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Exploring various Ancient Ratios gathered from great authors and researchers to see where fractional ratios collide with whole numbers!!.

 

Below…are works in progress! : I will continue to update other ratios for analysis as time permits….

 

Current Functions employed:

MOD(1)-MOD(13) BY:

Array of Pi variants

Array of Phi variants

Array of various other key ratios

SQRTS-10

Primes

These will be explored and updated as time permits.

 

Current Historical Pi fractional ratios utilized:

 

DIMENSION pi_variants(34,1)

pi_variants(1,1)='256/81'     && 2000 B.C.E.                                 Egyptians use p = 256/81 = 3.1605. 

pi_variants(2,1)='81/256'

pi_variants(3,1)='22/7'       &&Old School Pi [AE Pi too??!!][Archimedes Approximates this ... Pietre reccomends intended Pi=22/7]

pi_variants(4,1)='7/22'

pi_variants(5,1)='864/275'    &&1220  Leonardo de Pisa (Fibonacci) finds p = 3.141818.... , John Michell Pi [tDoP] Foundational Pi!

pi_variants(6,1)='275/864'

pi_variants(7,1)='(99/70)+(97/56)'        &&Suggested AE ROOT PI Apprx

pi_variants(8,1)='(1/((99/70)+(97/56)))'

pi_variants(9,1)='(140/99)+(97/56)'       &&Suggested AE ROOT PI Apprx 2

pi_variants(10,1)='(1/((140/99)+(97/56)))'

pi_variants(11,1)='355/113'         &&Tsu Ch'ung-chih (1500 ago)              3.141592920353... 0.0000002667...

pi_variants(12,1)='113/355'

pi_variants(13,1)='25/8'            &&2000 B.C.E..  Babylonians use p = 25/8 = 3.125. 

pi_variants(14,1)='8/25'

pi_variants(15,1)='3/1'       &&1100 B.C.E. Chinese use p = 3.  550 B.C.E.    Old Testament implies p = 3. 

pi_variants(16,1)='1/3'

pi_variants(17,1)='223/71'          && 250 B.C.E.  Archimedes uses a 96-sided polygon to establish 223/71 < pi < 22/7. He also uses a spiral to square the circle.   

pi_variants(18,1)='71/223'    &&

pi_variants(19,1)='377/120'         && 150  Claudius Ptolemy uses p = 3°8'30" = 377/120 = 3.14166.... 

pi_variants(20,1)='120/377'   &&   

pi_variants(21,1)='142/45'          && 250  Wang Fau uses p 142/45 = 3.1555.... 

pi_variants(22,1)='45/142'    &&   

pi_variants(23,1)='157/50'          && 263  Liu Hui uses p = 157/50 = 3.14.  

pi_variants(24,1)='50/157'    &&   

pi_variants(25,1)='62832/20000'           && 530  Aryabhata uses p = 62832/20000 = 3.1416.   

pi_variants(26,1)='20000/62832'     &&   

pi_variants(27,1)='10.5/1'          && 650  Brahmagupta uses p = 10 1/2 = 3.162....   

pi_variants(28,1)='1/10.5'    &&   

pi_variants(29,1)='10.5/1'          && 650  Brahmagupta uses p = 10 1/2 = 3.162....   

pi_variants(30,1)='1/10.5'    &&         

pi_variants(31,1)='(4*ATAN(1))/1'         && Current True Pi   

pi_variants(32,1)='1/(4*ATAN(1))'   &&

pi_variants(33,1)='SQRT(2)+SQRT(3)'       && Current SQRT Pi  

pi_variants(34,1)='1/(SQRT(2)+SQRT(3))'   &&

pi_variants(35,1)='14148475504056880/4503599627370496'            && MATLAB quantity pi   

pi_variants(36,1)='4503599627370496/14148475504056880'      &&

pi_variants(37,1)='732/233'         &&   

pi_variants(38,1)='233/732'   &&

pi_variants(39,1)='19/6'            &&   

pi_variants(40,1)='6/19'      &&

pi_variants(41,1)='179/57'          &&   

pi_variants(42,1)='57/179'    &&

pi_variants(43,1)='201/64'          &&   

pi_variants(44,1)='64/201'    &&

pi_variants(45,1)='201/64'          &&   

pi_variants(46,1)='64/201'    &&

pi_variants(47,1)='245/78'          &&   

pi_variants(48,1)='78/245'    &&

pi_variants(49,1)='245/78'          &&   

pi_variants(50,1)='78/245'    &&

pi_variants(51,1)='267/85'          &&   

pi_variants(52,1)='289/92'    &&

pi_variants(53,1)='311/99'          &&   

pi_variants(54,1)='99/311'    &&

pi_variants(55,1)='333/106'         &&   

pi_variants(56,1)='106/333'   &&

pi_variants(57,1)='52518/16717'           &&   

pi_variants(58,1)='16717/52518'     &&

pi_variants(59,1)='52873/16830'           &&   

pi_variants(60,1)='16830/52873'     &&

pi_variants(61,1)='52163/16604'           &&   

pi_variants(62,1)='16604/52163'     &&

pi_variants(63,1)='53228/16943'           &&   

pi_variants(64,1)='16943/53228'     &&

pi_variants(65,1)='53583/17056'           &&   

pi_variants(66,1)='17056/53583'     &&

pi_variants(67,1)='17169/53938'           &&    

pi_variants(68,1)='53938/17169'     &&

pi_variants(69,1)='54293/17282'           &&   

pi_variants(70,1)='17282/54293'     &&

 

** Pi  pattern of convergence in Pi approximation **

 

Rhind Papyrus 256:81 Statistics…

http://www.2dcode-r-past.com/Geometry/AE_ratios/256-81_81-256.htm

 

Rhind #1 Root 2 Approximation 140/99

http://www.2dcode-r-past.com/Geometry/AE_ratios/140-99_99-140.htm

 

Rhind #1 Root 2 Approximation 99/70

http://www.2dcode-r-past.com/Geometry/AE_ratios/99-70_70-99.htm

 

Root 3 Approximation 97/56 [John Michell tDoP]

http://www.2dcode-r-past.com/Geometry/AE_ratios/97-56_56-97.htm

 

6/7 [Stephen Dail – “…6/7 as in Royal and Common cubit units…”]

http://www.2dcode-r-past.com/Geometry/AE_ratios/6-7_7-6.htm

 

480/378 [Derek Skhane – “…GP Tan…”]

http://www.2dcode-r-past.com/Geometry/AE_ratios/480-378_378-480.htm

 

22/7 [Old School Pi….!] OR 4x Inverse Seked Ratio of 11:14.

http://www.2dcode-r-past.com/Geometry/AE_ratios/22-7_7-22.htm

 

864/275 [Foundational Number ][Fibonacci Pi][John Michell tDoP]

http://www.2dcode-r-past.com/Geometry/AE_ratios/864-275_275-864.htm

 

360/256 [Ratio resulting from [4] Rhind Papyrus 256:81 Pi Tangles in Base 360 Geometry]

http://www.2dcode-r-past.com/Geometry/AE_ratios/360-256_256-360.htm

 

256/243 [ Derek Skhane / Pythagorean Limma ] [Binds with 256:81 Rhind Papyrus current accepted AE Pi]

http://www.2dcode-r-past.com/Geometry/AE_ratios/256-243_243-256.htm

 

531441:524288 [ Derek Skhane Comma of Pythagoras][Binds with 256:81 Rhind Papyrus current accepted AE Pi]

http://www.2dcode-r-past.com/Geometry/AE_ratios/531441-524288.htm

 

756/440  Khufu BASE12 : RC [ Derek Skhane Units for GP][Egyptian Cubit]

http://www.2dcode-r-past.com/Geometry/AE_ratios/756-440.htm

 

89/55  [ Assem Dief AE Phi Approximation]

http://www.2dcode-r-past.com/Geometry/AE_ratios/89-55.htm

 

37/27  [Pythagorean minor third {32:27} [Derek Skhane]]

http://www.2dcode-r-past.com/Geometry/AE_ratios/37-27.htm

 

64/63  [Eye of Horus Fraction {64:63} [Derek Skhane]]

http://www.2dcode-r-past.com/Geometry/AE_ratios/64-63.htm

 

76/47  [Lucas Approximation Fraction]

http://www.2dcode-r-past.com/Geometry/AE_ratios/76-47.htm

 

256/234

http://www.2dcode-r-past.com/Geometry/AE_ratios/256-234.htm

 

9072/440 [ Ancient Royal Cubit ]

http://www.2dcode-r-past.com/Geometry/AE_ratios/256-234.htm

 

((99/70)+(97/56)) Potential AE Root2+Root3 Pi approximation utilizing Rhind#1 Root 2 approximation units.

http://www.2dcode-r-past.com/Geometry/AE_ratios/97-56_99-70.htm

 

((140/99)+(97/56)) Potential AE Root2+Root3 Pi approximation utilizing Rhind#1 Root 2 approximation units.

http://www.2dcode-r-past.com/Geometry/AE_ratios/97-56_140-99.htm

 

 

8/9  AE RMP#50 Fraction Ahmes used to solve..

http://www.2dcode-r-past.com/Geometry/AE_ratios/9-8.htm

 

How did Archimedes arrive at this particular approximation?  No puzzle has exercised more fascination

upon writers interested in the history of mathematics... 

 

Archimedes (265/153) < sqrt(3) < (1351/780)         

http://www.2dcode-r-past.com/Geometry/AE_ratios/265-153.htm

 

Archimedes (265/153) < sqrt(3) < (1351/780)

http://www.2dcode-r-past.com/Geometry/AE_ratios/1351-780.htm

 

577/408 SQRT 2 APPRX

http://www.2dcode-r-past.com/Geometry/AE_ratios/577-408.htm

 

989/571 SQRT 3 APPRX

http://www.2dcode-r-past.com/Geometry/AE_ratios/989-571.htm

 

(97/56))/ ((99/70) SQRT 3 /SQRT 2 APPRX [AE VESICA]

http://www.2dcode-r-past.com/Geometry/AE_ratios/97-56_d_99-70.htm

 

22/7 x 89/55 Old School Pi x Old School Phi

http://www.2dcode-r-past.com/Geometry/AE_ratios/22-7_89-55.htm

 

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