AP's 256th book:: Exploring Pi further and can we get it from pure physics// math research
4m views
Skip to first unread message
Subscribe

Archimedes Plutonium's profile photo
Archimedes Plutonium
1:22 AM (19 hours ago)



to
Make this my 256th book of science.

AP's math puzzles August2023// Determining Pi in Ancient Greek times Alright, it is brought to my attention a challenging puzzle. And I put the setting in history of Ancient Greek times, that of Socrates, Plato,Euclid and Archimedes.
Archimedes Plutonium's profile photo
Archimedes Plutonium<plutonium....@gmail.com>
Aug 9, 2023, 5:40:44 PM (2 days ago)



to Plutonium Atom Universe
AP's math puzzles August2023// Determining Pi in Ancient Greek times

Alright, it is brought to my attention a challenging puzzle.

And I put the setting in history of Ancient Greek times, that of Socrates, Plato,Euclid and Archimedes.

They had no decimal number systems, and used words for numbers, their alphabet substituted for numbers-- alpha, beta, gamma etc. Very awkward.

In modern times we have measuring rulers of meters, broken down into 100 centimeters or a thousand millimeters. And based on the emission line of Krypton-86. Later to the length of light traveled in terms of time seconds.

Now in Ancient Greek times, what would their Ruler lengths be marked in??? Would anyone have their own special ruler?? Where a person gets a straight enough piece of wood, then makes notches along a length and manufacture their own ruler.

Some bright ones would mark distance as 1 and have 10 notches before they got to 2.

So we are back in Ancient Greece with Euclid and Archimedes and we just made our own wood Ruler with 10 evenly spaced marks between 0 and 1 and 1 and 2. Our ruler goes out to 4, or 40 of the smaller markers.

Now we build a Square with side 1, or perimeter of 4.

We use a compass and build a inscribed circle in square.

Now we are tasked with a challenging problem of estimating the number pi, not by getting a small chain to lay out along the circumference and measure the distance length of chain and compare to diameter length of 1. This method tells us it is between 3.1
and 3.2.

Instead, we measure pi by a equilateral triangle whose side is also the same as the side of the square that encloses the circle.

Problem Challenge

Make a layout of the equilateral triangle such that it indicates that you need a larger distance length than 1 for the sides of the equilateral triangle. In fact you need 3.14 / 3 = 1.04. But this is so close to side 1, that the Ancient Greeks would have
settled for pi being 3.

Second Challenge

What in the environment of Ancient Greeks could have stood in as a replacement (instead of Krypton) as a measuring ruler.

I remember Archimedes talking about poppy seeds. But perhaps their size varies depending on where grown. I am looking for something that the whole of Ancient Greek and Ancient World could have used for a standard measuring ruler gradation. And come to
think of it, there really is not much other than perhaps poppy seeds. Say, 100 poppy seeds makes length 1, and 1,000 poppy seeds makes a length of 10.

AP

Archimedes Plutonium's profile photo
Archimedes Plutonium<plutonium....@gmail.com>
Aug 11, 2023, 11:13:45 AM (14 hours ago)



to Plutonium Atom Universe
Alright, get out a sheet of Graph paper and that has 10 small squares long and high. Then with compass draw a circle inside, (5,5) center inscribed inside square.

Now at (0,0) and (10,10) call it 1 unit. Our circle is circumference of 3.14. Our square is perimeter 4.

Now we proceed to graph in the square a equilateral triangle 60-60-60. The points are (0,0), (10,0) (5, 8.66). From this we can see that pi needs a equilateral triangle whose sides are a tiny bit more than 1 unit but needs 3.14/3 = 1.047.

If pi were square root of 10 = 3.162277 we need 1.054

Now what is the Sigma Error here? 1.054/1.047 = 0.6% good enough to claim equality.

So, now, suppose I was in Ancient Greek times, amoung Socrates, Plato, Euclid and Archimedes and wanted to know what pi was for the first time.

Naturally I would get a fine chain to lay out along the circumference of a circle which I then lay out on a gauged ruler to find a 3.1.

But what if I did not have a good enough chain nor a good enough graduated ruler for this is Ancient Greek times.

What if I wanted to know pi from just reasoning? I would make the square with circle inside and see that the equilateral triangle of 3 sides of unit is putting pi at 3 with a tiny bit more.

But what if I wanted to find this tiny bit more from pure reasoning? Is there anything I could do?

Yes, suppose I knew the decimal number system and could find the square root of 10. It is 3.162277....

I could play around multiplying to figure the square root of 10 is indeed 3.162277.....

And so, if asked-- what is pi, Mr. Plutonium, what is pi. I would answer in Ancient Greek times, it is 3.16.

AP
Archimedes Plutonium's profile photo
Archimedes Plutonium<plutonium....@gmail.com>
Aug 11, 2023, 4:06:01 PM (9 hours ago)



to Plutonium Atom Universe
On Friday, August 11, 2023 at 12:01:10 PM UTC-5, Jim Pennino wrote:

Archimedes Plutonium <plutonium....@gmail.com> wrote:

AP's math puzzles August2023// Determining Pi in Ancient Greek times

As usual, AP is full of it.

See:

https://qz.com/637633/the-history-of-why-pi-equals-3-1415926

Both Archimedes and Ptolemy used inside and outside polygons around
a circle to calculate pi.

Archimedes got to three accurate digits, i.e. 3.14.

Ptolemy got to 5 accurate digits, i.e. 3.1416.

<snip remaining AP nonsense>

As usual, Pennino with his 1/2 marble brain of science. What was the "ruler" used in Ancient Greece for either Ptolemy or Archimedes? A ruler in which you can speak of a 1/10, a 14/100, a 141/1000.

So, according to Pennino, we then suppose Archimedes had his own ruler based on something, and Ptolemy likewise built his own ruler based on something.

Drawing polygons inside and outside a circle may give you 3.1 in Ancient Greek times, but will not give you 3.14.

Flexible wire, or a flexible twig or branch shoot will give you 3.1, but never 3.14. Not even a chain will give you 3.14.

But the Ancient Greeks did know of square roots and in one day could have calculated square root of 10 to be 3.162. Even calculated it as 3.1622, then 3.16227.

The tools and implements available in Ancient Greek times, even a ruler with gradation marks could not give Archimedes or Ptolemy 3.14 the 14/100 for precision.

Now, the question is for a modern day computer is the question of whether square root of 10 has its widest off the mark reading of pi at 3.162 but thereafter the Sigma Error is best at 0.63%.

So we have 3.16/3.14 = 0.63% error, while 3.162/3.141 = 0.66%, 3.1622/3.1415 = 0.65%, 3.16227/3.14159 = 0.65%

Everyone who ever looks into Ancient Greek math history, never reports about the crude instruments of that past time. And so, what these historians end up doing is projecting our Modern Day results upon the Ancient Greek mathematicians and erroneously
reporting that Ptolemy got to 3.14159. When in reality they only got to 3.1.

The best mechanical way of deriving pi is by a chain laid over the quarter-circle that is able to be taut chain and with a almost modern day like ruler.

So, we have to ask, when in history, did mathematicians have the almost Modern Day Ruler of equal gradation marks? Who of the Ancient Greeks speaks of a carefully designed and made gradation Ruler?? And when do we have chains? I suppose a bead chain
would do to give 3.1 but not 3.14.

AP
Archimedes Plutonium's profile photo
Archimedes Plutonium<plutonium....@gmail.com>
Aug 11, 2023, 5:42:24 PM (8 hours ago)



to Plutonium Atom Universe
Alright, on my drawing table I have a graph paper with a square and inside the square is a circle inscribed and then from the two points on bottom of square (0,0) and (1,0) are two vertex points of a equilateral triangle trying to form a equilateral
triangle which gives 3 metric distance units of pi being 3.1415..... So I am looking to see how to make the equilateral triangle be 3.1415/3 to fit the circumference of circle exactly.

The circle center is (0.5,0.5). And the point (0.5,0.866) is significant as the vertex of a equilateral triangle that gives 3 of the value of pi.

Now I look to the Physics constants of their prefix digits. I look to see if any prefix digits are 8.66.

There is a very important constant of 8.854*10^-12 F*m^-1 electric permittivity

There is another important constant of physics 1.256*10^-6 N*A^-2 magnetic permeability

I am looking to see if those two physics constants are linked to the value of pi.

AP

Archimedes Plutonium's profile photo
Archimedes Plutonium<plutonium....@gmail.com>
Aug 11, 2023, 6:18:13 PM (7 hours ago)



to Plutonium Atom Universe

I pegged Jim as a engineer. Engineers rarely understand theoretical science, physics or math. And the history of pi is a good way of testing a engineer who pretends to know science.

On Friday, August 11, 2023 at 5:16:14 PM UTC-5, Jim Pennino wrote:

Archimedes Plutonium <plutonium....@gmail.com> wrote:

On Friday, August 11, 2023 at 12:01:10 PM UTC-5, Jim Pennino wrote:

Archimedes Plutonium <plutonium....@gmail.com> wrote:

AP's math puzzles August2023// Determining Pi in Ancient Greek times

As usual, AP is full of it.

See:

https://qz.com/637633/the-history-of-why-pi-equals-3-1415926

Both Archimedes and Ptolemy used inside and outside polygons around
a circle to calculate pi.

Archimedes got to three accurate digits, i.e. 3.14.

Ptolemy got to 5 accurate digits, i.e. 3.1416.

<snip remaining AP nonsense>

As usual, Pennino with his 1/2 marble brain of science. What was the "ruler" used in Ancient Greece for either Ptolemy or Archimedes? A ruler in which you can speak of a 1/10, a 14/100, a 141/1000.

No ruler, just mathematics that you will never understand.

So, according to Pennino, we then suppose Archimedes had his own ruler based on something, and Ptolemy likewise built his own ruler based on something.

Nope, pure mathematics.

Drawing polygons inside and outside a circle may give you 3.1 in Ancient Greek times, but will not give you 3.14.

Except it did and does.

<snip remaing drivel>

Jim has a attitude problem and probably holds him up in ever understanding science.

--- quoting Wikipedia ---
The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes. This polygonal algorithm dominated for over 1,000 years, and as a result π is
sometimes referred to as Archimedes's constant. Archimedes computed upper and lower bounds of π by drawing a regular hexagon inside and outside a circle, and successively doubling the number of sides until he reached a 96-sided regular polygon. By
calculating the perimeters of these polygons, he proved that
223
/
71
< π <
22
/
7
(that is, 3.1408 < π < 3.1429). Archimedes' upper bound of
22
/
7
may have led to a widespread popular belief that π is equal to
22
/
7
. Around 150 AD, Greek-Roman scientist Ptolemy, in his Almagest, gave a value for π of 3.1416, which he may have obtained from Archimedes or from Apollonius of Perga. Mathematicians using polygonal algorithms reached 39 digits of π