Prof. Wildberger I can say that in 2012 I discovered the Stern Brocot Tree and I was totally enlightened by your videos. My magic numbers were a continuous series: 1,2,1,2,2,3,2,1,2,3,1,2,3,2,2,1... which I introduced in The On-Line Encyclopedia of Integer Sequences search. For me it was a miracle because it proved to be a resolution of a problem which I had struggled from first gymnasium years about repeated mixtures of prime-color-spectrum proportions on color-circle. So, at a certain point I simplified the model of adding proportions only by having 0 (zero) and 1 as extreme limits of a line of prime-colors mixtures (let say red and yellow, where red has zero parts of yellow and yellow has 1 part of yellow). And the great surprise was that I discovered a fractal by bisecting the segment only to the zero part, because it consists only on adding zero to one (without making fractions). My purpose at that moment was to obtain color-circles of mixtures where you don't have a number of additions (or bisections) multiple of 2 or/and 6, but this proved to be unrealistic because Stern Brocot Tree, I think, is not a continuous series, so doesn't offers finite intermediary values between his numbers, other than by bisection. But still, for me is a beautiful construction because it unifies Fibonacci numbers with Golden Ratio. Actually I have now a new obsession: if we would build a structure made by bisection but without adding parts or dividing fractions, but inducing Golden Ratio sectioning (instead having number-gape-number-gape on a level, we would have M-m-M-m-M), could we find an organic system of packing, like in natural structures like buds, and see the connection between cycloidal movement of water and Golden Ratio? OK, I admit is maybe too much... I wish you to keep be inspired by Stern Brocot flame!
Coupled oscillators will phase lock into Farey fractions and if perturbed, will make MOD1 transitions to the neighboring MOD1 ratio in Farey space. MOD1 is when you have n/m and p/q and n x q - m x p = 1. The upper levels are, empirically, more STABLE than lower levels, so if perturbed a 1:2 coupled oscillator state might transition to 1:1. And 3:5 might transition to either 1:2 or 2:3 (both are MOD1 neighbors) but more likely will go to the more stable 1:2 state. These states are also known as resonances (aka "Arnold tongues", after the mathematician Arnold). Physics is full of experiments on inanimate rhythmic systems of coupled oscillators (e.g., circuits) that show state transitions that follow the Farey tree. It has also been looked at in biological systems, e.g., spontaneously beating chick heart embryo cells that phase lock to applied oscillatory currents ..and humans moving their limbs. Interestingly, the most consonant musical intervals (chords) are also the most stable according to this framework: 1:2 (octave) is more consonant than 2:3 (fifth)...than 3:4 (fourth)...so human aesthetics and ideas of beauty have a physical basis as described by number theory! metaffordance.com/papers/treffner-JEP-1993.pdf
great comment! And here is the key part of that cite: "The frequency ratio at the outset of a trial often changed during the trial. Consistent with the general theory, shifts were toward unimodular ratios of the Farey tree, and Fibonacci ratios tended to shift more than non-Fibonacci ratios. "
@@perceivingacting , thx. I like your "Occlusion information - Gibson". very nice presentation! that 'Old vs. New Car Crash - Toyota 1998 vs. 2015' pretty nice to see as well. thx!
Prof. Wildberger I can say that in 2012 I discovered the Stern Brocot Tree and I was totally enlightened by your videos. My magic numbers were a continuous series: 1,2,1,2,2,3,2,1,2,3,1,2,3,2,2,1... which I introduced in The On-Line Encyclopedia of Integer Sequences search. For me it was a miracle because it proved to be a resolution of a problem which I had struggled from first gymnasium years about repeated mixtures of prime-color-spectrum proportions on color-circle. So, at a certain point I simplified the model of adding proportions only by having 0 (zero) and 1 as extreme limits of a line of prime-colors mixtures (let say red and yellow, where red has zero parts of yellow and yellow has 1 part of yellow). And the great surprise was that I discovered a fractal by bisecting the segment only to the zero part, because it consists only on adding zero to one (without making fractions). My purpose at that moment was to obtain color-circles of mixtures where you don't have a number of additions (or bisections) multiple of 2 or/and 6, but this proved to be unrealistic because Stern Brocot Tree, I think, is not a continuous series, so doesn't offers finite intermediary values between his numbers, other than by bisection. But still, for me is a beautiful construction because it unifies Fibonacci numbers with Golden Ratio. Actually I have now a new obsession: if we would build a structure made by bisection but without adding parts or dividing fractions, but inducing Golden Ratio sectioning (instead having number-gape-number-gape on a level, we would have M-m-M-m-M), could we find an organic system of packing, like in natural structures like buds, and see the connection between cycloidal movement of water and Golden Ratio? OK, I admit is maybe too much... I wish you to keep be inspired by Stern Brocot flame!
Coupled oscillators will phase lock into Farey fractions and if perturbed, will make MOD1 transitions to the neighboring MOD1 ratio in Farey space. MOD1 is when you have n/m and p/q and n x q - m x p = 1. The upper levels are, empirically, more STABLE than lower levels, so if perturbed a 1:2 coupled oscillator state might transition to 1:1. And 3:5 might transition to either 1:2 or 2:3 (both are MOD1 neighbors) but more likely will go to the more stable 1:2 state. These states are also known as resonances (aka "Arnold tongues", after the mathematician Arnold). Physics is full of experiments on inanimate rhythmic systems of coupled oscillators (e.g., circuits) that show state transitions that follow the Farey tree. It has also been looked at in biological systems, e.g., spontaneously beating chick heart embryo cells that phase lock to applied oscillatory currents ..and humans moving their limbs. Interestingly, the most consonant musical intervals (chords) are also the most stable according to this framework: 1:2 (octave) is more consonant than 2:3 (fifth)...than 3:4 (fourth)...so human aesthetics and ideas of beauty have a physical basis as described by number theory!
metaffordance.com/papers/treffner-JEP-1993.pdf
great comment! And here is the key part of that cite:
"The frequency ratio at the outset of a trial
often changed during the trial. Consistent with the general theory, shifts were toward unimodular
ratios of the Farey tree, and Fibonacci ratios tended to shift more than non-Fibonacci ratios.
"
@@ariisaac5111 You're welcome. See my website for more publications.
@@perceivingacting , thx. I like your "Occlusion information - Gibson". very nice presentation! that 'Old vs. New Car Crash - Toyota 1998 vs. 2015' pretty nice to see as well.
thx!
Old but gold