An Algorithm to Approximate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI=(Extended Version)by Dr.phil. Friedrich FUTSCHIKretiredThe NETHERLANDSe-mail: fflo@tele2.nl

This worksheet demonstrates the use of MAPLE , by showing how to approximate the ratio between the perimeter and the diameter -- or between the area and the squared radius -- of a circle. Since the 18th century this ratio is known as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= , and since 1882 it is known to be a transcendent number. This is an irrational number, which satisfies no algebraic equation but can be calculated, by means of infinite series, to any desired number of decimal places. Using computers, by the end of the 20th century more than 200,000,000,000 digits were known.Following the traces of Archimedes (c. 250 BC), his idea was implemented by the author, using the modern means of a computer and the analytic computation system MAPLE.

The author was using his computer system: Intel Pentium II / 64 MB RAM / 8 GB hard disk / MS-Windows98 (oct.1998)and (therefore) the Classic Worksheet of MAPLE 9.01 .

Archimedes inscribed and circumscribed a circle with hexagons, divided each of the triangular sectors in half and obtained 12-gons, one inscribed, the other circumscribed. He then divided each of the triangular sectors of the 12-gons in half and obtained 24-gons, etc. etc. until he obtained 96-sided regular polygons. Calculating the sum of the areas of the in- and circumscribing triangles, with patience and without computers, he arrived at a lower and an upper bound for the area of the circle, whence at an approximate average value for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= of about 3.1418 .

The necessary data for plotting this figure are as follows: restart;with(plots):with(plottools):T:=textplot([2.5,1,`Figure 1`],align=RIGHT):c:=circle([1,1],1,color=red):p:=point([1,1],symbol=CIRCLE,symbolsize=10,color=red):A:=point([1-sqrt(2)/2,1+sqrt(2)/2],symbol=CIRCLE,symbolsize=10,color=black):B:=point([1+(sqrt(2))/2,1-sqrt(2)/2],symbol=CIRCLE,symbolsize=10,color=black):l1:=line([1,1],[1+(sqrt(2))/2,1+(sqrt(2))/2]):l2:=line([1+sqrt(2)/2,1-sqrt(2)/2],[1-sqrt(2)/2,1+sqrt(2)/2]):l3:=line([1-sqrt(2)/2,1+sqrt(2)/2],[1+sqrt(2)/2,1+sqrt(2)/2]):l4:=line([1,1],[1,2]):l5:=line([1+sqrt(2)/2,1+sqrt(2)/2],[1,2]):l6:=line([1-sqrt(2)/2,1+sqrt(2)/2],[1,2]):l7:=line([1-sqrt(2)/2,1+sqrt(2)/2],[1-sqrt(2)/2,1-sqrt(2)/2]):l8:=line([1,1],[1+sin(Pi/8),1+cos(Pi/8)]):l9:=line([1-sqrt(2)/2,1-sqrt(2)/2],[1+sqrt(2)/2,1-sqrt(2)/2]):l10:=line([1+sqrt(2)/2,1-sqrt(2)/2],[1+sqrt(2)/2,1+sqrt(2)/2]):l11:=line([1,2],[1+sin(Pi/8),1+cos(Pi/8)]):l12:=line([1+sin(Pi/8),1+cos(Pi/8)],[1+sqrt(2)/2,1+sqrt(2)/2]): display((c,p,A,B,l1,l2,l3,l4,l5,l6,l7,l8,l9,l10,l11,l12,T),TEXT([0.25,1.77],'`A`',FONT(COURIER,BOLD),COLOR(HUE,0.9)),TEXT([1.75,0.25],'`B`',FONT(COURIER,BOLD),COLOR(HUE,0.9)),TEXT([0.7,1.65],'`s[k]/2`'),TEXT([1.075,1.85],'`h[k]`'),TEXT([0.8,1.85],'`s[k+1]`'),TEXT([1.65,1.85],'`s[k+2]`'),TEXT([0.55,1.35],'`r`',FONT(COURIER,BOLD)),TEXT([0.85,1.35],'`alpha[k+1]`'),TEXT([0.94,0.96],'`O`',FONT(COURIER,BOLD),COLOR(HUE,0.9)),scaling=constrained);

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N.B.: One should probably change the outcome of the construction of Figure1 by changing the scaling and/or the size to achieve the same result as hereunder.

Figure1 illustrates the method for inscribed polygons and serves as a starting point for the author's 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Taking the diameter AB in Figure 1 as a starting point, it is possible to interpret this diameter as a "2-gon", i.e. having two sides AB and BA. Halving these two sides renders the square shown. Then follow the steps of dividing the angular sectors into two equal sub-sectors and construction of the inscribed octogon. Successively bisecting the central angles one creates a series of polygons with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRImtGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI= sides, k=(1,2,3,4,.................LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEpaW5maW5pdHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn) . So, e.g. for k=1..10 the first ten of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRImtGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI=-gons are: n:=seq(2^k,k=1..10);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJuR0YoNiwiIiMiIiUiIikiIzsiI0siI2siJEciIiRjIyIkNyYiJUM1NyxGL0YwRjFGMkYzRjRGNUY2RjdGOA==

Note that for k = 1 the number of sides equals two, i.e. AB and BA from the "2-gon"!

We define s[k] as one side of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRImtGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI=-gon and h[k] as indicated in Figure 1. Then the following formulae can be derived:firstly by using Euclid's theorem about intersecting chords (Euclid III.3) restart;equ1:=(s[k]/2)^2=h[k]*(2*r-h[k]);sols1:=[solve(equ1,h[k])];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcXUxR0YoLywkKiYjIiIiIiIlRjIpJkkic0dGKDYjSSJrR0YoIiIjRjJGMiomJkkiaEdGKEY3RjIsJiomRjlGMkkickdGKEYyRjJGOyEiIkYyNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZzb2xzMUdGKDckLCZJInJHRigiIiIqJiNGMSIiI0YxKSwmKiYiIiVGMSlGMEY0RjFGMSokKSZJInNHRig2I0kia0dGKEY0RjEhIiJGM0YxRjEsJkYwRjFGMkZANyNGLg==

whereby of course h[k] must be smaller than the radius r, i.e.h[k]:=sols1[2];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiaEdGKDYjSSJrR0YoLCZJInJHRigiIiIqJiNGMyIiI0YzKSwmKiYiIiVGMylGMkY2RjNGMyokKSZJInNHRihGL0Y2RjMhIiJGNUYzRkA3I0Yx

secondly using Pythagoras' wellknown relation equ2:=(s[k+1])^2=(h[k])^2+(s[k]/2)^2;sols2:=solve(equ2,s[k+1]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVlcXUyR0YoLyokKSZJInNHRig2IywmSSJrR0YoIiIiRjZGNiIiI0Y2LCYqJCksJkkickdGKEY2KiYjRjZGN0Y2KSwmKiYiIiVGNilGPEY3RjZGNiokKSZGMjYjRjVGN0Y2ISIiRj5GNkZIRjdGNkY2KiYjRjZGQkY2RkVGNkY2NyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZzb2xzMkdGKDYkKiQpLCYqJkkickdGKCIiIiksJiomIiIlRjQpRjMiIiNGNEY0KiQpJkkic0dGKDYjSSJrR0YoRjpGNCEiIiNGNEY6RjRGQSomRjpGNEY5RjRGNEZCRjQsJEYvRkE3JEYvRkQ=

renders the (positive) solution fors[k+1]:=sols2[1];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+Jkkic0dGKDYjLCZJImtHRigiIiJGMkYyKiQpLCYqJkkickdGKEYyKSwmKiYiIiVGMilGNyIiI0YyRjIqJCkmRi42I0YxRj1GMiEiIiNGMkY9RjJGQiomRj1GMkY8RjJGMkZDRjI3I0Yz

This formula is the wellknown recursive relationship, which allows the consecutive computation of the side s[k] of any LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRImtGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI=-gon, inscribed in a circle with radius r, and by using only the side s[k-1] of the previous LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZtaW51cztGJ0YyLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZGLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlUtRi82JFEiMUYnRjJGMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMg==-gon.

Defining this sequence is being done in MAPLE by using the option "remember", so that MAPLE does not have to recompute functional values previously computed, thus easing the computation for large values of k. We take a circle with r = 1 and get: r:=1:k:=`k`:s:=`s`:s:=proc(k) option remember; s(k):=sqrt(2-sqrt(4-s(k-1)^2)) end;We remember that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0Y+LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPg== = 2.r , and are able to create a list of sidelengths for k = 1....n ; e.g. for n = 6: s(1):=2: SideOneToSix:=seq(s(k),k=1..6);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJzR0YoZio2I0kia0dGKEYoNiNJKXJlbWVtYmVyR0YoRig+LUYtRi8tSSVzcXJ0R0YlNiMsJiIiIyIiIi1GNjYjLCYiIiVGOiokKS1GLTYjLCZGMEY6RjohIiJGOUY6RkRGREYoRihGKDcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS1TaWRlT25lVG9TaXhHRig2KCIiIyokKUYvIyIiIkYvRjMqJCksJkYvRjNGMCEiIkYyRjMqJCksJkYvRjMqJCksJkYvRjNGMEYzRjJGM0Y3RjJGMyokKSwmRi9GMyokKSwmRi9GM0Y7RjNGMkYzRjdGMkYzKiQpLCZGL0YzKiQpLCZGL0YzRkFGM0YyRjNGN0YyRjM3KEYvRjBGNEY4Rj5GRA==

Obviously the general formula for the side LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImtGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== of the inscribed LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRImtGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI=-gon can be put as follows:Formula(1) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImtGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== = ( 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 etc.) , whereby the "etc." indicates that the number "two" must figure (k-1)-times under the first square root. E.g., if we leave out the "etc." in the Formula(1) as represented here, k would be 10 and the formula would hold for the side LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEjMTBGJy9GNlEnbm9ybWFsRidGPi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj4= of a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRjU2JFEjMTBGJ0Y4LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Y4RitGOA=== 1,024-gon.

We can see from Figure 1 that the perimeter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiUEYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtRi82JlEia0YnRjJGNUY4L0Y5USVib2xkRicvJStmb250d2VpZ2h0R0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGQEZC of the inscribed LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRImtGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI=-gon and the area 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 of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RIitGJ0YyLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZGLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlUtRi82JFEiMUYnRjJGMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMg==-gon are both smaller than the respective quantities of the circle: restart;P[k]:=2^k*s[k]<2*r*pi;A[k+1]:=2^k*r*s[k]/2<r^2*pi;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiUEdGKDYjSSJrR0YoMiomKSIiI0YwIiIiJkkic0dGKEYvRjUsJCooRjRGNUkickdGKEY1SSNwaUdGKEY1RjU3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiQUdGKDYjLCZJImtHRigiIiJGMkYyMiwkKiojRjIiIiNGMilGN0YxRjJJInJHRihGMiZJInNHRig2I0YxRjJGMiomKUY5RjdGMkkjcGlHRihGMjcjRjM=

and, when again taking radius r = 1 subs(r=1,{%,%%});

PCQyKiYpIiIjSSJrRzYiIiIiJkkic0dGKDYjRidGKSwkKiZGJkYpSSNwaUdGKEYpRikyLCQqKCNGKUYmRilGJUYpRipGKUYpRi8=

both render the same inequality 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 . Already Archimedes knew that, for a circle the ratio between the perimeter and the diameter, and the ratio between the area and the squared radius have the same value. And because the infinity-gon's perimeter certainly equals the circle's circumference we know 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 So we can use Formula(2) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtRi82JVEia0YnL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZtaW51cztGJ0YyLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZGLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlUtRi82JFEiMUYnRjJGMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMg==.( 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 etc.) in order to calculate successive approximations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtRi82JVEia0YnL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 for the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= . Once again: the number "two" appears (k-1)-times under the first square root.

Let us produce an array which presents, side by side, the number of sides of the first 15 polygons in our series of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRImtGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI=-gons, and the successive approximations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtRi82JVEia0YnL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI=. For Plist the number of "Digits" has to be increased, in order to let MAPLE calculate with the required accuracy, when using both, Formula(2) and the procedure from the section "Defining the Sequence of s[k]" above.We pair both lists, zip them together and present the result in the array "polygons_2",indicating that we use the series of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRImtGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI=-gons. Side_list:=[seq(2^k,k=1..20)]: Digits:=30:k:=`k`:s:=`s`:s:=proc(k) option remember; s(k):=sqrt(2-sqrt(4-s(k-1)^2)) end: s(1):=2:Plist:=[evalf(seq(2^(k-1)*s(k),k=1..20))]: pair:=(Side_list,Plist)->[Side_list,Plist]: polygons_2:=array(zip(pair,Side_list,Plist));

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

We can use Formula(2) to calculate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI='s value for any k, for instance for the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRjU2JFEkMTAwRidGOC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGOEYrRjg=-gon with its number of sides equal to n[100]:=2^100;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkibkdGKDYjIiQrIiJAd2A/Lm5cLCVII0ctZ113RSI3I0Yx

We only have to use Formula(2) , choosing the number of "Digits" sufficiently high (about 1.3 times k) and reduce the range for k to just one number, e.g. k = 100. In order to check the accuracy of the result, we compare it with MAPLE's own 65-digit number for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI=: Digits := 125:k:=`k`:s:=`s`:s:=proc(k) option remember; s(k):=sqrt(2-sqrt(4-s(k-1)^2)) end; s(1):=2:pi[100]:=evalf(2^99)*evalf(seq(s(k),k=100));Digits:=65:evalf(Pi);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJzR0YoZio2I0kia0dGKEYoNiNJKXJlbWVtYmVyR0YoRig+LUYtRi8tSSVzcXJ0R0YlNiMsJiIiIyIiIi1GNjYjLCYiIiVGOiokKS1GLTYjLCZGMEY6RjohIiJGOUY6RkRGREYoRihGKDcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkjcGlHRig2IyIkKyIkImhyaDowMzU8Kj4mKikzJDMkRyRmUUl5KzkpKj08SGV5Ij5RS05rUFRcKDQjZTV2JCpScHI+JSlHXXpLUVZFWVFLeiplYEVmVEohJEMiNyNGMQ==

JCJcb0JmV1woNCNlNXYkKlJwcj4lKUddektRVkVZUUt6KmVgRWZUSiEjaw==

and we notice that our approximation is accurate up to the 60th digit, i.e. up to and including the group of figures ....7494.. !

Going back to Figure 1 we can easily see the relationship between s[k] and the angle 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 at the centre O of the circle: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 we can create yet another interesting table for the series of 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 and their sines. We proceed as in the above section, were we described the "First Results", by producing an array which presents, side by side, the e.g., first twelve 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 in our series of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRImtGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI=-gons, and their sines. We use Formula(1) and the procedure from the section "Defining the Sequence of s[k]" above, then we pair both lists, zip them together and present the result in the array "SineList_polygons_2", indicating again that we are using the series of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRImtGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI=-gons: restart; alpha_list:=[seq(pi/2^k,k=1..12)]: k:=`k`:s:=`s`:s:=proc(k) option remember; s(k):=sqrt(2-sqrt(4-s(k-1)^2)) end: s(1):=2:Side_list:=[seq(1/2*s(k),k=1..12)]: pair:=(alpha_list,Side_list)->[alpha_list,Side_list]: SineList_polygons_2:=array(zip(pair,alpha_list,Side_list));

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

An interesting table, and this "tree" can be extended ad infinitum!

If we imagine or evaluate the lengths of the sides of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRImtGJy8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjI=-gon by means of Formula(1) , it is obvious 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 .E.g., if we choose k = 100 , we get restart;Digits:=65: k:=`k`:s:=`s`:s:=proc(k) option remember; s(k):=sqrt(2-sqrt(4-s(k-1)^2)) end: s(1):=2:Sidelength[100]:=evalf(seq(s(k),k=100));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkrU2lkZWxlbmd0aEdGKDYjIiQrIiQiXG9uSEJOWzZ2KXpUUT9SQWg7IkdEeHUrV1JHY3cuITM4Y2NcISMlKjcjRjE=

which is already rather short.Since the other factor in Formula(2) , being LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZtaW51cztGJ0YyLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZGLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlUtRi82JFEiMUYnRjJGMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMg== , is going to be Limit(2^(k-1),k=infinity)=limit(2^(k-1),k=infinity);

Ly1JJkxpbWl0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQpIiIjLCZJImtHRigiIiJGLiEiIi9GLUkpaW5maW5pdHlHRiZGMQ==

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 is becoming an expression (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEpaW5maW5pdHlGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn times LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yv) . Which, of course in mathematics is nothing strange, and we know that here this expression equals LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= .There is, however a rather astonishing conclusion from all this. If we putLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiTkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImtGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== = ( 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 etc.) , Formula(1) can be written as k:=`k`:s:=`s`:s(k)^2 = 2-N(k);N(k) = 2-s(k)^2;

LyokKS1JInNHNiI2I0kia0dGJyIiIyIiIiwmRipGKy1JIk5HRidGKCEiIg==

Ly1JIk5HNiI2I0kia0dGJSwmIiIjIiIiKiQpLUkic0dGJUYmRilGKiEiIg==

whence Limit(s(k)^2,k=infinity)=2-Limit(N(k),k=infinity);

Ly1JJkxpbWl0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJCktSSJzR0YoNiNJImtHRigiIiMiIiIvRi9JKWluZmluaXR5R0YmLCZGMEYxLUYkNiQtSSJOR0YoRi5GMiEiIg==

therefor -- see above -- 2 - 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 = 0 ,and that means, the rational number 2 must be equal to 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 = Limit (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEnJnJhcnI7RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtRiw2JVEpaW5maW5pdHlGJ0Y0RjdGPkYrRj4=) ( 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 etc.) !!Let us check this at k = 100 : We can take LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVErU2lkZWxlbmd0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUkjbW5HRiQ2JFEkMTAwRicvRjZRJ25vcm1hbEYnRj4vJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ from above and we find 2-(Sidelength[100])^2;

JCJcb0RWdioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKioqKj4hI2s=

which is already close to 2 . A very interesting relation indeed, between the rational number 2 , infinitely many irrational square roots and the irrational, transcendent number LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= .

Another possibility to use the same approach consists of starting with an equilateral triangle, inscribed in the circle. Now Figure 2 illustrates the method for the inscribed 3.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GIzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZtaW51cztGJ0YyLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZGLyUpc3RyZXRjaHlHRkYvJSpzeW1tZXRyaWNHRkYvJShsYXJnZW9wR0ZGLyUubW92YWJsZWxpbWl0c0dGRi8lJ2FjY2VudEdGRi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlUtRi82JFEiMUYnRjJGMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMg==-gons (k=1....n) , and serves as a starting point for calculations.N.B.: For k = 2....6 one is "passing" Archimedes' endevour, which ended at the 96-sided regular polygon.

The necessary data for plotting this figure are as follows: with(plots):with(plottools):T:=textplot([2.5,1,`Figure 2`],align=RIGHT):c:=circle([1,1],1,color=red):p:=point([1,1],symbol=CIRCLE,symbolsize=10,color=red):A:=point([1-sqrt(3)/2,1.5],symbol=CIRCLE,symbolsize=10,color=black):B:=point([1+sqrt(3)/2,1.5],symbol=CIRCLE,symbolsize=10,color=black):C:=point([1,0],symbol=CIRCLE,symbolsize=10,color=black):l1:=line([1-sqrt(3)/2,1.5],[1+sqrt(3)/2,1.5]):l2:=line([1-sqrt(3)/2,1.5],[1,0]):l3:=line([1+sqrt(3)/2,1.5],[1,0]):l4:=line([1-sqrt(3)/2,1.5],[1,2]):l5:=line([1+sqrt(3)/2,1.5],[1,2]):l6:=line([1,0],[1,2]):l7:=line([1-sqrt(3)/2,1.5],[1,1]):l8:=line([1,1],[1+sin(Pi/6),1+cos(Pi/6)]):l9:=line([1,2],[1+sin(Pi/6),1+cos(Pi/6)]):l10:=line([1+sqrt(3)/2,1.5],[1+sin(Pi/6),1+cos(Pi/6)]): display((c,p,A,B,C,l1,l2,l3,l4,l5,l6,l7,l8,l9,l10,T),TEXT([0.1,1.6],'`A`',FONT(COURIER,BOLD),COLOR(HUE,0.9)),TEXT([1.9,1.6],'`B`',FONT(COURIER,BOLD),COLOR(HUE,0.9)),TEXT([1,-0.15],'`C`',FONT(COURIER,BOLD),COLOR(HUE,0.9)),TEXT([0.65,1.55],'`s[k]/2`'),TEXT([1.075,1.7],'`h[k]`'),TEXT([0.7,1.75],'`s[k+1]`'),TEXT([1.8,1.75],'`s[k+2]`'),TEXT([0.6,1.15],'`r`',FONT(COURIER,BOLD)),TEXT([0.8,1.25],'`alpha[k+1]`'),TEXT([0.94,0.96],'`O`',FONT(COURIER,BOLD),COLOR(HUE,0.9)),scaling=constrained);

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One should probably change the outcome of the construction of Figure2 by changing the scaling and/or the size to achieve the same result as hereabove.

The approximation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= can now be started by means of the triangle ABC , from which there follow the regular inscribed polygons with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbW5HRiQ2JVEkMy4yRicvJSVib2xkR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtaUdGJDYmUSJrRidGMi8lJ2l0YWxpY0dRJXRydWVGJy9GNlEnaXRhbGljRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnL0Y2USVib2xkRicvJStmb250d2VpZ2h0R0ZF sides, by successively bisecting the central angles. From the above figure we find for the triangle: restart; G1:=(s/2)^2+(2*r-h)^2=s^2;sols1:=[solve(G1,h)];h[1]:=sols1[2];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHMUdGKC8sJiomIyIiIiIiJUYyKUkic0dGKCIiI0YyRjIqJCksJiomRjZGMkkickdGKEYyRjJJImhHRighIiJGNkYyRjIqJEY0RjI3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZzb2xzMUdGKDckLCYqJiIiIyIiIkkickdGKEYyRjIqKCNGMkYxRjIpIiIkRjVGMkkic0dGKEYyRjIsJkYwRjJGNCEiIjcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiaEdGKDYjIiIiLCYqJiIiI0YwSSJyR0YoRjBGMCooI0YwRjNGMCkiIiRGNkYwSSJzR0YoRjAhIiI3I0Yx

G2:=simplify((s/2)^2-h[1]*(2*r-h[1]));sols2:=[solve(G2,s)];s[1]:=sols2[2];h[1]:=subs(s=s[1],h[1]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSNHMkdGKCwmKiQpSSJzR0YoIiIjIiIiRjMqKCkiIiQjRjNGMkYzRjFGM0kickdGKEYzISIiNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSZzb2xzMkdGKDckIiIhKiYpIiIkIyIiIiIiI0Y0SSJyR0YoRjQ3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+Jkkic0dGKDYjIiIiKiYpIiIkI0YwIiIjRjBJInJHRihGMDcjRjE=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiaEdGKDYjIiIiLCQqJiNGMCIiI0YwSSJyR0YoRjBGMDcjRjE=

and then we can go on as in section "The Basic Idea" above, which is leading to the identical recurrence formula : s[k+1]:=(2*r^2-r*(4*r^2-s[k]^2)^(1/2))^(1/2);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+Jkkic0dGKDYjLCZJImtHRigiIiJGMkYyKiQpLCYqJiIiI0YyKUkickdGKEY3RjJGMiomRjlGMiksJiomIiIlRjJGOEYyRjIqJCkmRi42I0YxRjdGMiEiIiNGMkY3RjJGQ0ZERjI3I0Yz

The perimeter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiUEYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtRi82JlEia0YnRjJGNUY4L0Y5USVib2xkRicvJStmb250d2VpZ2h0R0ZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGQEZC of the inscribed LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEkMy4yRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMg==-gon and the area 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 of the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEkMy4yRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiYtSSNtaUdGJDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvRjNRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiK0YnRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkYvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGVS1GLzYkUSIxRidGMkYyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yy-gon give the following successive relations for the respective quantities of the circle and both the same approximation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI=: P[k]:=3*2^(k-1)*s[k]<2*r*pi;A[k+1]:=3*2^(k-1)*r*s[k]/2<r^2*pi;

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiUEdGKDYjSSJrR0YoMiwkKigiIiQiIiIpIiIjLCZGMEY1RjUhIiJGNSZJInNHRihGL0Y1RjUsJCooRjdGNUkickdGKEY1SSNwaUdGKEY1RjU3I0Yx

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkiQUdGKDYjLCZJImtHRigiIiJGMkYyMiwkKiojIiIkIiIjRjIpRjgsJkYxRjJGMiEiIkYySSJyR0YoRjImSSJzR0YoNiNGMUYyRjIqJilGPEY4RjJJI3BpR0YoRjI3I0Yz

For a circle with radius r = 1 subs(r=1,{%,%%});

PCQyLCQqKCIiJCIiIikiIiMsJkkia0c2IkYnRichIiJGJyZJInNHRiw2I0YrRidGJywkKiZGKUYnSSNwaUdGLEYnRicyLCQqKCNGJkYpRidGKEYnRi5GJ0YnRjM=

i.e., both render the same inequality LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2J0YrLUYjNiYtSSVtc3VwR0YkNiUtSSNtbkdGJDYkUSQzLjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvRjtRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnRjovJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRk0vJSlzdHJldGNoeUdGTS8lKnN5bW1ldHJpY0dGTS8lKGxhcmdlb3BHRk0vJS5tb3ZhYmxlbGltaXRzR0ZNLyUnYWNjZW50R0ZNLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGZm4tRjc2JFEiMkYnRjpGOi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRkg2LVExJkludmlzaWJsZVRpbWVzO0YnRjpGS0ZORlBGUkZURlZGWC9GZW5RJjAuMGVtRicvRmhuRmNvLUklbXN1YkdGJDYlLUYsNiVRInNGJ0ZCRkUtRiM2JEY/RjovJS9zdWJzY3JpcHRzaGlmdEdGXm9GOi1GSDYtUSI8RidGOkZLRk5GUEZSRlRGVkZYL0ZlblEsMC4yNzc3Nzc4ZW1GJy9GaG5GY3AtRiw2JVEjcGlGJy9GQ0ZNRjpGOkYrRjo= .

The following expression sequence can be programmed for the lengths of the sides of successive LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEkMy4yRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMg==-gons: restart;k:=`k`:s:=`s`: s:=proc(k) option remember; s(k):=sqrt(2-sqrt(4-s(k-1)^2)) end; s(1):=sqrt(3): Sidelist:=[seq(s(k),k=1..10)]:

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJzR0YoZio2I0kia0dGKEYoNiNJKXJlbWVtYmVyR0YoRig+LUYtRi8tSSVzcXJ0R0YoNiMsJiIiIyIiIi1GNjYjLCYiIiVGOiokKS1GLTYjLCZGMEY6RjohIiJGOUY6RkRGREYoRihGKDcjRi4=

E.g., the 10th member of this list equals SideLenght[9]:=Sidelist[9];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+JkkrU2lkZUxlbmdodEdGKDYjIiIqKiQpLCYiIiMiIiIqJCksJkY0RjUqJCksJkY0RjUqJCksJkY0RjUqJCksJkY0RjUqJCksJkY0RjUqJCksJiIiJUY1KiQpLCYqJiNGNUY0RjUpIiInRk1GNUY1KiZGTUY1KUY0Rk1GNSEiIkY0RjVGUkZNRjVGNUZNRjVGNUZNRjVGNUZNRjVGNUZNRjVGNUZNRjVGUkZNRjU3I0Yx

Doing a little work for MAPLE, we check the last square root simplify(4-(1/2*6^(1/2)-1/2*2^(1/2))^2);

LCYiIiMiIiIqJCkiIiQjRiRGI0YkRiQ=

and when substituting SideLength:=subs(4-(1/2*6^(1/2)-1/2*2^(1/2))^2=2+3^(1/2),Sidelist);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SStTaWRlTGVuZ3RoR0YoNywqJCkiIiQjIiIiIiIjRjNGMywmKiZGMkYzKSIiJ0YyRjNGMyomRjJGMylGNEYyRjMhIiIqJCksJkY0RjMqJCksJkY0RjNGL0YzRjJGM0Y7RjJGMyokKSwmRjRGMyokKSwmRjRGM0Y/RjNGMkYzRjtGMkYzKiQpLCZGNEYzKiQpLCZGNEYzRkVGM0YyRjNGO0YyRjMqJCksJkY0RjMqJCksJkY0RjNGS0YzRjJGM0Y7RjJGMyokKSwmRjRGMyokKSwmRjRGM0ZRRjNGMkYzRjtGMkYzKiQpLCZGNEYzKiQpLCZGNEYzRldGM0YyRjNGO0YyRjMqJCksJkY0RjMqJCksJkY0RjNGZ25GM0YyRjNGO0YyRjM3I0Yu

we recognize that the term under the last square root in each member of the above list is equal to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSIrRidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZNLUkmbXNxcnRHRiQ2Iy1GMjYkUSIzRidGNUY1RitGNQ== .This allows the general formulae for the lengths of the sides and the approximation for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= as follows: Formula(3) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImtGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== = ( 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 etc.) , Formula(4) 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. ( 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 etc.) , whereby the number "two" must figure (k-2)-times under the first square root (e.g. if we leave out the "etc." , k would be 10, the formula would hold for a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1JI21uR0YkNiRRJDMuMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GNTYkUSI5RidGOC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGOEYrRjg== 1,536-gon).We create the array which presents, side by side, the number of sides of the first 12 polygons in our series of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEkMy4yRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMg==-gons, and the successive approximations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtRi82JVEia0YnL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI=. For Plist we use Formula(4) and the procedure from the section "Defining the Sequence of s[k]" above.We pair the sequences "number of sides" and Plist , zip them together and present the result in the array "polygons_3", indicating that here we use the series of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEkMy4yRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMg==-gons. restart;Digits:=20:k:=`k`:s:=`s`: Side_list:=[seq(3*2^(k-1),k=1..15)]: s:=proc(k) option remember; s(k):=sqrt(2-sqrt(4-s(k-1)^2)) end: s(1):=sqrt(3):Plist:=[evalf(seq(3*2^(k-2)*s(k),k=1..15))]: pair:=(Side_list,Plist)->[Side_list,Plist]: polygons_3:=array(zip(pair,Side_list,Plist));

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

The number LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtSSNtbkdGJDYkUSM5NkYnRjVGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRjU= = 3.141031........... shows that good old Archimedes was right for the first 4 digits, already 2,300 years ago !

As befor, now from Figure 2 we can see the relationship between s[k] and the angle 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 at the centre O of the circle: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 we can create the table for the series of 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 and their sines. We produce the array which presents, side by side, e.g. the first ten 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 in our series of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEkMy4yRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnL0YzUSdpdGFsaWNGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMg==-gons, and their sines. We use Formula(3) and present the results in the array "SineList_polygons_3", indicating that we are using the series of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbW5HRiQ2JFEkMy4yRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiYtSSNtaUdGJDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvRjNRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEoJm1pbnVzO0YnRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkYvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGVS1GLzYkUSIxRidGMkYyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yy-gons: restart; alpha_list:=[seq(pi/(3*2^(k-1)),k=1..10)]:k:=`k`:s:=`s`:s:=proc(k) option remember; s(k):=sqrt(2-sqrt(4-s(k-1)^2)) end: s(1):=sqrt(3):Side_list:=[1/2*seq(s(k),k=1..10)]:sidelength:=subs(4-(1/2*6^(1/2)-1/2*2^(1/2))^2=2+3^(1/2),Side_list): pair:=(alpha_list,sidelength)->[alpha_list,sidelength]: SineList_polygons_3:=array(zip(pair,alpha_list,sidelength));

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

We mentioned already that the number of "Digits" to be used, in order to get the maximally possible number of correct digits in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtRi82JVEia0YnL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 must be chosen about 30% higher than k. And known was that the number of correct digits by means of this algorithm equals 60% of k , i.e. 3 new correct digits for an increase of 5 steps in k (see Figure 3).

Calculated were the values of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtRi82JVEia0YnL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 for the k's of "k_list1" and "k_list2" , which were put together into "k_list" . The corresponding times (in sec), which the computer needed for each calculation, were measured by means of the computer and put in "t_list1" and "t_list2" . The numbers of correct digits in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtRi82JVEia0YnL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 were determined and put into "dig_list" .N.B.: For the sake of LeastSquare CurveFitting hereunder the obvious values 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 were included in "k_list2" , "t_list2" and "dig_list" respectively.. restart;k_list1:=[50,80,100,150,200,210,300,375,510,520,530,540,550,650];k_list2:=[0,500,800,1000];t_list1:=[8,19.3,31.4,72.3,152,171,391.6,668.4,1516.4,1595,1675.4,1757.3,1840.7,2910];t_list2:=[0,1610,7273,13650];dig_list:=[0,30,48,60,90,120,126,180,226,301,307,313,319,325,331,391,481,600];k_list:=sort([op(k_list1),op(k_list2)]);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShrX2xpc3QxR0YoNzAiI10iIyEpIiQrIiIkXSIiJCsjIiQ1IyIkKyQiJHYkIiQ1JiIkPyYiJEkmIiRTJiIkXSYiJF0nNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SShrX2xpc3QyR0YoNyYiIiEiJCsmIiQrKSIlKzU3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSh0X2xpc3QxR0YoNzAiIikkIiQkPiEiIiQiJDkkRjIkIiRCKEYyIiRfIiIkciIkIiU7UkYyJCIlJW8nRjIkIiZrXiJGMiIlJmYiJCImYW4iRjIkIiZ0diJGMiQiJjIlPUYyIiU1SDcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSh0X2xpc3QyR0YoNyYiIiEiJTU7IiV0cyImXU8iNyNGLg==

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlkaWdfbGlzdEdGKDc0IiIhIiNJIiNbIiNnIiMhKiIkPyIiJEUiIiQhPSIkRSMiJCwkIiQyJCIkOCQiJD4kIiREJCIkSiQiJCJSIiQiWyIkKyc3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSdrX2xpc3RHRig3NCIiISIjXSIjISkiJCsiIiRdIiIkKyMiJDUjIiQrJCIkdiQiJCsmIiQ1JiIkPyYiJEkmIiRTJiIkXSYiJF0nIiQrKSIlKzU3I0Yu

pair:=(k_list,dig_list)->[k_list,dig_list]:reldig:=array(zip(pair,k_list,dig_list)):KD_relation:=convert(reldig,listlist);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSxLRF9yZWxhdGlvbkdGKDc0NyQiIiFGMDckIiNdIiNJNyQiIyEpIiNbNyQiJCsiIiNnNyQiJF0iIiMhKjckIiQrIyIkPyI3JCIkNSMiJEUiNyQiJCskIiQhPTckIiR2JCIkRSM3JCIkKyYiJCwkNyQiJDUmIiQyJDckIiQ/JiIkOCQ3JCIkSSYiJD4kNyQiJFMmIiREJDckIiRdJiIkSiQ3JCIkXSciJCJSNyQiJCspIiQiWzckIiUrNSIkKyc3I0Yu

KDP:=plot(KD_relation,style=POINT,symbol=CIRCLE,symbolsize=10):with(CurveFitting):LS:=LeastSquares(KD_relation,k):LS:=unapply(evalf(LS),k);

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T:=plots[textplot]([290,620,'`Figure 3`'],align=RIGHT): LSP:=plot(LS(k),k=1..1000,color=blue,linestyle=DASHDOT):plots[display]([KDP,LSP,T],labels=[k,`correct digits`]);

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a perfect straight line, which indeed, shows experimentally that this algoritm renders 3 extra correct digits for every increase of 5 of the number k .

There are two distinct sets of measurements: The values for curve 1 were all taken successively in one session, i.e. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiQtRi82JVEia0YnL0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1 was calculated, then k was augmented to the next value and the calculation was started immediately again. Most probably inside MAPLE there were still data which could be used for each successive calculation. For curve 2 each calculation was done in a separate session, so obviously this took longer because nothing which MAPLE could use was left in memory from a similar previous calculation. pair1:=(k_list1,t_list1)->[k_list1,t_list1]:pair2:=(k_list2,t_list2)->[k_list2,t_list2]:relt1:=array(zip(pair1,k_list1,t_list1)):relt2:=array(zip(pair2,k_list2,t_list2)):KT_relation1:=convert(relt1,listlist);KT_relation2:=convert(relt2,listlist);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS1LVF9yZWxhdGlvbjFHRig3MDckIiNdIiIpNyQiIyEpJCIkJD4hIiI3JCIkKyIkIiQ5JEY2NyQiJF0iJCIkQihGNjckIiQrIyIkXyI3JCIkNSMiJHIiNyQiJCskJCIlO1JGNjckIiR2JCQiJSVvJ0Y2NyQiJDUmJCIma14iRjY3JCIkPyYiJSZmIjckIiRJJiQiJmFuIkY2NyQiJFMmJCImdHYiRjY3JCIkXSYkIiYyJT1GNjckIiRdJyIlNUg3I0Yu

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SS1LVF9yZWxhdGlvbjJHRig3JjckIiIhRjA3JCIkKyYiJTU7NyQiJCspIiV0czckIiUrNSImXU8iNyNGLg==

KT_rel1:=plot(KT_relation1,thickness=1,style=POINT,symbol=CIRCLE,symbolsize=10):KT_rel2:=plot(KT_relation2,thickness=1,style=POINT,symbol=CROSS,symbolsize=10): with(CurveFitting):LS1:=LeastSquares(KT_relation1,k,curve=a*k^3+b*k^2):LS2:=LeastSquares(KT_relation2,k,curve=a*k^3+b*k^2):LS1:=unapply(LS1,k);LS2:=unapply(evalf(LS2),k);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRMUzFHRihmKjYjSSJrR0YoRig2JEkpb3BlcmF0b3JHRihJJmFycm93R0YoRigsJiomXTNGNjA4NTk0MjFENEVBMTYiIiIpRjAiIiNGN0Y3KiZdM0VERjRBNjM1Nzk0OTdFOEY3KUYwIiIkRjdGN0YoRihGKDcjRi4=

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRMUzJHRihmKjYjSSJrR0YoRig2JEkpb3BlcmF0b3JHRihJJmFycm93R0YoRigsJiomJCIrTWgvNGYhIzgiIiIpRjAiIiNGOUY5KiYkIis3RGc3OCEjOUY5KUYwIiIkRjlGOUYoRihGKDcjRi4=

LST1:=plot(LS1(k),k=1..1000,color=blue,linestyle=DASH):LST2:=plot(LS2(k),k=0..1000,color=cyan,linestyle=DASH):T1:=plots[textplot]([740,3330,'`curve 1`'],align=RIGHT): T2:=plots[textplot]([690,9550,'`curve 2`'],align=RIGHT): T:=plots[textplot]([360,14520,'`Figure 4`']): plots[display]([LST1,LST2,KT_rel1,KT_rel2,T1,T2,T],labels=[k,`time in sec`]);

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These relations, of course are valid only for the author's computer system, be it only because of its clock speed.

Archimedes' method of exhaution, by using MAPLE 9.01 (Classic) was transformed into a recursive relationship, which can be used to approximate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= to a(ny) desired accuracy. The nescessary procedure was established and described. Interesting results were presented.Within the algorithm for determining the irrational, transcendent number LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= , an intriguing relation between the rational number 2 and an infinite succession of square roots under square roots with only 2's in them, has been discovered.

Char, B. W. e.a., Maple 9 Learning Guide, Maplesoft 2003Abell, Martha. L. and Braselton, James P., Maple V By Example, Academic Press, 2nd EditionDuschek, Adalbert, Vorlesungen \303\274ber h\303\266here Mathematik, Erster Band, Springer 1949Heilbron, J. L., Geometry Civilized, Clarenton Press, Oxford 1998Bronstein, I. N. and Semendjajew, K. A., Taschenbuch der Mathematik, B.G. Teubner, 25th Edition 1991www.britannica.com, Pi_Britannica.htm, Internet 2005Delahaye, Jean-Paul, Het fascinerende getal LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI=, Natuurwetenschap en Techniek 2004 (translated from Le Fascinant nombre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjcGlGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI=, Berlin - pour la Science 1997)

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The following License Agreement is included in this submission for the Application to be posted on the Maple Application Center and accepted by the undersigned.

has developed an Application (the "Application") using Maple and wishes to submit the Application to Maplesoft, a division of Waterloo Maple Inc. for publication by Maplesoft or its agents. 1. The Author grants to Maplesoft a non-exclusive, perpetual License and right to use all or any part of the Application, to modify or adapt the Application for the promotion of Maplesoft. 2. The Author represents that the Application is the original work of the Author, the Author has the right to grant a License to Maplesoft with respect to the Application, no other party has any right, title or interest in or to the Application, and that the use of the Application by Maplesoft will not infringe any intellectual property rights of any third party. 3. The Author agrees to execute such documents and take such other action as Maplesoft may request in order to give effect to the intention of this License Agreement. This License Agreement shall be binding upon and inure to the benefit of the Author, and all heirs, executors, administrators, successors and permitted assigns of the Author.

The author can state, to the best of his knowledge that he has never encountered the demonstrated algorithm and some of the conclusions, to the extent described herein, except for the basic principles which, of course, date back to the Greeks at least 23 centuries. Done at EINDHOVEN , The NETHERLANDS , in March 2005 , by Dr.F.FUTSCHIK