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height:90pt'> <v:imagedata src="Ratios_files/image001.gif" o:title="2decode-r-past_logo"/> </v:shape><![endif]--><![if !vml]><img width=800 height=120 src="Ratios_files/image001.gif" v:shapes="_x0000_i1026"><![endif]><o:p></o:p></span></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></b></p> <p class=MsoNormal><span style='font-size:11.0pt;font-family:Georgia'>| <a href="http://www.2dcode-r-past.com/">Home </a>| <a href="http://www.2dcode-r-past.com/contactus.htm">Contact </a>| <a href="http://www.2dcode-r-past.com/credo.htm">Credos [<span style='mso-bidi-font-family: Georgia'>È<span class=SpellE>k<span style='font-family:"Lucida Sans Unicode"'>~</span>eÐd<span style='font-family:Tahoma'>*</span>o</span>Ð]):</span></a>| <a href="http://www.2dcode-r-past.com/Geometry/index.htm">Sacred Geometry</a> | <a href="http://www.2dcode-r-past.com/Geometry/Numbers.htm">Exploring Numbers</a> | <a href="http://www.2dcode-r-past.com/Great_Pyramid/index.htm">Great Pyramid</a> | <a href="http://www.2dcode-r-past.com/links.htm">Links</a> |<o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></i></b></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></i></b></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>Exploring various Ancient Ratios gathered from great authors and researchers </span></i></b><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"'>to see where fractional ratios collide with<span style='color:red'> <b style='mso-bidi-font-weight:normal'><u>whole numbers<span class=GramE>!!<span style='font-weight:normal;text-decoration:none;text-underline:none'>.</span></span></u></b></span><o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>Below& are <b style='mso-bidi-font-weight:normal'><u>works in progress</u></b>! : I will continue to update other ratios for analysis as time permits& .<o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family: "Courier New"'>Current Functions employed:<o:p></o:p></span></u></b></p> <p class=MsoNormal style='text-indent:.5in'><span class=GramE><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"'>MOD(</span></span><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"'>1)-MOD(13) BY:<o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>Array of Pi variants<o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>Array of Phi variants<o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>Array of various other key ratios <o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>SQRTS-10<o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>Primes<o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>These will be explored and updated as time permits.<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>Current Historical Pi fractional ratios utilized:<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span style='font-size:10.0pt;font-family:"Courier New"; color:blue'>DIMENSION </span><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>34,1)<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>1,1)='256/81' <span style='mso-tab-count:1'>    </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp; 2000 B.C.E.<span style='mso-spacerun:yes'>                                 </span>Egyptians use p = 256/81 = 3.1605.<span style='mso-spacerun:yes'>  </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>2,1)='81/256'<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>3,1)='22/7'<span style='mso-tab-count:2'>       </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;Old School Pi [AE Pi too??!!][Archimedes Approximates this ... <span class=SpellE>Pietre</span> <span class=SpellE>reccomends</span> intended Pi=22/7]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>4,1)='7/22'<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>5,1)='864/275'<span style='mso-tab-count:1'>    </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;1220<span style='mso-spacerun:yes'>  </span>Leonardo de Pisa (Fibonacci) finds p = 3.141818.... , John <span class=SpellE>Michell</span> Pi [<span class=SpellE>tDoP</span>] Foundational Pi!<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>6,1)='275/864' <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>7,1)='(99/70)+(97/56)'<span style='mso-tab-count:2'>        </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;Suggested AE ROOT PI <span class=SpellE>Apprx</span></span><span style='font-size:10.0pt;font-family:"Courier New"; color:black'><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>8,1)='(1/((99/70)+(97/56)))' </span><span style='font-size:10.0pt;font-family:"Courier New";color:green'><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>9,1)='(140/99)+(97/56)'<span style='mso-tab-count:2'>       </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;Suggested AE ROOT PI <span class=SpellE>Apprx</span> 2<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>10,1)='(1/((140/99)+(97/56)))' <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>11,1)='355/113'<span style='mso-tab-count:2'>         </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span class=SpellE>Tsu</span> <span class=SpellE>Ch'ung-chih</span> (1500 ago)<span style='mso-tab-count:2'>        </span><span style='mso-tab-count:1'>      </span>3.141592920353...<span style='mso-tab-count: 1'> </span>0.0000002667...<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>12,1)='113/355' <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>13,1)='25/8'<span style='mso-tab-count:2'>            </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;2000 B.C.E..<span style='mso-spacerun:yes'>  </span>Babylonians use p = 25/8 = 3.125.<span style='mso-spacerun:yes'>  </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>14,1)='8/25' <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>15,1)='3/1'<span style='mso-tab-count:2'>       </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;1100 B.C.E. Chinese use p = 3.<span style='mso-spacerun:yes'>  </span><span class=GramE>550 B.C.E.</span><span style='mso-spacerun:yes'>    </span>Old Testament implies p = 3.<span style='mso-spacerun:yes'>  </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>16,1)='1/3' <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>17,1)='223/71'<span style='mso-tab-count:2'>          </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp; 250 B.C.E.<span style='mso-spacerun:yes'>  </span>Archimedes uses a 96-sided polygon to establish 223/71 &lt; pi &lt; 22/7. He also uses a spiral to square the circle.<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>18,1)='71/223' <span style='mso-tab-count:1'>   </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>19,1)='377/120'<span style='mso-tab-count:2'>         </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp; 150<span style='mso-spacerun:yes'>  </span>Claudius Ptolemy uses p = 3°8'30&quot; = 377/120 = 3.14166....<span style='mso-spacerun:yes'>  </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>20,1)='120/377' <span style='mso-tab-count:1'>  </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-tab-count: 1'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>21,1)='142/45'<span style='mso-tab-count:2'>          </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp; 250<span style='mso-spacerun:yes'>  </span>Wang <span class=SpellE>Fau</span> uses p 142/45 = 3.1555....<span style='mso-spacerun:yes'>  </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>22,1)='45/142' <span style='mso-tab-count:1'>   </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-tab-count: 1'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>23,1)='157/50'<span style='mso-tab-count:2'>          </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp; 263<span style='mso-spacerun:yes'>  </span>Liu <span class=SpellE>Hui</span> uses p = 157/50 = 3.14.<span style='mso-spacerun:yes'>   </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>24,1)='50/157' <span style='mso-tab-count:1'>   </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-tab-count: 1'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>25,1)='62832/20000'<span style='mso-tab-count:2'>           </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp; 530<span style='mso-spacerun:yes'>  </span><span class=SpellE>Aryabhata</span> uses p = 62832/20000 = 3.1416.<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>26,1)='20000/62832' <span style='mso-tab-count:1'>    </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-tab-count: 1'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>27,1)='10.5/1'<span style='mso-tab-count:2'>          </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp; 650<span style='mso-spacerun:yes'>  </span><span class=SpellE>Brahmagupta</span> uses p = 10 1/2 = 3.162....<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>28,1)='1/10.5' <span style='mso-tab-count:1'>   </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-tab-count: 1'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>29,1)='10.5/1'<span style='mso-tab-count:2'>          </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp; 650<span style='mso-spacerun:yes'>  </span><span class=SpellE>Brahmagupta</span> uses p = 10 1/2 = 3.162....<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>30,1)='1/10.5' <span style='mso-tab-count:1'>   </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-tab-count: 2'>          </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>31,1)='(4*ATAN(1))/1'<span style='mso-tab-count:2'>         </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp; Current True Pi<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>32,1)='1/(4*ATAN(1))' <span style='mso-tab-count:1'>  </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>33,1)='SQRT(2)+SQRT(3)'<span style='mso-tab-count:2'>       </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp; Current SQRT Pi<span style='mso-spacerun:yes'>   </span><o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span class=SpellE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>34,1)='1/(SQRT(2)+SQRT(3))' <span style='mso-tab-count:1'>  </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;</span><span style='font-size: 10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>35,1)='14148475504056880/4503599627370496'<span style='mso-tab-count:2'>            </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp; MATLAB quantity pi<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>36,1)='4503599627370496/14148475504056880' <span style='mso-tab-count:1'>     </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>37,1)='732/233'<span style='mso-tab-count:2'>         </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>38,1)='233/732' <span style='mso-tab-count:1'>  </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>39,1)='19/6'<span style='mso-tab-count:2'>            </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>40,1)='6/19' <span style='mso-tab-count:1'>     </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>41,1)='179/57'<span style='mso-tab-count:2'>          </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>42,1)='57/179' <span style='mso-tab-count:1'>   </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>43,1)='201/64'<span style='mso-tab-count:2'>          </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>44,1)='64/201' <span style='mso-tab-count:1'>   </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>45,1)='201/64'<span style='mso-tab-count:2'>          </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>46,1)='64/201' <span style='mso-tab-count:1'>   </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>47,1)='245/78'<span style='mso-tab-count:2'>          </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>48,1)='78/245' <span style='mso-tab-count:1'>   </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>49,1)='245/78'<span style='mso-tab-count:2'>          </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>50,1)='78/245' <span style='mso-tab-count:1'>   </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>51,1)='267/85'<span style='mso-tab-count:2'>          </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>52,1)='289/92' <span style='mso-tab-count:1'>   </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>53,1)='311/99'<span style='mso-tab-count:2'>          </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>54,1)='99/311' <span style='mso-tab-count:1'>   </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>55,1)='333/106'<span style='mso-tab-count:2'>         </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>56,1)='106/333' <span style='mso-tab-count:1'>  </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>57,1)='52518/16717'<span style='mso-tab-count:2'>           </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>58,1)='16717/52518' <span style='mso-tab-count:1'>    </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>59,1)='52873/16830'<span style='mso-tab-count:2'>           </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>60,1)='16830/52873' <span style='mso-tab-count:1'>    </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>61,1)='52163/16604'<span style='mso-tab-count:2'>           </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>62,1)='16604/52163' <span style='mso-tab-count:1'>    </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>63,1)='53228/16943'<span style='mso-tab-count:2'>           </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>64,1)='16943/53228' <span style='mso-tab-count:1'>    </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>65,1)='53583/17056'<span style='mso-tab-count:2'>           </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>66,1)='17056/53583' <span style='mso-tab-count:1'>    </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>67,1)='17169/53938'<span style='mso-tab-count:2'>           </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp; <span style='mso-spacerun:yes'>   </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>68,1)='53938/17169' <span style='mso-tab-count:1'>    </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=SpellE><span style='font-size:10.0pt; font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>69,1)='54293/17282'<span style='mso-tab-count:2'>           </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;<span style='mso-spacerun:yes'>    </span><o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span class=SpellE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>pi_<span class=GramE>variants</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:black'>(</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>70,1)='17282/54293' <span style='mso-tab-count:1'>    </span></span><span style='font-size:10.0pt; font-family:"Courier New";color:green'>&amp;&amp;</span><span style='font-size: 10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>**<a href="http://www.2dcode-r-past.com/Geometry/SQ/pi_converge.htm"> <span class=GramE>Pi <span style='mso-spacerun:yes'> </span>pattern</span> of convergence in Pi approximation</a> **<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"'>Rhind</span></span><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"'> Papyrus 256:81 Statistics& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/256-81_81-256.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/256-81_81-256.htm</a><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"'>Rhind</span></span><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"'> #1 Root 2 Approximation 140/99<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/140-99_99-140.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/140-99_99-140.htm</a><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"'>Rhind</span></span><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"'> #1 Root 2 Approximation 99/70<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/99-70_70-99.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/99-70_70-99.htm</a><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>Root 3 Approximation 97/56 [John <span class=SpellE>Michell</span> <span class=SpellE>tDoP</span>]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/97-56_56-97.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/97-56_56-97.htm</a><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>6/7 [</span><span style='font-size:10.0pt;font-family:Arial;color:black'>Stephen <span class=SpellE>Dail</span>   & 6/7 as in Royal and Common cubit units&  </span><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"'>]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/6-7_7-6.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/6-7_7-6.htm</a><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>480/378 [</span><span style='font-size:10.0pt;font-family:Arial;color:black'>Derek Skhane   & </span><span lang=EN-GB style='mso-ansi-language:EN-GB'>GP Tan</span><span style='font-size: 10.0pt;font-family:Arial;color:black'>&  </span><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/480-378_378-480.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/480-378_378-480.htm</a><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>22/7 [</span><span style='font-size:10.0pt;font-family:Arial;color:black'>Old School Pi& .!] <span class=GramE>OR 4x Inverse <span class=SpellE>Seked</span> Ratio of 11:14.</span></span><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/22-7_7-22.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/22-7_7-22.htm</a><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>864/275 [Foundational <span class=GramE>Number ][</span>Fibonacci Pi][John <span class=SpellE>Michell</span> <span class=SpellE>tDoP</span>]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/864-275_275-864.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/864-275_275-864.htm</a><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>360/256 [Ratio resulting from [4] <span class=SpellE>Rhind</span> Papyrus 256:81 Pi Tangles in Base 360 Geometry]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/360-256_256-360.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/360-256_256-360.htm</a><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><o:p>&nbsp;</o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'>256/243 <span class=GramE>[<span style='font-family:"Courier New";color:green'> Derek</span></span></span><span style='font-size:10.0pt;font-family:"Courier New";color:green'> Skhane / Pythagorean <span class=SpellE>Limma</span> ]</span><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New";color:black'> [Binds with 256:81 <span class=SpellE>Rhind</span> Papyrus current accepted AE Pi]</span><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/256-243_243-256.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/256-243_243-256.htm</a></span><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none; text-underline:none'><o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><o:p>&nbsp;</o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New";color:black'>531441:524288 </span><span class=GramE><span style='font-size:10.0pt;font-family:Georgia; mso-bidi-font-family:"Courier New"'>[</span><span style='font-size:10.0pt; font-family:"Courier New";color:green'> Derek</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:green'> Skhane </span><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"; color:blue'>Comma of </span><span style='font-size:10.0pt;font-family:Georgia; mso-bidi-font-family:"Courier New";color:black'>Pythagoras][Binds with 256:81 <span class=SpellE>Rhind</span> Papyrus current accepted AE Pi]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/531441-524288.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/531441-524288.htm</a></span><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none; text-underline:none'><o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><o:p>&nbsp;</o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New";color:black'>756/440<span style='mso-spacerun:yes'>  </span><span class=SpellE>Khufu</span> BASE12 : RC </span><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"'>[</span><span style='font-size:10.0pt;font-family:"Courier New";color:green'> Derek Skhane Units for GP][</span><span style='font-size:10.0pt;font-family:Georgia; mso-bidi-font-family:"Courier New";color:blue'>Egyptian Cubit</span><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"; color:black'>]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/756-440.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/756-440.htm</a><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New";color:black'>89/<span class=GramE>55<span style='mso-spacerun:yes'>  </span><span style='color:windowtext'>[</span></span></span><span style='font-size:10.0pt;font-family:"Courier New";color:green'> <span class=SpellE>Assem</span> <span class=SpellE>Dief</span> </span><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"; color:black'>AE Phi Approximation]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/89-55.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/89-55.htm</a><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New";color:black'>37/<span class=GramE>27<span style='mso-spacerun:yes'>  </span><span style='color:windowtext'>[</span></span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>Pythagorean minor third {32:27} [Derek Skhane]</span><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New";color:black'>]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/37-27.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/37-27.htm</a><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New";color:black'>64/<span class=GramE>63<span style='mso-spacerun:yes'>  </span><span style='color:windowtext'>[</span></span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>Eye of <span class=SpellE>Horus</span> Fraction {64:63} [Derek Skhane]</span><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"; color:black'>]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/64-63.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/64-63.htm</a><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New";color:black'>76/<span class=GramE>47<span style='mso-spacerun:yes'>  </span><span style='color:windowtext'>[</span></span></span><span style='font-size:10.0pt;font-family:"Courier New";color:black'>Lucas Approximation Fraction</span><span style='font-size:10.0pt;font-family:Georgia; mso-bidi-font-family:"Courier New";color:black'>]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Georgia;mso-bidi-font-family:"Courier New"'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/76-47.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/76-47.htm</a><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><o:p>&nbsp;</o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'>256/234<o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/256-234.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/256-234.htm</a> <o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><o:p>&nbsp;</o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'>9072/440 <span class=GramE>[ Ancient</span> Royal Cubit ]<o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/256-234.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/256-234.htm</a> <o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><o:p>&nbsp;</o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'>((99/70<span class=GramE>)+</span>(97/56)) Potential AE Root2+Root3 Pi approximation utilizing Rhind#1 Root 2 approximation units.<o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/97-56_99-70.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/97-56_99-70.htm</a> <o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><o:p>&nbsp;</o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'>((140/99<span class=GramE>)+</span>(97/56)) Potential AE Root2+Root3 Pi approximation utilizing Rhind#1 Root 2 approximation units.<o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/97-56_140-99.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/97-56_140-99.htm</a> <o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><b style='mso-bidi-font-weight:normal'><span style='color:windowtext;text-decoration: none;text-underline:none'><o:p>&nbsp;</o:p></span></b></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><o:p>&nbsp;</o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'>8/<span class=GramE>9<span style='mso-spacerun:yes'>  </span>AE</span> RMP#50 Fraction <span class=SpellE>Ahmes</span> used to solve..<o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/9-8.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/9-8.htm</a> <o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><o:p>&nbsp;</o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span style='font-size:10.0pt;font-family:"Courier New"; color:green'>How did Archimedes arrive at this particular approximation?<span style='mso-spacerun:yes'>  </span>No puzzle has exercised more fascination <o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span class=GramE><span style='font-size:10.0pt;font-family:"Courier New";color:green'>upon</span></span><span style='font-size:10.0pt;font-family:"Courier New";color:green'> writers interested in the history of mathematics...<span style='mso-spacerun:yes'>  </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><o:p>&nbsp;</o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span style='font-size:10.0pt;font-family:"Courier New"; color:green'>Archimedes (265/153) &lt; <span class=SpellE><span class=GramE>sqrt</span></span><span class=GramE>(</span>3) &lt; (1351/780)<span style='mso-tab-count:1'>     </span> <span style='mso-tab-count:1'>     </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;mso-layout-grid-align:none; text-autospace:none'><span class=MsoHyperlink><span style='color:windowtext; text-decoration:none;text-underline:none'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/265-153.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/265-153.htm</a><o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><o:p>&nbsp;</o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:"Courier New";color:green'>Archimedes (265/153) &lt; <span class=SpellE><span class=GramE>sqrt</span></span><span class=GramE>(</span>3) &lt; (1351/780)<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/1351-780.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/1351-780.htm</a><o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><o:p>&nbsp;</o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:"Courier New";color:green'>577/408 SQRT 2 APPRX<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/577-408.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/577-408.htm</a><o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><o:p>&nbsp;</o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:"Courier New";color:green'>989/571 SQRT 3 APPRX<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/989-571.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/989-571.htm</a><o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><o:p>&nbsp;</o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'>(97/56))/ ((99/70) </span></span><span style='font-size:10.0pt;font-family:"Courier New"; color:green'>SQRT 3 /SQRT 2 APPRX [AE VESICA]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/97-56_d_99-70.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/97-56_d_99-70.htm</a><o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><o:p>&nbsp;</o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:"Courier New";color:green'>22/7 x 89/55 Old School Pi x Old School Phi<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/22-7_89-55.htm">http://www.2dcode-r-past.com/Geometry/AE_ratios/22-7_89-55.htm</a><o:p></o:p></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none;text-underline:none'><o:p>&nbsp;</o:p></span></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:10.0pt;font-family:Georgia;mso-bidi-font-family:"Courier New"'><span style='mso-spacerun:yes'>               </span>(<b style='mso-bidi-font-weight: normal'>Copyright 2dCODE-R-PAST.com 2008<span class=GramE>-)</span><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none; text-underline:none'><o:p></o:p></span></span></b></span></p> </div> </body> </html>