tetrahedron Posted April 4, 2019 Report Share Posted April 4, 2019 Except for 'cardinal' points 0, 90, 180, and 270 degrees, ALL multiples of 18 degrees on a circle will give sine or cosine values which are plus or minus half either the smaller Golden Ratio (sqrt5-1)/2=0.618033989... or the larger (sqrt5+1)/2=1.618033989... The cardinal points act like symmetry axes in the assignment of particular values. In an article I recently read about the synthesis of superheavy elements it was mentioned that deformed shell closures occurred at 108 protons and 126 neutrons. Interestingly both of these counts are non-cardinal multiples of 18 degrees, and so have the relationships outlined above. I've started wondering whether other similar relationships, hidden under surface values, might pertain to other shell-closure combinations in deformed nuclei. But I don't yet have data on what these counts for shell closures would be. sin 18= 0.309016994... cos 18= 0.951056516...sin 36= 0.587785252... cos 36= 0.809016994...sin 54= 0.809016994... cos 54= 0.587785252...sin 72= 0.951056516... cos 72= 0.309016994...sin 90= 1.000000000... cos 90= 0.000000000...sin 108= 0.951056516... cos 108= -0.309016994...sin 126= 0.809016994... cos 126= -0.587785252...sin 144= 0.587785252... cos 144= -0.809016994sin 162= 0.309016994... cos 162= -0.951056516...sin 180= 0.000000000... cos 180= -1.000000000...sin 198= -0.309016994... cos 198= -0.951056516...sin 216= -0.587785252... cos 216= -0.809016994...sin 234= -0.809016516... cos 234= -0.587785252...sin 252= -0.951056516... cos 252= -0.309016994...sin 270= -1.000000000... cos 270= 0.000000000...sin 288= -0.951056516... cos 288= 0.309016994...sin 306= -0.809016994... cos 306= 0.587785252...sin 324= -0.587785252... cos 324= 0.809016994...sin 342= -0.309016994... cos 342= 0.951056516...sin 360= 0.000000000... cos 360= 1.000000000... Jess Tauber Quote Link to comment Share on other sites More sharing options...
tetrahedron Posted April 4, 2019 Author Report Share Posted April 4, 2019 Interestingly, all of these fractional values (except for + or - 0 or 1) relate to the Golden Ratio. All you have to do is double the fractional value, then square. The resulting numbers then either directly relate to Phi (the smaller, larger, larger squared, etc.) or are the complements when subtracting from whole number values, such as 1-Phi, 2-Phi, 3-Phi, etc. Jess Tauber Quote Link to comment Share on other sites More sharing options...
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