Finding Pi by Archimedes' Method (Follow-up)
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- เผยแพร่เมื่อ 8 เม.ย. 2012
- The previous video on this topic ( • Finding Pi by Archimed... ) contained a logical shortcut for the sake of simplicity of the presentation. This one shows the extra extra steps needed to make the method more logically correct. Anyone with a background in high school algebra who knows about similar figures should be able to follow the argument.
See my website: mathwithoutborders.com
This is a great way to use a spreadsheet. Thanks for the demonstration.
Excellent tutorials and demonstration of use of Excel
Kudos for including the circumscribed series of polygons, as Archmedes did; most YT videos that show his method only work the inscribed ones.
Incidentally, the excess found with the circumscribed polygons is, in the limit, exactly twice the defect found with the inscribed ones.
This can be seen by looking at the Taylor series for sin and tan. Because the approximations they give for π are:
P(ins, n)/2r = n sin(π/n) = π - ⅙π³/n² + O(1/n⁴)
P(cir, n)/2r = n tan(π/n) = π + ⅓π³/n² + O(1/n⁴)
Fred
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The belief seems to be that the Greeks never found calculus, but they were just a tiny step away. Maybe some of their findings have been lost or obscured.
Maybe they did but were just not driven by the need for it.
We are still tyring to piece it all together.
Then u have to calculate in how rods and cones work. And what do u define as the center. Keeping in mind that a cathode ray tube has infinite bandwith...
So, multiplying 6, the base number of sides, by 2^n (starting with n = 0) means that we have relative convergence at n = 24. Something very interesting about this.
THE NATURAL CIRCLE AND ITS SQUARE
ABSOLUTE
INCONTROVERTIABLE SIMPLE ARITHMETIC
Given a "Diametric Distance" of 120-centimetres.
1. Multiply the 120-centimetre Diametric Distance by 3.
2. The length of distance to the Circle's Edge is 360-centimetres.
3. The length of distance to the Circle's Edge is 360 degrees.
4. Each degree of distance to the Circle's Edge is 1-centimetre in length.
Squaring the Circle
5. Multiply the 120-centimetre “Diametric Distance” by 4, the perimeter length of the Circles Square is 480-centimetres.
6. The Circle is 360-centimetres and 360 Degrees in length, which is three-quarters of the length to its 480-centimetre perimeter Square.
Simply
Three times the length of…A Line…is the length of the lines Circle.
Four times the length of… A Line…is the length of the lines Square.
Questions
1. When we look at the shape of a bright yellow full moon as it is being silhouetted against the dark background of the night sky, does the full moon have a circumference - circumferential outline?
Answer
No, it does not; the full moon is a yellow coloured round circular area of shape; which is being contrasted against the greater surrounding area, of the darkness of the night sky. to produce a round silhouetted circular shape that does not possess an outline.
2. If we take a black marker pen and draw a black circle at the centre of a sheet of yellow A4 paper, does the yellow round circular shape in the middle of the paper have a circumference - outline?
Answer
No, it does not; the yellow round circular shape of area in the middle of the paper is being contrasted against the surrounding area of blackness belonging to the circumferential thickness of another circumventing black circular shape. And the circumferential thickness of the area to the black circular shape is its turn is being contrasted against the lighter background of the rest of the yellow A4 paper.
Question
3. When we look at a tree in the brightness of daylight, does the shape of the tree possess an outline?
Answer
No, it does not; the darker area belonging to the shape of the tree is being contrasted against the greater surrounding area, of the brightness of daylight and the blueness of the sky.
Simply
Shapes are not geometric; they are the visual forms of things that exist in nature, which are made visually manifest by the presence of a surrounding and contrasting background. And the surrounding and contrasting backgrounds are made visually manifest according to six aspects of visibility; shades of darkness, shades of brightness, shades over distance, shades of perspective, shades of colour, shades of texture.
In nature as opposed to Euclidean applied geometry and mathematics in physics, there is no such thing as a circumference outline or a line.
*Sidebar*
All things in nature are comprised of primal electromagnetic particles, larger particles, and larger groups of sub-atomic particles which form atoms and all of these particles *invisibly coexist at the quantum level*, in a perpetual state of interactive motion.
In order for the particles and the atoms at the *quantum level* to be able to manifest at the molecular level of visible structures, there has to be a vastly larger gravitational body present, which first draws them into its gravitational field; and then gravitationally compresses and ***aligns the atoms together interactively*** to form solid molecular structures.
At the level of our 20-20 vision molecular structures (e.g. elemental crystals and solid bodies) do appear to possess straight linear aspects to their structures, however as any electron microscope will confirm appearances, are deceptive.
However, we do not need an electron microscope to confirm that this is the nature of all things, all we need to do is look out into the night sky toward the constellations of the stars. And there, although we see what appear to be the stars formed into linear shapes and patterns, there are no actual lines between them, for it is we are who are responsible for aligning them in the imagination of our minds- eye.
Concerning two-millennia of disingenuous Grecian-Roman Euclidean education (brainwashing).
Quote: Stuart Close
For those who believe no proof is necessary, for those who do not believe no proof is possible (You can take a horse to water, but you cannot make it drink).
Reality Versus Fiction
The genius of stupidity is that the stupid are too stupid to realise, that they are too stupid to be the geniuses; they stupidly assume themselves to be.
The genius of intelligence is when the intelligent reach a point whereby they are so humbled in the face of the awe-inspiring intelligence of our Cosmic Mother Nature, as to realise. There is no such thing as to any one of us being a genius, for a tendency toward genius, lies-only within the realms of the ingenuity and the genius of our Cosmic Mother Nature.
www.fromthecircletothesphere.net
www.geometry-mass-space-time-.com
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To 15 significant figures, an arbitrarily chosen precision level.
Fewer figures ==> fewer steps (= n)
More figures ==> more steps
"Relative convergence" is just that-relative. There's nothing special about 15 figures; it's just what the application-computer combination he's using, gives.
Each additional (doubling) step puts you 4 times as close to π.
So 5 steps gains you 3 more digits.
Fred
Thanks :)
could you possibly provide me with your first and last name so I am able to reference this video in an essay on Archimedes for my university mathematics course? many thanks
Notice the name of the channel is MathWithoutBorders? Google it and you will find the website of the guy where his name and education informed
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I personally prefer the other method because it converged faster but this is also really nice.
which application are you using? anyone, any idea?
looks like GeoGebra
How to find Pi ? Draw a circle of radius 0.5. Draw another radius at 90 degrees. Join 2 ends. We get chord 0.707106781...Half of it is 0.35355339...Take 7 radii equal to 3.5 and subtract 0.35355339.. and we get 3.146446609...It is Circumference and also Pi (14 - root2)/4.. Truth is a roaring fire. --- SJR.
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You were doing just fine until, "It is Circumference..." - It isn't.
It's pretty close, but not equal. Too large by a bit more than 1 part in 1000. Still good for many practical purposes.
Archimedes proved more than 2000 years ago, that π lies between these two fractions:
223/71 < π < 22/7
And your value lies outside that range:
22/7 < 7/2 - 1/(2√2)
and is therefore greater than π.
"Truth is a train. You can either get aboard, or get run over by it."
Fred
Where as it is logical to accept value of Pi to be between the value calculated using internal and external polygons, how do we explain that Pi has been calculated in excess of 20 digits by mathematicians. Could it be due to our calculation of sqrt...sundar.
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