - Calculating e and The Moebius Strip

04/13/2008 3:37 PM

Sorry JJ, no pain intended.

How's the weather in Georgia?

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#39
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Re: Number Theory - Calculating e and The Moebius Strip

04/14/2008 12:02 AM

Call it what you like except a Moebius Strip. The construct in the drawing is nothing but a twisted or distorted torus. It has an inside and an outside and encloses a volume.

A Moebius Strip has ONE side, NO inside, therefore NO volume can exist.

Most posters to this thread would flat Flunk a course in Topology.

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#40
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Re: Number Theory - Calculating e and The Moebius Strip

04/14/2008 10:23 AM

Actually SS, you would flunk a course in Reading Comprehension. Any one of the three sides of a triangular tube could be a moebius strip. No reason why not. But the other two sides would be just going along for the ride and could not be moebius strips at the same time.

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#41
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Re: Number Theory - Calculating e and The Moebius Strip

04/14/2008 10:50 AM

If one side were a Moebius strip, how could you add the second or third strip, given that you have only a single edge to attach them to? A sort of "Y" shaped cross section? I confess that I cannot envision that sufficiently well to see whether the other strips in such a situation would, or would not be, Moebius strips on their own. Suppose we start with the strip cross-section and label it "A". We attach strip "B". It seems clear that the second apparent edge of "A" (in that cross-section) is merely another point along the first; must not the same be true for "B"? And so, doesn't adding "B" merely make "A" wider? Then where do we attach "C"? No time to play and make pseudo- Moebius Strips.

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#42
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Re: Number Theory - Calculating e and The Moebius Strip

04/14/2008 11:52 AM

Hi Ron,

Let's start with a single sheet of material. One dimension is considerably longer than the other. Take each end in hand and let them form a circular ring. Now, rotate one end through 180 degrees and glue or weld them together. You have a moebius strip.

Now, do the same with one side of a triangular tube. Where's the problem?

Make a ring out of the tube. If the side selected to be the moebius strip is horizontal at the point of contact, the other two sides form a 'vee' below it. All vertices meet. Now, rotate one end 180 degrees and glue or weld the ends together. Again you have a moebius strip, but the vertex outside the moebius strip is below it on one side and above it on the other.

One can form a moebius strip out of any side of a triangular tube (and by extension any multi-sided tube), but only one side at a time. The other sides will not abut end to end and cannot form moebius strips simultaneously.

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#43
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Re: Number Theory - Calculating e and The Moebius Strip

04/14/2008 12:37 PM

So now, "A" and "B" each go only halfway around the Moebius Strip, and do not join - quite a different concept than I thought was intended by earlier posts (#15, e.g.) regarding triangular tubes. But using this method, it is clear that the number of sides of the tube doesn't matter; they will simply not return to the starting point, but each be half the length of the MS edge (more or less). A square tube is no more difficult than a three-sided one, and clearly this must be general as we increase the count. We can extend the concept downward to a "D" shaped cross-section with only two sides, or even have a single line on a tube rejoin to give a zero-width Moebius with its twist implied by a series of local directional arrows (such as the diameter of the tube) along its length. Is the resulting figure-eight edge at the join topologically equivalent to having a "conventional" Moebius Strip with the edge touching itself at some point? Seems like it must be . . .

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#45
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Re: Number Theory - Calculating e and The Moebius Strip

04/14/2008 2:48 PM

Post #18 contemplated joining a triangular toroid, presumably on all three sides, after giving one end a rotation of 1/3 or 2/3 of a turn (i.e. 120 or 240 degrees). Either way, there would be two surfaces, one inside and the other outside. But there would be no Moebius strips. With a rotation of 0 or 360 degrees, there would be six surfaces, three inside and three outside.

I doubt that a line would qualify as a Moebius strip as it has no surface.

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#44
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Re: Number Theory - Calculating e and The Moebius Strip

04/14/2008 2:35 PM

Hi ba/ael,

My original post concerning a triangular tube was not meant to suggest a 180º turn but rather a 120º turn such that all sides align. If one takes it to the extreme of a circular tube, i.e., a toroid what do you get? I suspect a reduction in mobius properties. As S says (paraphrased), you deviate more and more from a "one-sided" object. I believe S will agree that this may be mathematically possible but not practical in the real world sense.

-John

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#46
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Re: Number Theory - Calculating e and The Moebius Strip

04/14/2008 2:58 PM

Hi JJ,

I was posting a reply to Ron when your post came through. I understood what you were originally suggesting. But in order to obtain a Moebius strip out of any plane, you must do a rotation of 180o. The 120o turn will not produce any Moebus strips.

As for the circular tube, there are clearly two surfaces, one inside and one outside. Both surfaces are curved, but so is the surface of a Moebius strip. The circular tube has no "Moebius properties" at all.

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#20
In reply to #8

Re: Number Theory - Calculating e and The Moebius Strip

03/20/2008 4:05 PM

Well, if you "built" a toroid by inflating a hollow mobius strip, and even filled it up with the same type of material as the original mobius strip, would not the resulting toriod still have an "inside" and an "outside"?... (that is, until you fused the filler material with the containing material, to then create a homogeneously solid toroid?...)

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#21
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Re: Number Theory - Calculating e and The Moebius Strip

03/20/2008 10:12 PM

"Well, if you "built" a toroid by inflating a hollow mobius strip," IT CAN'T BE DONE!!!

You have a thorough misunderstanding of a mobius strip. It is a SURFACE. IF it were hollow it would be neither a mobius strip nor a single sided surface

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#22
In reply to #21

Re: Number Theory - Calculating e and The Moebius Strip

03/20/2008 10:37 PM

Now, just hold on a minute S,

While I agree that, technically, it is a surface but lets be somewhat pragmatic here.

In your description, is a strip of paper twisted and reattached to itself, a mobius strip?

If you agree then you must admit that no matter how thin the strip is, it still has edges. Therefore, as I said earlier, if we could blow the thing up it would, at some point, become a toroid!

I have trouble imagining a true surface, as you say, that could be twisted, or shaped, in any fashion.

Thus a mobius strip is nothing more than a flattened toroid.

-John

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#23
In reply to #22

Re: Number Theory - Calculating e and The Moebius Strip

03/20/2008 10:53 PM

"In your description, is a strip of paper twisted and reattached to itself, a mobius strip?"

Welllll NO. A twisted piece of paper is a representation of a mobius strip. The mobius is a one sided SURFACE with no thickness. It is a 'Topological' entity which does not exist in 3-dimension reality no more that the Klein bottle does. The glass models made in CA are 3-D representations. <kleinbottles.com> These cute representations of the real thing have not volume yet you can fill one with water. How's for a paradox?

Sorry to have confused you! SS

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#24
In reply to #23

Re: Number Theory - Calculating e and The Moebius Strip

03/20/2008 11:19 PM

Hi S, No confusion. Just thinking about the concept. Granted, all we have is representations of theoretical entities, as you say. Suppose we turn it around. Lets take a donut (toroid), shrink it to infinite flatness, cut it, twist it, reattach it to itself, and since it has infinite flatness, it becomes a "surface?".

Wait! Before we shrink it, lets first cut it, then reattach it to itself, the shrink it to infinite flatness. Do we have a mobius "surface"?

-John

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#9

Re: Number Theory - Calculating e and The Moebius Strip

02/10/2008 4:44 PM

There is more to 'e' than meets the 'i' !

The NATURAL NUMBER 'e'

  • Every symbol has its history—the principal whole numbers 0 and 1: the chief mathematical relations + and =: Pi the discovery of Hippocrates: 'i' the sign for the 'impossible' square root of -1; and 'e' the base of Naperian logarithms. -----"The World of Mathematics" James R. Newman, Pg. 151.

Wikipedia has an article on 'e' with considerable information

The Mathematical Constan 'e'

e = 2.718281828459045235360287471352662497757+
The base of an exponential function equal to its own derivative: Sigma 1/n!

Among many other things, e is also the limit of (1 + 1/n ) n as n tends to infinity.

The letter e may now no longer be used to denote
anything other than this positive universal constant.


The base of an exponential function equal to its own derivative: sigma 1/n!

http://home.att.net/~numericana/answer/constants.htm#e

Natural Logarithms and 'e'

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#11
In reply to #9

Re: Number Theory - Calculating e and The Moebius Strip

02/10/2008 7:15 PM

Thanks Stan,

There is a lot of history in the first and third links. I couldn't get the second one to work. In the third one is a list of The Most Important Inventions. Do you agree?

S

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#14
In reply to #11

Re: Number Theory - Calculating e and The Moebius Strip

02/10/2008 7:29 PM

"In the third one is a list of The Most Important Inventions. Do you agree?"

Yes. Some of more importance than others, and some less. Life would be a bit harder as you take away even the seemingly insignifigant ones.

If i am not mistaken the second link is redundant.

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#29
In reply to #11

Re: Number Theory - Calculating e and The Moebius Strip

04/01/2008 10:14 PM

Said list is very interesting, even if it doesn't include arrowhead/spearpoint, integrated circuit, or Civil Service (the last of which was also invented by the Chinese). Civil Service is a more crucial invention than is generally appreciated. Historically, the Chinese came to realize that government positions should not all be filled by family members and political appointees. So, they decided that it might be better if certain, lower level goverment functionaries were selected on the basis of merit, rather than transient social/political expedience. They set up testing procedures to evaluate applicants on the basis of the highest scores thereon. And lo, and behold, things started to work a little better, a little bit more logically, and a little bit more affordably, so the practice of promotion on the basis of merit was expanded. After all, it sounded like a good idea and it worked, at first. But who tended to do better in the selection processes? Since the evaluations tended to favor those applicants with a well-rounded background, the selectees tended to be those applicants with the equivalent of a liberal arts education of those times. And how did that work in the long-run? Well, human nature being what it was, once a person got the job, they tended to become more and more concerned with keeping their particular job, and stifling competition for that job. Thus, the modern variety of government bureacracy was born. Thus, the expanding Chinese Empire of that era became more and more consumed with job security, and therefore, risk-avoidance, that their empire, which may have stretched from India to Peru, slowly stifled itself by concerning itself with maintaining the status quo of sqelching internal threats, and geneally scientifically progressing less and less until the outside world imposed Colonialism upon them a couple of centuries ago. Thus, a simple little idea like Civil Service slowly put the leading cultural and scientific civilization of it's time in a functional coma. This is important because, if it hadn't, we might all be speaking some dialect of Chinese by now...

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#13

Re: Number Theory - Calculating e and The Moebius Strip

02/10/2008 7:28 PM

Great. I was looking for one of you math types. Can someone tell me if it is coincidence that the value of the Feigenbaum Number is approximated by 10/(Pi-1)?

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#19
In reply to #13

Re: Number Theory - Calculating e and The Moebius Strip

02/13/2008 9:07 PM

Hi Guest,

I don't consider myself a math type, but I think it's coincidence. Do you use a Feigenbaum constant in your work? If so, tell us about it.

S

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#26

Re: Number Theory - Calculating e and The Moebius Strip

03/24/2008 4:32 PM

Anonymus, Guest, et all.

It's caused by too much Balsamic Vinegar on too much green salad stuff.

A mobius strip is technically a Topological delicacy as is a klein bottle.

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