[trushinalexander49 (OP)]

Can you help me please with question assuming attached file. We assume that the radius is equal to 1. It is seen from the drawing that the sum of 2 others described will be selected on either side of the inscribed n-gon. But according to the well-known theorem, any side of a triangle is less than the sum of 2 other sides, which obviously implies a constant less than the inscribed n-gon in comparison with the described one. But we also see from the drawing:
2 * x [3] = (1-x [3]) * sqrt (2), whence x [3] = sqrt (2) / (2 + sqrt (2)), also x [1] = 1-sqrt (1- (2-sqrt (2)) / 4) and x [2] = sqrt (1 + 1 / (3 + 2 * sqrt (2))) - 1. Since x [1]> x [2] and each side of the inscribed n-gon has a side equal to the sum of the two sides of the triangle containing x [1] and x [2] for the described n-gon, we conclude that the limit of the described n -gon, although like the inscribed n-gon has a circle, the circumscribed n-gon tends to 2 * pi faster than the inscribed n-gon. As far as I know, no one paid attention to this fact earlier, trying to approve the concept of calculating the circumference of a circle in the wrong way by finding the supposedly zero limit of the difference between the variable perimeters of the described and inscribed polygons, assuming this while increasing the number of their sides. in this regard, I am sure that the limit of the difference between the circumscribed and inscribed polygons is not equal to 0. Am I right? Thanks a lot!

[alancalverd]

This seems to be a classic example of the failure of discrete approximations to converge to the analytic value.

Take the simplest example of a 45° right-angled triangle ABC where B = 90° and AB = BC = 1. We know the length of the hypotenuse CA is √2.

Now approximate the shape with n smaller triangles of side 1/n:  the path along and up the steps is 2 however large we make n.

This has important implications in education: you can't teach analytical geometry with a computer!

[trushinalexander49 (OP)]

you are right, but you misunderstood - in your example there is nothing that shows the fact of a faster tending to the limit of one than the other - I meant that because of the desire of the described polygon faster than the one inscribed to the same limit, the limit of the difference of their variables perimeters with the same number of sides and their simultaneous increase is not equal to 0.

[alancalverd]

It doesn't matter which tends to what. Neither tends to a circle.

[vhfpmr]

This has important implications in education: you can't teach analytical geometry with a computer!

I once said that 4/(0.4recurring) = 9, and got into an argument with someone who said that he had tested it with his calculator, and insisted that it was only an approximation.

[chiralSPO]

alancalverd is quite right here. One must be very careful when setting up limits, that the actual result is meaningful and accurate.

There is a commonly-taught numerical approximation for the area under the curve which can be found by adding the areas of increasingly thinner rectangular slices of heights that touch the curve above them. The narrower the rectangle, the closer the approximate area is to the real one. In the limit, this is simply the definite integral. However, this rectangles approach will never converge to the correct path length of the curve (unless the curve is a straight line with zero slope, in which case the approximation will be perfect). The sum of those edges will always just be the length of the domain over which the value is being calculated.

Switching to the trapezoid method (allowing for different edge lengths and heights within a slice by setting the height of each side to the value of the function at that position--essentially making the slice contain more information) makes only a slight difference for area approximation, allowing the approximated area to converge to the real area slightly faster than the simpler method (and the limit is, in both cases the actual integral). But by allowing for slopes of the line segments approximating the curve to change, this also will allow the path length to be approximated.

[alancalverd]

But you can teach a lot of physics and maths with an analog computer!

[charles1948]

But you can teach a lot of physics and maths with an analog computer!

When you mention "analog" computers, I've often wondered whether our modern fixation with "digital" computers comes simply from this:

Our present technical inability to make big computers operate reliably with anything more than crude binary "on" and "off" switches.  And what disturbs me even more - the switches used are transistors!

The transistor is capable of "analog" functions, just as its predecessor, the "thermionic valve" or "vacuum tube" was.
To see the potential of this beautiful device wasted, by using it as a simple switch, is disappointing.

[alancalverd]

The problem with analog computers is a lack of stability. "Zeroes" tend to drift, so integration loses accuracy, and differentiation makes the system vulnerable to noise. There were some very sophisticated analog machines around in the 1950s, including some wonderful naval artillery and antiaircraft gunnery directors, but credibly simulating a generalised system with more than half a dozen variables or operators was very much a matter of luck.

[evan_au]

The transistor is capable of "analog" functions

I was introduced to "hands on" analog and digital computers in the same summer holiday program when I was at high school.
- These analog computers did not use transistors as a building block, but the basic building block was an "operational amplifier", a circuit consisting of many transistors. It also needed precision resistors and capacitors, and wired up with a lot of flying leads. You could hope for an accuracy of perhaps 0.1% (providing the integrators did not drift away to infinity, as Alan observed). Looking back, I think that pickup of Radio Frequency Interference in the wiring would be a real problem for differentiators....
- They used BASIC for the introduction to digital computers, using a digital approximation to an integrator. We quickly learned that low-order integrators are not very useful! But overall, the digital simulation could provide more decimal places (if your algorithm could generate more decimal places!)
- The subject of the exercise was simulating the behavior of a nuclear reactor during transients (and included a visit to 2 research reactors)
- As someone who had played with electronics for many years, the analog computer came more naturally to me. Others found the digital computer easier.
- A time-shared BASIC system could handle far more students at once than an analog computer.
- We know where this ended - digital computers have taken over, and the analog computers are now museum pieces
https://en.wikipedia.org/wiki/Analog_computer#Electronic_analog_computers

However, we know that digital computers cannot match the energy efficiency of the human brain for many real-world problems (eg self-driving cars). Part of this is due to the analog aspects of neurons
- Researchers are looking at low-accuracy analog computing using transistors in a neural network with pre-computed weights
- Unfortunately, our current techniques for computing the weights requires high-power, high-precision digital calculations (unlike the human brain, which dynamically refines the weights without using any more power)

[evan_au]

I am sure that the limit of the difference between the circumscribed and inscribed polygons is not equal to 0. Am I right?

I ran the arithmetic through a spreadsheet, assuming a unit circle (radius r=1 unit).
- Perimeter of a unit circle = 2πr = 2π
- Area of a unit circle = πr² = π
- I did cheat, using trigonometry functions to determine the lengths of sides, instead of exclusively Pythagoras' theorem (no ancient Greek geometer would accept using an infinite series!)

Polygon_pi.png (23.02 kB . 759x232 - viewed 4567 times)

It seems to me that the perimeter of the inscribed & circumscribed polygons do converge to a good estimate of π
- And the difference between the inscribed and circumscribed polygon perimeter does converge to 0
- The average of inscribed and circumscribed polygons converges even faster
- The same trend is visible when you look at the polygon areas
See: https://en.wikipedia.org/wiki/Approximations_of_%CF%80#Polygon_approximation_to_a_circle

Switching to the trapezoid method

Effectively, the polygons do converge to the curve of the circle, so the errors of rectangular integration don't apply here.
- But I did calculate π using both the area and the perimeter. Both work ok.

[trushinalexander49 (OP)]

Thank you very much for your computer check, but if I may say so, I have a strong belief that, although there is a really very small number in the limit, it is still not zero, and for some reason, imagine that we took the described and inscribed polygons with a fixed equal number of sides and steel exactly at the same time to gradually double the number of sides of these polygons, leading their variable perimeters to the length of the circle - but then if the number of doublings "necessary" to "reach" the circle for the described polygon is n, then for the inscribed, due to the different speed of tending to the circle, it will definitely not be n , but there will be some functional dependence on n (albeit artificially selected, but definitely not equal to n) -could you share your thoughts on this matter-what do you think? Thank you so much!

[chiralSPO]

By definition, you can never reach the limit.

So you're right that for any finite value of n, there will be a disagreement between inscribed and circumscribed polygon models of a circle. But as n increases, the difference between these two models decreases.

And it's not just that the difference between them gets smaller (we can define δ = perimeter of circumscribed polygon – perimeter of inscribed polygon). There is no number n for which the inscribed polygon will have a greater perimeter than the circumscribed polygon (ie δ cannot be less than zero). And there is no limit greater than zero for this difference. Find me a number greater than zero, and I will tell you which value of n allows δ to be less than that number, whether it's 0.001, or 0.000000000000000001 or whatever.

[trushinalexander49 (OP)]

What exactly this number is is an important but another question, I will definitely consider it, but we have (if we combine each of the statements) an indisputable fact that 1. for fixed initial polygons, we simultaneously (!) Double their sides, 2. variable perimeter of the described polygon always larger than the inscribed of the same name, 3. just in the limit they are equal to each other and equal to the length of the circle but 4. just with the simultaneous doubling of the sides in the limit, this is due to the different speed of tending to the circle a different number of doublings, due to the faster tendency the limit of the described polygon, its sides once at infinity of the same rank (which has to be indisputably considered with the simultaneous doubling of the sides) will catch up or overtake the sides of the inscribed, but this means (when catching up), due to the constant largeness of the described polygon, reaching in the limit of the circumference, but the same rank infinity inevitably means the same absolute time of striving of the sides to the point, and hence the classic the problem of the speed of convergence of points - it follows from it that when finding the limit of the difference in perimeters, we will inevitably stumble at infinity of the same rank on catching up or overtaking the corresponding sides in the length-overtaking due to the constant largeness of the described polygon, there cannot be maybe due to the disproportionate speed reduction of the sides - and it follows that the ranks of the infinity of the described and inscribed polygons are different, that is, in the limit with the same number of doublings - paradoxically at first glance - but it is logical that when the described polygon "reached" circle-inscribed has not yet reached-about a specific numerical value, I will definitely investigate

[evan_au]

1. for fixed initial polygons, we simultaneously (!) Double their sides

This method of estimating π by using inscribed and circumscribed polygons dates back to ancient Greek mathematician Archimedes.
- The Ancient Greeks only approved of doing geometry with compass and straight-edge.
- They knew of no way to trisect a general angle (and now we know that there is no such method)
- But they did know how to draw a hexagon in a circle, or a hexagon outside a circle
- They knew how to bisect a general angle or a general line
- So this method repeatedly bisects the angles of a hexagon, giving polygons with 12, 24, 48 etc sides for both inner & outer polygons
- The Greeks did not have the option of using different numbers of polygons inside & outside, since they did not know how to construct them
- ...and it wouldn't have made much difference anyway, since a straightedge and compass cannot give accurate numerical values - the width of your pencil line introduces inaccuracy.
- Numerical calculations were extremely tedious with their number system - the abacus would be more effective.
- They could construct square roots geometrically, but there is no finite numerical method to calculate general square roots (as needed for Pythagoras' Theorem)

the different speed of tending to the circle

The Ancient Greeks didn't know that.
- We have the advantage that we know the "right" answer, which my spreadsheet shows as 3.14159265359...
- We can compare the calculated error for inscribed & circumscribed polygons, and see that they do converge at different rates.

Polygon_errors.png (20.83 kB . 657x235 - viewed 4498 times)

For the ancient Greeks, it was a huge advance to be able to estimate π as "between 3¹⁰/₇₁ and 3¹/₇", (3.1408... and 3.1428) using a 96-sided polygon.
- It was good that they could get an upper and lower bound
- This would have been enough to exclude the "Indiana Pi Bill", which implied a value for pi around 3.2
See: https://en.wikipedia.org/wiki/Indiana_Pi_Bill

Mathematicians today go to extreme lengths to derive upper or lower limits on mathematical conditions, if they can't prove an exact answer
- So the Greek method of calculating π is actually very successful, yielding a very narrow bounds for a practical calculation (96 sides)
- And it would have yielded the exact answer as the number of sides approached infinity.
- We only know the "right" answer because our computers uses a method that only gives the right answer as the number of calculations approaches infinity (which can never be reached). So we have no reason to criticize the ancient Greeks!

 Editorial comment: Alexander, try breaking your post into paragraphs and sentences - it is a bit breathless, as written.

[syhprum]

In the early eighties I worked with HELL scanners that used a very elaborate analogue computer to convert the RGB scans to CMYK to produce printing plates.
The chief problem was not zero drift but the jittery signals produced the 50 or so consumer grade potentiometers
When I first got to service one I was much relieved to note that the potentiometers were of an open construction so that they could be squirted full of WD40 that got everything working smoothly for a month or so.
I came to HELL from Muirhead that used simple analogue computers but always had a speaker on line so that you could always detect a jittery potentiometer which I tried to convince HELL to incorporate but the Germans would not listen to a technician from a primative third world country

[evan_au]

the simplest example of a 45° right-angled triangle ABC

This is also a case that the Ancient Greeks could have constructed, since they knew how to construct a square inside & outside a given circle.

Here are the results for successively doubling the sides of a square:

Square_Polygon.png (20.77 kB . 658x235 - viewed 4360 times)

It does converge to the right value. But the recorded result from Archimedes started with a hexagon, perhaps because a square clearly does not fill a circle, but a hexagon starts fairly close

PS: Inscribing a square inside a given circle is very different from producing a square with the same area as a given circle, which can't be done with straightedge & compass (in a finite number of steps).

[charles1948]

I wonder whether "pi" is all that important anyway.

It just seems to be a number.  Why should a particular number have any significance?

I mean, the number of fundamental forces in the Universe is currently supposed to be 4.

Suppose we found the number should be increased to 5.  Why would that be important?

[chiralSPO]

I wonder whether "pi" is all that important anyway.

It just seems to be a number.  Why should a particular number have any significance?

There are some numbers that appear to be of particular significance purely mathematically. Pi shows up basically anywhere curves are involved. Another number, e, appears to be even more ubiquitous, appearing across a range of seemingly unrelated contexts. Here's a nifty video that goes through several examples:

[evan_au]

I wonder whether "pi" is all that important anyway.

In it's traditional role as "the ratio of the circumference to the diameter of a circle", π comes out of Euclid's geometry.

However, that is not the only kind of geometry.
- If you imagine the shape of the Earth as an ideal ellipsoid, and draw a circle on it, the ratio of the circumference to the diameter of that circle is < π
- Last I heard, cosmologists were unsure: If you drew a circle on the scale of the observable universe, whether the ratio of the circumference to the diameter of that circle would be less or more than π

Why should a particular number have any significance?

Some numbers are important for life as we know it.
- For example, if the attraction between protons & neutrons was just a little stronger, all hydrogen would have fused in the Big Bang, and we would not have hydrogen to make our organic molecules
- If it were a little weaker, repulsion between protons would have dominated, and we would not have the carbon we need for organic molecules.

There are debates about how much variation is possible, but the universe would look very different if some of these numbers varied even a little.
See: https://en.wikipedia.org/wiki/Fine-tuned_universe#Examples

I mean, the number of fundamental forces in the Universe is currently supposed to be 4.
Suppose we found the number should be increased to 5

The discovery here would be discovery of the existence a 5th force, not creation of a 5th force.
- If such a force exists now, we are living quite happily with it, and its discovery would win a Nobel prize.
- But the discovery won't suddenly destroy all life on Earth (unless we start to use it against each other)