Bonsai, first to address your Telescope question, I would hold fire.
Study the mattter before definite decisions.
Are you really trying to compete with the 4 x 8.2m Very Large Telescope in the high Atacama Desert of Chile?
THAT'S my idea of a proper Telescope!
Mine own results have been more about effort against adversity than anything else:
4 planets in one shot is not bad!
Naturally I am more interested in the, IMO, absolutely correct Standard Model of Physics:
Built upon the Rock of Amelie Noether's ideas about Symmetry.
With every Symmetry comes a Conservation Law.
However, even I admit to difficulties with the Neutral Kaon.
https://www.quantamagazine.org/a-new-kind-of-symmetry-shakes-up-physics-20230418/
My current efforts in reconciling the Euler Zeta (2) function with Pure Geometry:
https://www.quantamagazine.org/two-students-unravel-a-widely-believed-math-conjecture-20230810/
I can only hope this is not one of those Mathematical Dead Ends:
One can only hope for that Kavli Prize in 2024!
https://www.kavliprize.org/prizes/astrophysics/2010
I am, after all, too old for a Fields Medal.
Study the mattter before definite decisions.
Are you really trying to compete with the 4 x 8.2m Very Large Telescope in the high Atacama Desert of Chile?
THAT'S my idea of a proper Telescope!
Mine own results have been more about effort against adversity than anything else:
4 planets in one shot is not bad!
Naturally I am more interested in the, IMO, absolutely correct Standard Model of Physics:
Built upon the Rock of Amelie Noether's ideas about Symmetry.
With every Symmetry comes a Conservation Law.
However, even I admit to difficulties with the Neutral Kaon.
https://www.quantamagazine.org/a-new-kind-of-symmetry-shakes-up-physics-20230418/
My current efforts in reconciling the Euler Zeta (2) function with Pure Geometry:
https://www.quantamagazine.org/two-students-unravel-a-widely-believed-math-conjecture-20230810/
I can only hope this is not one of those Mathematical Dead Ends:
One can only hope for that Kavli Prize in 2024!
https://www.kavliprize.org/prizes/astrophysics/2010
I am, after all, too old for a Fields Medal.
I didn't understand a "bit" of it. 😀
Only a mathematician could conjure up an infinite number of infinities!
Attachments
Very interesting latest blog post from Ethan Seagal. Looks like we may have overestimated the number of stars in the universe.
https://bigthink.com/starts-with-a-bang/overestimated-stars-in-universe/
https://bigthink.com/starts-with-a-bang/overestimated-stars-in-universe/
If I am interpreting the article correctly, we should reduce our estimate of about 2.1 × 10^21 stars in the observable universe to 8.0 × 10^19.
However, I think Isaac Asimov described the situation more poetically:
Yup. We have to remember we are LOOKING BACK IN TIME so what we see now is actually not what the universe is now.
Hilbert's Infinite Hotel my young protege!
https://en.wikipedia.org/wiki/David_Hilbert
"Nobody shall expel us from the (Infinite) Paradise that Cantor created!"
Generally considered an extremely competent Mathematician from the Gottingen school.
Actually solved General Relativity before our own dear Albert Einstein. But only did it after attending a lecture by Einstein, who was struggling with the Mathematics of the Metric Tensor, therefore never claimed credit for it.
As he said, any young Mathematician at Gottingen would have seen it immediately!
In fact my own personal Mathematical Hero from Gottingen is Felix Klein:
https://en.wikipedia.org/wiki/Felix_Klein
I read one of his books in Portsmouth University library, and I was stunned by his multidimensional ability.
It was one of those "Why didn't I see this before" moments.
For us mere loudspeaker and signal and Information theory ground-dwellers there is little beyond the Hilbert Transform and Claude Shannon's work:
https://en.wikipedia.org/wiki/Hilbert_transform
See, the Truth of Life is staring you in the face.
All is symmetry. But occasionally broken and perhaps irreversible. 😀
Estimating the number of stars in the observable universe is made more difficult by the fact that the universe is expanding.
Light from galaxies that are ever further away continues to reach Earth for the first time, meaning that we are able to see more and more of the universe as time passes.
According to Encyclopedia Britannica: Given the constant expansion of the universe, the observable universe expands another light-year every Earth year.
I was referring to Cantor's tower of infinities. https://plus.maths.org/content/cantor-and-cohen-infinite-investigators-part-ii
Mathematicians have known for well over a century now that infinity isn’t just one thing, it is infinitely many. There is an unending tower of ever greater infinities stretching up all the way to… well, whatever you’d like to call it.
Bonsai's link made reference to German mathematician Georg Cantor who used diagonalization to prove that some infinities are larger than others.
He discovered that the “real” numbers (most with never-ending digits, like 3.14159…) outnumber “natural” numbers like 1, 2 and 3, even though there are infinitely many of both.
As an Information Theorist, I happen to know a thing or two about the subject:
How come we got these incredible pictures?
Absolutely the 24D Leech Lattice.
https://en.wikipedia.org/wiki/Leech_lattice
No other way to do it. 🙂
How come we got these incredible pictures?
Absolutely the 24D Leech Lattice.
https://en.wikipedia.org/wiki/Leech_lattice
No other way to do it. 🙂
All fine and dandy..
BUT
Can we design a gravitron heterogeneous discordant audio system that uses orthogonal gravity to create audio waves?
You know.. the Gravityneous Wall of Sound?
Can we make virtual planar sources of audio waves.... virtual Magnepans that, when excited, also serve as portals through multi dimensions to parts far and away in our Universe? Just make sure to turn the stereo off when you walk around that part of the room!
BUT
Can we design a gravitron heterogeneous discordant audio system that uses orthogonal gravity to create audio waves?
You know.. the Gravityneous Wall of Sound?
Can we make virtual planar sources of audio waves.... virtual Magnepans that, when excited, also serve as portals through multi dimensions to parts far and away in our Universe? Just make sure to turn the stereo off when you walk around that part of the room!
Bonsai's link, to which I referred, may hold the answer to your question.
https://www.quantamagazine.org/alan-turing-and-the-power-of-negative-thinking-20230905/
Georg Cantor used diagonalization to prove that some infinities are larger than others in 1873.
Alan Turing adapted Cantor’s version of diagonalization to the theory of computation in 1936.
Computer scientists have since continued to explore diagonalization and its blind spots.
Turing’s diagonalization proof - the infinite list of possible algorithms - it's all above my pay grade! 🤓
I believe the Leech lattice can be described in terms of those pesky octonions!
The octonionic Leech lattice is defined as the set of triples (x,y,z) of octonions. 😵
Frankly, my understanding of the Leech lattice is no greater than my understanding of tonyEE's Gravityneous Wall of Sound! 😀
Add Kurt Gödel in the use of diagonalization.
Do not miss how Albert Einstein helped K.G. to pass immigration in the US.
I read that Teleparallel Gravity (TG) is making a comeback.
In TG, any entity with mass or energy twists up spacetime around it, and that twisting instructs other objects how to move.
TG attributes gravitation to the twisting of spacetime, while General Relativity (GR) attributes it to the curvature of spacetime.
The two approaches, one based on twistiness and the other based on curvature, are mathematically equivalent, but because Einstein developed the curvature-based language first, it's much more widely used.
A team of theoretical physicists has recently explored how TG would give black holes 'hair'.
Since the 'hair' would carry information about the black hole inside it, we'd be able to understand more about black holes without having to dive inside them!
Full story here: https://www.space.com/twisty-theory-of-gravity-says-information-can-escape-black-holes
In TG, any entity with mass or energy twists up spacetime around it, and that twisting instructs other objects how to move.
TG attributes gravitation to the twisting of spacetime, while General Relativity (GR) attributes it to the curvature of spacetime.
The two approaches, one based on twistiness and the other based on curvature, are mathematically equivalent, but because Einstein developed the curvature-based language first, it's much more widely used.
A team of theoretical physicists has recently explored how TG would give black holes 'hair'.
Since the 'hair' would carry information about the black hole inside it, we'd be able to understand more about black holes without having to dive inside them!
Full story here: https://www.space.com/twisty-theory-of-gravity-says-information-can-escape-black-holes
If you ran a prime sieve for the age of the universe you still wouldn't have found all of them.
https://www.geeksforgeeks.org/sieve-of-eratosthenes/
Pure maths view of numerical infinity. 1 can always be added to what ever number you happen to choose. This leads to some rules in handling them - eg 1/n large tends to 0 etc. That one is ok. There are others.
LOL Interesting - take any prime and some number can always be added to it that will produce a none prime eg any smaller prime. 😉 The more you find the worse it gets.
https://www.geeksforgeeks.org/sieve-of-eratosthenes/
Pure maths view of numerical infinity. 1 can always be added to what ever number you happen to choose. This leads to some rules in handling them - eg 1/n large tends to 0 etc. That one is ok. There are others.
LOL Interesting - take any prime and some number can always be added to it that will produce a none prime eg any smaller prime. 😉 The more you find the worse it gets.