1. Saw this video about some surfer dude in Hawaii (another video, same lines) who is making the claim that reality is a simulation. Talks about some eight dimensional geometric crystal called E8 that converts to a 4D matrix/universe then a 3D universe/matrix for us observers. Now there was a ratio between 4D & 3D size wise and they called it the golden ratio (.618). Apparently this ratio is found elsewhere in mathematics/geometry or at least keeps popping up in calculations for various geometric shapes I think. Not sure and always skeptical about such things so I wonder if someone here could explain exactly what is meant by Golden Ratio, what it is exactly and if it really is something that should be taken seriously?  2.

3. Originally Posted by zinjanthropos Saw this video about some surfer dude in Hawaii (another video, same lines) who is making the claim that reality is a simulation. Talks about some eight dimensional geometric crystal called E8 that converts to a 4D matrix/universe then a 3D universe/matrix for us observers. Now there was a ratio between 4D & 3D size wise and they called it the golden ratio (.618). Apparently this ratio is found elsewhere in mathematics/geometry or at least keeps popping up in calculations for various geometric shapes I think. Not sure and always skeptical about such things so I wonder if someone here could explain exactly what is meant by Golden Ratio, what it is exactly and if it really is something that should be taken seriously?
A ratio such that a+b/a = a/b

In terms of a line divided into 2 lengths a and b, the total length divided by the longer section equals the longer section divided by the shorter section. The ratio expressed in this way is 1.618...... but its reciprocal is 0.618......

This ratio is said to have a natural or organic feel to it and I understand it has often been used in architecture, e.g. for the proportions of the sides of a rectangle, as it seems aesthetically pleasing.

So yes it most definitely is a thing.  4. Originally Posted by zinjanthropos Saw this video about some surfer dude in Hawaii (another video, same lines) who is making the claim that reality is a simulation. Talks about some eight dimensional geometric crystal called E8 that converts to a 4D matrix/universe then a 3D universe/matrix for us observers. Now there was a ratio between 4D & 3D size wise and they called it the golden ratio (.618). Apparently this ratio is found elsewhere in mathematics/geometry or at least keeps popping up in calculations for various geometric shapes I think. Not sure and always skeptical about such things so I wonder if someone here could explain exactly what is meant by Golden Ratio, what it is exactly and if it really is something that should be taken seriously?
The golden ratio, often denoted by phi, is less famous than pi or e, but it deserves its place among the pantheon of cool numbers. Here are a phew phun phacts about phi:

1) The ratio of consecutive Fibonacci numbers approaches phi.
2) Draw a rectangle whose sides are in a ratio phi. Call the long and short sides L and W, respectively. Cut off a W x W square; what remains is a smaller rectangle whose sides are in a ratio phi.
3) Stradivarius used phi to set the dimensions of his instruments.
4) Debussy was somewhat obsessed with Fibonacci numbers. Some musicologists have argued that some of his compositions were informed by phi.
5) Da Vinci's art was influenced by the golden ratio ("the divine proportion")
6) Fibonacci numbers were discovered during a study of population growth (of rabbits, orginally). If one takes an ordered list of populations of cities, for example, the populations will frequently be in a ratio surprisingly close to phi.

As with pi and e, numerology often creeps into the literature on phi (including, arguably, some of the above). But phi is deserving of a wider appreciation.  5. An interesting property of the Golden Ratio is that, relative to the size of the denominator, it has the worst rational approximation of any number.  6. Originally Posted by KJW An interesting property of the Golden Ratio is that, relative to the size of the denominator, it has the worst rational approximation of any number.
Thanks very much, KJW. I never knew that.

Best wishes for the new year!  7. Originally Posted by tk421 Best wishes for the new year!
Thank you, and best wishes for the new decade to you too. Originally Posted by tk421  Originally Posted by KJW An interesting property of the Golden Ratio is that, relative to the size of the denominator, it has the worst rational approximation of any number.
Thanks very much, KJW. I never knew that.
Strictly speaking, the more correct statement is provided in this Wikipedia article.

You may be interested in continued fractions in general.

At the opposite end of the spectrum, the numbers with the best rational approximations (relative to the size of the denominator) are the Liouville numbers, which are all transcendental.  8. Originally Posted by zinjanthropos what is meant by Golden Ratio, what it is exactly and if it really is something that should be taken seriously?

Well, it is real and should be taken seriously, but it is not a ratio between 4D & 3D size in an "eight dimensional geometric crystal called E8 that converts to a 4D matrix/universe then a 3D universe/matrix for us observers."  9. Originally Posted by Ken Fabos  Originally Posted by zinjanthropos what is meant by Golden Ratio, what it is exactly and if it really is something that should be taken seriously?

Well, it is real and should be taken seriously, but it is not a ratio between 4D & 3D size in an "eight dimensional geometric crystal called E8 that converts to a 4D matrix/universe then a 3D universe/matrix for us observers."
Ken... good to hear from you. I’m not the best at explaining things I see/hear etc, so all that stuff is in the video. I was more interested in the golden ratio than matrices   10. Originally Posted by KJW An interesting property of the Golden Ratio is that, relative to the size of the denominator, it has the worst rational approximation of any number.
What does this mean precisely?  11. Originally Posted by anticorncob28  Originally Posted by KJW An interesting property of the Golden Ratio is that, relative to the size of the denominator, it has the worst rational approximation of any number.
What does this mean precisely?
This is best answered by the "A property of the golden ratio φ" section of the Wikipedia article "Continued fraction".

The continued fraction representation of the Golden Ratio ϕ = [1;1,1,1,1,1,1,1,1,1,1,1,...]. Truncation of the representation at increasing lengths provides a sequence of increasingly better rational approximation of the number. But because the numerical values in the representation of ϕ never exceeds 1, the rational approximations in the sequence will be consistently worse than the corresponding rational approximations for numbers whose continued fraction representation contains numerical values that are greater than 1.  Bookmarks
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