ñ ø à				PHI

à IS THE APEX ANGLE OF AN ISOSCELES TRIANGLE WHOSE SIDE DIVIDED BY ITS BASE
   	    GIVES THE FOLLOWING ROOTS OF POLYNOMIALS

+- AND +- VARY TOGETHER, WHILE +- AND -+ VARY INDEPENDENTLY

  			ODD SIDED POLYGONS

		< SQUARE ROOT >    ( PARENTHESIS )
No. of
ROOTS POLYGON	ROOTS               POLYNOMIALS
1     3-GON
à60ø		R1=1                X-1

2     5-GON	R1&2=(1ñ<5>)/2      X^2ñX-1
  or, R1=<1/(<5>-1)+1/(1-1/<5>)>
à36ø	    or, R1=(1+<5>)/2
 and, R2=1-<1/(<5>-1)+1/(1-1/<5>)>
à108ø	    or, R2=(1-<5>)/2

3     7-GON
à		R1=                 X^3-2X^2-X+1, or
à		R2=                 X^3-X^2-2X+1
à		R3=

4     9-GON     		    (X^3-3X^2+1)*(X+1)

5    11-GON			    (


			EVEN SIDED POLYGONS
No. of
ROOTS POLYGON	ROOTS               POLYNOMIALS
1     4-GON
à90ø		R1=1/<2>                                 2X^2-1

2     6-GON
à60ø		R1=1                                     X-1
à120ø  		R2=1/<3>                                 3X^2-1

3     8-GON     R1&3=<1ñ1/<2>>                           2X^4-4X^2+1
à45ø		R1=<1+1/<2>>
à90ø		R2=1/<2>                                 2X^2-1
à135ø		R3=<1-1/<2>>)

4     10-GON	R2&4=<(ñ1+<5>)/2<5>>                     5X^4-5X^2+1
		or, =<(5ñ<5>)/10>
à72ø   		R2=<<1/(<5>-1)+1/(1-1/<5>)>/<5>>
		or=<(1+<5>)/2<5>>
		  =<(5+<5>)/10>
à144ø 		R4=<1-<1/(<5>-1)+1/(1-1/<5>)>/<5>>
		  =<(-1+<5>)/2<5>>
		  =<(5-<5>)/10>

5     12-GON	R1&5=<2ñ<3>>                             X^4-4X^2+1
à30ø		R1=<2+<3>>
à60ø		R2=1
à90ø		R3=1/<2>
à120ø		R4=1/<3>
à150ø		R5=<2-<3>>

6     14-GON
à		R1=
à		R2=
à		R3=
à		R4=
à		R5=
à		R6=

8     16-GON    R1,3,5,7=<2*(2+-<2>)*(2-+<2+-<2>>)>/2    2X^8-16X^6+20X^4-8X^2+1
	    	    or, =1/<2+-1/<1-+1/<2>>>
		R2,6=<4ñ2*<2>>/2                         2X^4-4X^2+1
		R4=<2>/2                                 2X^2-1
à22.5ø          R1=<2*(2+<2>)*(2+<2+<2>>)>/2
à67.5ø		R3=<2*(2-<2>)*(2+<2-<2>>)>/2
à112.5ø		R5=<2*(2-<2>)*(2-<2-<2>>)>/2
à157.5ø		R7=<2*(2+<2>)*(2-<2+<2>>)>/2