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Comment by andrewflnr | Hacker News Reader

> Universal Symbolic Mirrors of Natural Laws Within Us; Friendly Reminders of Inclusion to Forgive the Dreamer of Separation

Are we really upvoting this on HN? Truly the end times have come.

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https://ar5iv.labs.arxiv.org/html/1712.01826

> In this work, I propose a rigorous approach of this kind on the basis of algorithmic information theory. It is based on a single postulate: that universal induction determines the chances of what any observer sees next. That is, instead of a world or physical laws, it is the local state of the observer alone that determines those probabilities. Surprisingly, despite its solipsistic foundation, I show that the resulting theory recovers many features of our established physical worldview: it predicts that it appears to observers as if there was an external world that evolves according to simple, computable, probabilistic laws. In contrast to the standard view, objective reality is not assumed on this approach but rather provably emerges as an asymptotic statistical phenomenon. The resulting theory dissolves puzzles like cosmology’s Boltzmann brain problem, makes concrete predictions for thought experiments like the computer simulation of agents, and suggests novel phenomena such as “probabilistic zombies” governed by observer-dependent probabilistic chances. It also suggests that some basic phenomena of quantum theory (Bell inequality violation and no-signalling) might be understood as consequences of this framework.

You're welcome

> Natural Laws Within Us

We did some statistical analysis on the golden ratio and its use in art. It does indeed seem that artists gravitate away from regular geometry such as squares, thirds etc and towards recursive geometry such as the golden ratio and the root 2 rectangle. Most of our research was on old master paintings, so it might be argued that this was learned behavior, however one of our experiments seems to show that this preference is also present in those without any knowledge of such prescribed geometries.

Is that really true ?

Golden ratio is very specific, whereas any proportional that is vaguely close to 1.5 (equivalently, 2:1) gets called out as an example of golden ratio.

The same tendency exists among wannabe-mathematician art critics who see a spiral and label it a logarithmic spiral or a Fibonacci spiral.

Certainly some art critics and artists over-apply and over-think so-called 'golden' geometry. What I think is happening is very simple... that artists avoid regularity (e.g. two lights of the same color and intensity, exact center placement, exact placement at thirds, corner placement, two regions at the same angle, two hue spreads of equal sides on opposite sides of the RYB hue wheel etc etc). These loose 'rules' of avoidance can be confused with 'rules' of prescription such as color harmony, golden section etc.

What does that have to do with the barely-coherent woo of "Friendly Reminders of Inclusion to Forgive the Dreamer of Separation"?

No, we're upvoting the solid and novel (to many of us) mathematical derivation. I don't really mind what woo-woo statements sacred geometry enthusiasts make as long as the math checks out.

The chord through the midpoints of two sides of an inscribed equilateral triangle cuts a diameter in the golden ratio. This interesting method gives a purely geometric construction of positive Phi without using Fibonacci numbers.

> This interesting method gives a purely geometric construction of positive Phi without using Fibonacci numbers.

There's nothing particularly interesting about that; phi is (1 + √5)/2. All numbers composed of integers, addition, subtraction, multiplication, division, and square roots can be constructed by compass and straightedge.

I was somewhat surprised to learn that phi is _merely_ (1 + √5)/2, I didn't have a good conception of what it was at all but I didn't think it was algebraic.

Phi is conceptually defined like so:

    Suppose you have a rectangle whose side length ratio is ϕ. You draw a line across the rectangle which divides it into a square and another rectangle.

    Then the side length ratio of the new, smaller rectangle is also ϕ.

The diagram is straightforward to set up:

       a        b
    +-----+--------+
    |     |        |
    |  ϕa-|        |
    |     |        |-b
    |     |        |
    +-----+--------+
     \            /
      -----  -----
           \/
           ϕb

This gives us a system of two equations:

    ϕa = b
    ϕb = a + b

If you substitute b = φa into the other one, you get

    ϕ(ϕa) = a + ϕa

And since a is just an arbitrary scaling factor, we have no problem dividing it out:

    ϕ² = 1 + ϕ

Since we defined φ by reference to the length of a line, we know that it is the positive solution to this equation and not the negative solution.

(Side note: there are two styles of lowercase phi, fancy φ and plain ϕ. They have their own Unicode points.

HN's text input panel displays ϕ as fancy and φ as plain. This is reversed in ordinary text display (a published comment, as opposed to a comment you are currently composing). And it's reversed again in the monospace formatting. (Which matches the input display.)

The ordinary text display appears to be incorrect, going by the third usage note at https://en.wiktionary.org/wiki/%CF%95 )

HN's text input panel displays ϕ as fancy and φ as plain. This is reversed in ordinary text display (a published comment, as opposed to a comment you are currently composing). And it's reversed again in the monospace formatting. (Which matches the input display.)

I'm glad you posted this. I'm not a unicode expert and have always assumed these weird dichotomies were some sort of user/configuration error on my part. Realizing the unicode glitches are actaully at the website end instead of between my ears is quite a relief.

To be more specific, that usage note strongly suggests that the problem is in the font used by HN. The font is what complies or doesn't comply with the Unicode standard. We can also say that HN has a problem, but HN's problem is "they're using a noncompliant font for monospaced text".

(On further investigation, I got the characters backwards, and HN's ordinary display is correct while the monospaced display isn't.)

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