Going beyond the Golden Ratio
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Going beyond the Golden Ratio
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Re: Going beyond the Golden Ratio
#2Author here. Happy to try to answer any questions any one might have on this post or topic. )
Re: Going beyond the Golden Ratio
#3850/10 equals 85, not 8.5. 425/5 is equal to 85, not 8.5.
Re: Going beyond the Golden Ratio
#4850/10 equals 85, not 8.5. 425/5 is equal to 85, not 8.5.
LOL! You're totally right. It should be 85/10 and 425/50. Now fixed.
Re: Going beyond the Golden Ratio
#5I recently saw this [0] Numberphile video that touches some of the similar stuff at the end of this article, with the spirals being animated.
Re: Going beyond the Golden Ratio
#6Author here. Happy to try to answer any questions any one might have on this post or topic. )
Fascinating post. I had always assumed that the 3rd most irrational number would be the third metallic mean given by n = (n+ sqrt(n^2+4))/2, and subsequently the fourth metallic mean etc. The metallic means also pack the disks nicely. I have recently had my interest in them sparked after I came across solution to point vortex equilibria involving them.
Do you know what the metallic means are bounded by? Are they as bad as the silver ratio/(1+sqrt(2))?
These most irrational numbers, (9+sqrt(221))/10, (13+sqrt(1517))/26... how interesting that they are not just the simple generalization of the continued fraction for the golden ratio.
Re: Going beyond the Golden Ratio
#7Author here. Happy to try to answer any questions any one might have on this post or topic. )
Fascinating post. I had always assumed that the 3rd most irrational number would be the third metallic mean given by n = (n+ sqrt(n^2+4))/2, and subsequently the fourth metallic mean etc. The metallic means also pack the disks nicely. I have recently had my interest in them sparked after I came across solution to point vortex equilibria involving them. Do you know what the metallic means are bounded by? Are they as b…
What I find fascinating is that there seem to be so many valid ways to generalize the Golden Ratio.
As you say, the "metallic means" [1] are quite well-known, and relate to the recurrence relation via: T(n) = m *T(n-1)+ T(n-2), for some constant integer m. For example, m=1 is the golden ratio, m=2 is the silver ratio,...
But one of my other posts [2], generalizes the Golden ratio via the "Harmonious Numbers", as defined by the lagged recurrence, T(n+m) = T(n)+T(n-1), for some constant m. In this case, m=1 relates to the Golden Ratio, and m=2 relates to the Plastic Number [3].
And then finally, this post explores generalizing it via a completely different perspective, that of "Lagrange Numbers".
It seems that we need to 'think outside the box' a litte when generalizing the Golden ratio, as there is not single obvious way to generalise continued fractions.
[1] https://en.wikipedia.org/wiki/Metallic_mean
[2] http://extremelearning.com.au/unreasonable-effectiveness-of-...
Re: Going beyond the Golden Ratio
#8I recently saw this [0] Numberphile video that touches some of the similar stuff at the end of this article, with the spirals being animated. [0] https://www.youtube.com/watch?v=sj8Sg8qnjOg
Absolutely! Everybody loves the numberphile videos. They frequently distil deep maths topics into very intuitive and visual explanations. ;)
Re: Going beyond the Golden Ratio
#9Author here. Happy to try to answer any questions any one might have on this post or topic. )
open google earth use ruler for below
miles from Angkor Wat to Giza pyramid 4754 miles. This multiplied by the glden ratio of 1.618 give 7692 miles which is the distance from Giza to Nazca . Now 7692 miles multiplied by the golden ratio again gives 12446, which is the distance from Nazca to Angkor Wat
why?
Re: Going beyond the Golden Ratio
#10Author here. Happy to try to answer any questions any one might have on this post or topic. )
Why not penalize the error by the cube or some other higher power of the denominator?