User:Lynkeus Artifex

FOLK MATHEMATICS  offers users a variety of applicable, but little-known, approximations. Many of these are believed to form part of the secret lore of esoteric organisations like Pythagoreans, Freemasons, Illuminati – and so forth. They are published here as the result of independent work by “Lynkeus Artifex”, a twentieth-century autodidact HERMETICIST, who came upon them by serendipity, but confirmed them as valid by simple arithmetic.

If these approximations are indeed part of the canon of masonic secrets, that secrecy is maintained by the “MASTER TYLER” of the Order, and by the Tylers of the various Lodges. Hence, these approximations deserve to be published under the name of “The Tyler Numbers”.

The “MASTER TYLER “ AND  HIS  TWELVE  APPRENTICES

6    20    21    (27)     34     48    55    59    76    83  90   96    (107)

This “COVEN” of integers enables practitioners of “folk mathematics” to find some very useful approximations of square roots and other irrationals without using difficult algorithms.

The number “34” is known as THE  MASTER  TYLER  because  when IT, and only IT – (34), acts as the denominator or numerator of a rational fraction with one of the other  tylers , a useful approximation results. We list them below, accurate to FIVE/SIX decimal places :

ITEM		accurate (a)	Tyler Approx'n (TA)	Error: ((a-TA)/a		Error (1 in X)

tangent 10 degrees	0.176327..... 6/34 =  0.176470.....	-0.000814.....		1  in  1228.5

Pentagon : side/circumdiameter	0.587785..... 20/34 = 0.58823.....	-0.000445.....		1 in  2247

1/Phi (1/1.618034)	0.618034..... 21/34 = 0.617647.....	 0.000387....		1 in  2584

1/(cube-root 2)		0.793700.... 27/34 = 0.79411....	-0.0005166....		1 in  1936 (cube-root 2)		1.259921... 34/27 = 1.25925...	 0.0005252...		1 in  1936

UNITY/& its sq.rt	1 - exactly	34/34 = 1 - exactly	 ZERO			INFINITY

square-root 2		1.414214.... 48/34 = 1.41176...	 0.00245.....		1 in  407

Phi (1.618034..)	1.618034... 55/34 = 1.617647...	 0.000387		1 in  2584

square-root 3		1.732051... 59/34 = 1.735294...	-0.001872...		1 in  534

square-root 5		2.236068... 76/34 = 2.235294...	 0.000346...		1 in  2889

square-root 6		2.44949... 83/34 = 2.441176...	 0.0033938..		1 in  295

square-root 7		2.645751... 90/34 = 2.647059...	-0.000494...		1 in  2023

square-root 8		2.828427... 96/34 = 2.823529...	 0.001731...		1 in  577.5 PI; (square-root 10)	3.141593.. 107/34 = 3.147058...	-0.005466...		1 in  183 (3.162278...	(107.5)/34 = 3.161764..	 0.0001622...		1 in  6165

(NOTE re PI)	(tyler 48) plus (tyler 59) equals 107, - and 107/34 = (3.147058...) ! Now, Pi to seven decimal places	is (3.1415927). Thus 107/34 is an approximation of Pi with an error of ( -0.005466...) or ( 1  in  183 ). The sum of accurate (sq.rt 2 + sq.rt 3) is  (3.1462644...) to seven decimal places – a well-known approximation for Pi. The Tyler relationships are best remembered as : (107/34) gives a useful value for Pi  -  while (107.5/34) gives an accurate approximation for (sq.rt 10)