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## Magic "Integer" Circles

 Some of you may be interested in the following "mathy beauty". Under all circles with integer radius 0 < r <= 10000 there are four "magic" ones with radius 5, 25, 65 and 325. These four have relatively many points of *integers* (x,y) such that x^2 + y^2 = r^2. (These points are NOT rounded values that only approximate the circle value but integer values that are *on* the circle). Clearly there are at least four such points: (-r,0),(0,r),(r,0),(0,-r) -- if the origin is (0,0). Now the relation of radius and number of such points may be such that we can visually accept a polygon of these points as circles. To comfort you (script for an option button): on mouseDown   put 5 &cr& 25 &cr& 65 &cr& 325 into me end mouseDown on mouseUp   put the label of me into R -- the radius = one of 5, 25, 65, 325   if there is no grc "poly" then create grc "poly"   set style of grc "poly" to "polygon"   set opaque of grc "poly" to "true"   set backColor of grc "poly" to "255,128,0"   set markerDrawn of grc "poly" to "true"   set markerPoints of grc "poly" to "0,0"   set markerLineSize of grc "poly" to "3"   set points of grc "poly" to integerPoints(R)   set loc of grc "poly" to the loc of this card end mouseUp -- the circle origin is (0,0), so set the loc of the polygon later on! function integerPoints rds   switch rds     case 5       -- a circle with radius r=5 contains these 12 points of integers:       put (4,-3)&cr&(3,-4)&cr&(0,-5)&cr&(-3,-4)&cr&(-4,-3)&cr&(-5,0)&cr& \           (-4,3)&cr&(-3,4)&cr&(0,5)&cr&(3,4)&cr&(4,3)&cr&(5,0) into p       break     case 25       -- a circle with radius r=25 contains these 20 points of integers:       put (24,-7)&cr&(20,-15)&cr&(15,-20)&cr&(7,-24)&cr&(0,-25)&cr& \           (-7,-24)&cr&(-15,-20)&cr&(-20,-15)&cr&(-24,-7)&cr&(-25,0)&cr& \           (-24,7)&cr&(-20,15)&cr&(-15,20)&cr&(-7,24)&cr&(0,25)&cr& \           (7,24)&cr&(15,20)&cr&(20,15)&cr&(24,7)&cr&(25,0) into p       break     case 65       -- a circle with radius r=65 contains these 36 points of integers:       put (63,-16)&cr&(60,-25)&cr&(56,-33)&cr&(52,-39)&cr&(39,-52)&cr& \           (33,-56)&cr&(25,-60)&cr&(16,-63)&cr&(0,-65)&cr&(-16,-63)&cr& \           (-25,-60)&cr&(-33,-56)&cr&(-39,-52)&cr&(-52,-39)&cr& \           (-56,-33)&cr&(-60,-25)&cr&(-63,-16)&cr&(-65,0)&cr&(-63,16)&cr& \           (-60,25)&cr&(-56,33)&cr&(-52,39)&cr&(-39,52)&cr&(-33,56)&cr& \           (-25,60)&cr&(-16,63)&cr&(0,65)&cr&(16,63)&cr&(25,60)&cr& \           (33,56)&cr&(39,52)&cr&(52,39)&cr&(56,33)&cr&(60,25)&cr& \           (63,16)&cr&(65,0) into p       break     case 325       -- a circle with radius r=325 contains these 60 points of integers:       put (323,-36)&cr&(315,-80)&cr&(312,-91)&cr&(300,-125)&cr& \           (280,-165)&cr&(260,-195)&cr&(253,-204)&cr&(204,-253)&cr& \           (195,-260)&cr&(165,-280)&cr&(125,-300)&cr&(91,-312)&cr& \           (80,-315)&cr&(36,-323)&cr&(0,-325)&cr&(-36,-323)&cr& \           (-80,-315)&cr&(-91,-312)&cr&(-125,-300)&cr&(-165,-280)&cr& \           (-195,-260)&cr&(-204,-253)&cr&(-253,-204)&cr&(-260,-195)&cr& \           (-280,-165)&cr&(-300,-125)&cr&(-312,-91)&cr&(-315,-80)&cr& \           (-323,-36)&cr&(-325,0)&cr&(-323,36)&cr&(-315,80)&cr& \           (-312,91)&cr&(-300,125)&cr&(-280,165)&cr&(-260,195)&cr& \           (-253,204)&cr&(-204,253)&cr&(-195,260)&cr&(-165,280)&cr& \           (-125,300)&cr&(-91,312)&cr&(-80,315)&cr&(-36,323)&cr& \           (0,325)&cr&(36,323)&cr&(80,315)&cr&(91,312)&cr&(125,300)&cr& \           (165,280)&cr&(195,260)&cr&(204,253)&cr&(253,204)&cr& \           (260,195)&cr&(280,165)&cr&(300,125)&cr&(312,91)&cr& \           (315,80)&cr&(323,36)&cr&(325,0) into p       break   end switch   return p &cr& line 1 of p -- close polygon end integerPoints Below are the possible number of integer points for all integers radius values r with 0 < r <= 10000. Each of these numbers is followed by the two smallest radii where these numbers occur. TMHO, of these the 12:5, 20:25, 36:65 and 60:325 are acceptable to serve alone as points of a polygon approximating a circle of radius r (see above). [number of integer points:radius1,radius2] 4:1,2 12:5,10 20:25,50 28:125,250 36:65,85 44:3125,6250 60:325,425 84:1625,2125 100:4225,7225 108:1105,1885 180:5525,9425 For the math behind that last computation see http://oeis.org/A046109_______________________________________________ use-livecode mailing list [hidden email] Please visit this url to subscribe, unsubscribe and manage your subscription preferences: http://lists.runrev.com/mailman/listinfo/use-livecode