some advanced maths...

[Marin85]

sphere eversion

I hope you´ll enjoy the video

Cheers

Marin

[mattbiernat]

amazing.... how about changing the sphere into infinitely small point and then turning it inside out? seems like much simpler solution. dunno if it is valid thou, i am no mathematician...

[jdhurst]

I think, but I have forgotten, that this comes from Topology. A like kind of theorem, for numbers, I posted elsewhere here. That theorem removes everything from something and leaves an infinite amount left over. I could not put my head around this one posted today.
... JDH

[Superego]

Pretty cool visualization of Smale's paradox. I'm not sure if reducing the sphere to a point would be valid, as a point technically has a dimension of 0. Maybe use a different type of dimension would aid, but topology really isn't my area so I'm not sure.

@ jdhurst: I may be wrong, but it sounds like you're talking about Cantor's set (which basically started the field of topology). You take a line segment and remove the middle third, and then the middle third from the remaining segments, and the middle third from those segments....and on and on. What's cool is that the number of points left is equal to the number of points you removed. I use the term "number" loosely because in actuality the set of points you're left with is uncountably infinite. Great stuff!

[egibbs]

Ok - time for a quiz.

You have a sphere with a hole drilled clean through it, leaving a donut-like ring. The hole is 10 inches from top lip to bottom lip.

(| |)

You know neither the diameter of the hole nor the diameter of the original sphere.

What is volume of the remaining donut-like ring? (You do have all the information you need to answer the question)

Ed Gibbs

[RealBlackStuff]

I'd say it's this:
Π = Pi = 3.14.......
r = radius = 10"
4/3Πr3
or perhaps clearer:
4/3 * Pi * (r to the power of 3)
You do the math.

[jdhurst]

<snip>
@ jdhurst: I may be wrong, but it sounds like you're talking about Cantor's set (which basically started the field of topology). You take a line segment and remove the middle third, and then the middle third from the remaining segments, and the middle third from those segments....and on and on. What's cool is that the number of points left is equal to the number of points you removed. I use the term "number" loosely because in actuality the set of points you're left with is uncountably infinite. Great stuff!

Yes - it was the Cantor-Ternary theorem I was thinking of. I studied real analysis, differential geometry and such like way back, but never got into Topology. All of that was decades ago, and I have forgotten most of it.

WRT the above, the number of sets you remove things is countably infinite, whereas the beginning set was uncountably infinite (the real closed interval 0 to 1), so what you have left over remains uncountable. But on top of that, the remaining set is closed (contains only closed sets because we removed only open ones), bounded (still between 0 and 1), compact (contains all of its limit points) and perfect (every point is a limit point). It is one of the niftiest sets going. The left over "zero" if I remember correctly was a calculation with Lebesgue Integration ... JDH

[Superego]

I studied real analysis, differential geometry and such like way back, but never got into Topology. All of that was decades ago, and I have forgotten most of it.

Good to see another mathematician on the board. My area is more applied mathematics....linear algebra, numerical analysis, scientific computation.

Regarding the Cantor set...yeah it's definitely nifty, especially when you consider that it has a non-integer dimension. Specifically, its dimension is (ln 2)/(ln 3). Not quite sure I follow what you mean by the left over "zero", but the set does have a Lebesgue measure of 0.

[jdhurst]

<snip>
Good to see another mathematician on the board. My area is more applied mathematics....linear algebra, numerical analysis, scientific computation.

Regarding the Cantor set...yeah it's definitely nifty, especially when you consider that it has a non-integer dimension. Specifically, its dimension is (ln 2)/(ln 3). Not quite sure I follow what you mean by the left over "zero", but the set does have a Lebesgue measure of 0.

With respect to the Cantor Ternary set, the Lebesge Integral is zero, but since it contains an uncountably infinite number of points, I don't know how else to describe it but "left over". Not well done on my part.

I studied applied mathematics as well. I remember well Anthony Ralston's text on Numerical Analysis. Also, some time back, Foreman S. Acton (IIRC) wrote a book, published by McGraw Hill that had the title in gold leif "Numerical Methods that Work". In depressed letters on the cover (i.e., not in gold leif) was the word "Usually". That was a riot. I saw the book in the library and now wish I had purchased a copy.

One of the interesting examples early in that book was also most interesting:

A prankster sees a length of rail one mile long pinned at each end. He (or She today) cuts the rail in two in the middle and adds in exactly one foot of rail. So the rail is now 5281 feet long and bows out from the straight line. How far out does it bow out?

Most people will figure a few inches. Answer tomorrow.

Edit: Saturday June 21 - Estimate the answer with straight segments, so: Answer = Sqrt[(2640.5 squared) - (2640 squared)] or 51 feet. Now, straight segments are a bit longer out than a graceful arc, so Newton's method (approximation using derivatives) will yield the correct answer of approximately 44 feet). Still the estimate gets in the ballpark quickly.

Numerical analysis was always fun. The big debate in computing back then was whether to use it or whether to try something more elementary.

To wit: A roadway will use a spiral to transition from straight to circle. That spiral is called a clothoid (IIRC) and balances the centriful forces from turning and from superelevation so that in theory, with a properly aligned vehicle doing the design speed, there is no need to steer. I have actually done just that. A clothoid is a beast to work with because the curve is an infinite power series expansion of both x and y coordinates. Intersecting two clothoids is not trivial if one uses pure methodolgy.

Now bridges intersecting other bridges or roadways are frequently within a clothoid, so working out the edges of the deck and working out vertical clearances can be a difficult exercise using diffential calculus (Newton's method or such like). If you run these calculations on a computer, and seed an approximation, the results sometimes run off into never-never-land.

So, back then, we decided to create roads and bridges with centrelines of 1-foot straight sections. Calculating intersections and clearances is now a simple and very fast sift-sort problem.

Where and when? I wrote the geometry programs for vertical and horizontal geometry on curved bridges across Highway 401 in Toronto back in the late 1960's for the Government of Ontario using Fortran IV on a 360-65 IBM Mainframe as a summer student (multiple years) when attending University. Very interesting work, and since the bridges are still there today, I think about this work a lot.
... JDH

[wearetheborg]

WRT the above, the number of sets you remove things is countably infinite, whereas the beginning set was uncountably infinite (the real closed interval 0 to 1), so what you have left over remains uncountable.

Thats not quite right. The number of sets being removed may be countable, but each of those sets is uncountable (eg, the interval (1/3,2/3). Hence it does not immediately follow that the lefover set is uncountable.

[jdhurst]

WRT the above, the number of sets you remove things is countably infinite, whereas the beginning set was uncountably infinite (the real closed interval 0 to 1), so what you have left over remains uncountable.

Thats not quite right. The number of sets being removed may be countable, but each of those sets is uncountable (eg, the interval (1/3,2/3). Hence it does not immediately follow that the lefover set is uncountable.

Like I said earlier, it has been a very long time, and I have forgotten much. However, with respect to the Cantor-Ternary set,
the language used can be found in Natanson - Theory of Functions of a Real Variable (as one source) as follows: "... By this process, we remove an open set G-zero from [0,1]; the set G-zero is the sum of a denumerable family of intervals ... The remaining set P-zero = C[0,1](G-zero) is perfect ... ". Natanson goes on "... The Cantor perfect set P-zero has power C (uncountable). The result from earlier shows that the Cantor set contains points in addition to the end-points of the removed intervals (which form a denumerable set).

So I said it wrong up above. However, the Cantor set is uncountably infinite.

Thank you for the correction. ... JDH