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#mathematics

25 posts17 participants0 posts today
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Dani Laura (they/she/he)

I have found an interesting geometric fact: suppose you have a hexagon of side 1 and duplicate and enlarge it by the golden ratio 𝜑; the distance from one vertex of the unit hexagon to a vertex of the bigger hexagon 60° apart is √2. Furthermore, if another hexagon reduced by 𝜑 is drawn inside, the distance from one vertex of the unit hexagon to a vertex of the smaller hexagon 120° apart is also √2 [first figure].
This boils down to the fact that a triangle of sides 1, √2, and 𝜑 has an angle of 60° opposite to side √2. That triangle is very remarkable as it contains the three more relevant algebraic geometric constants: √2, √3/2 (altitude to the bigger side) and 𝜑 [second figure]. Of course this can be also used to construct 𝜑 from a square and a triangle (I bet this is known). In the follow-up some artistic designs exploiting those facts.
#geometry #Mathematics #triangle #GoldenRatio

Three concentric hexagons, see post for details.
Triangle of sides 1, √2, and 𝜑, and altitude √3/2.
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Alexander Knochel

How does this translate to recent claims of a "three sigma" result indicating biomolecules on exoplanets as possibly observed by JWST?

I don't know how the researchers define their significance, and it is probably a complicated procedure analyzing the spectrum, but still -

Let's call the observed result again "R", and the null hypothesis that there are NO biomolecules on the planet "F".

A three sigma result would usually mean

P(R|F)=0.003, or in other words, if there are no biomolecules, the observed result only has a probability of 0.003.

As above, to obtain the *probability that there are biomolecules on the exoplanet* is the backwards claim of P(F|R), and to obtain this number, we again have to do

0.003*P(F)/P(R), meaning that we have to mulitply by the a priori probability that exoplanets are devoid of this biomolecule, and divide by the general probability of observing such spectra. On this admittedly naive level of analysis, it becomes clear that in order to make a statement about the probability of biomolecules on the exoplanet after the recent JWST result, one needs to factor in knowledge or a priori assumptions about the prevalence of such molecules and confounding factors.

End of thread

#Astronomy
#mathematics
#science

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Chris Purcell

Here's a stupid #geometry / #astronomy question. Planetary orbits are ellipses (under the obvious simplifying assumptions) which are conic sections. So what's the cone? There are infinitely many cones that fit of course. But is there one that best explains? To put it another way, is there a neat geometric argument that an orbit should be an ellipse, that doesn't require too much physics? #mathematics

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Alexander Knochel

-> Thread on what sigmas in science mean and what they don't mean, and how to correctly talk about and calculate the probability of claims using bayes law:

Apropos of recent events, here's the periodic reminder what "sigmas" mean for the probability of experimental results, because this is almost always presented wrong by popular texts and journalists
#Science
#mathematics

Thread 1/

BBC headline "Telescope finds promising hints of life on distant planet", artists impression of a blue backlit planet. In the article, the author talks about "three sigma" evidence for bio-associated molecules on the exoplanet
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