Divine Proportions: Rational Trigonometry To Un... May 2026

), a dimensionless ratio that measures the "separation" between two lines. Unlike angles, which are circular, spread is a rational function. For a right triangle with quadrances Q1cap Q sub 1 Q2cap Q sub 2 , and hypotenuse Q3cap Q sub 3

is a revolutionary approach to geometry developed by Dr. Norman J. Wildberger that replaces transcendental functions like tantangent

with purely algebraic concepts. By avoiding irrational numbers and infinite series, it allows for exact calculations over any field, not just the real numbers. 1. Replace distance with quadrance Divine Proportions: Rational Trigonometry to Un...

s=QoppositeQhypotenuses equals the fraction with numerator cap Q sub o p p o s i t e end-sub and denominator cap Q sub h y p o t e n u s e end-sub end-fraction The spread ranges from indicates parallel lines and indicates perpendicular lines. 3. Apply the Main Laws

(Q1+Q2+Q3)2=2(Q12+Q22+Q32)open paren cap Q sub 1 plus cap Q sub 2 plus cap Q sub 3 close paren squared equals 2 open paren cap Q sub 1 squared plus cap Q sub 2 squared plus cap Q sub 3 squared close paren : The rational equivalent of the Sine Law: ), a dimensionless ratio that measures the "separation"

: These laws work in any coordinate system, including those used in Einstein's Special Relativity (Minkowski space). ✅ Answer

Rational Trigonometry is a purely algebraic alternative to classical trigonometry that replaces distance and angle with and spread , allowing for exact geometric calculations without transcendental functions. Norman J

(Q1+Q2−Q3)2=4Q1Q2(1−s3)open paren cap Q sub 1 plus cap Q sub 2 minus cap Q sub 3 close paren squared equals 4 cap Q sub 1 cap Q sub 2 open paren 1 minus s sub 3 close paren Why This Matters : You never need to use a calculator for 2the square root of 2 end-root . All results are exact fractions.