I am, as ever, your humble servant:
Antoine Song and Conghan Dong proved that spaces with small amounts of mass must be nearly flat.
They used a concept known as 'scalar curvature' and simply chopped off the spiky bits!
https://www.wikiwand.com/en/Scalar_curvature
Hope that helps!
Antoine Song and Conghan Dong proved that spaces with small amounts of mass must be nearly flat.
They used a concept known as 'scalar curvature' and simply chopped off the spiky bits!
https://www.wikiwand.com/en/Scalar_curvature
Hope that helps!
Lot to get through, as usual...
@Bonsai, I have been doing due diligence on this "bargain" Telescope.
Alas, it is a Sheep in Wolf's clothing. "Caveat Emptor" as a Celestron Lawyer might argue. Sorry to SHOUT.
This Cooper chap seems well-informed about matters telescopic:
https://supercooper.jimdofree.com/choosing-telescopes-complete-essentials/
Some serious discussion for buyers here:
https://www.space.com/31229-best-beginner-telescopes.html
Looks like £200 is the proper entry point.
@Galu and @TNT:
https://www.wikiwand.com/en/Scalar_curvature
Why do those two names give me the giggles? I didn't understand much of it except the mention of Gaussian Curvature, which is an interesting topic. Looked a dubious technique to me. Cutting out the problematic parts. Keeping the bit that works...
@Bonsai, I have been doing due diligence on this "bargain" Telescope.
Alas, it is a Sheep in Wolf's clothing. "Caveat Emptor" as a Celestron Lawyer might argue. Sorry to SHOUT.
This Cooper chap seems well-informed about matters telescopic:
https://supercooper.jimdofree.com/choosing-telescopes-complete-essentials/
Some serious discussion for buyers here:
https://www.space.com/31229-best-beginner-telescopes.html
Looks like £200 is the proper entry point.
@Galu and @TNT:
https://www.wikiwand.com/en/Scalar_curvature
Why do those two names give me the giggles? I didn't understand much of it except the mention of Gaussian Curvature, which is an interesting topic. Looked a dubious technique to me. Cutting out the problematic parts. Keeping the bit that works...
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It gets a mention in a rather well known site - a fairly well grounded one.
https://stargazerslounge.com/topic/399172-celestron-ps1000-equatorial-reflector-telescope/
A poster points some one at Skywatcher. That's not bad advice. There 8" newtonians are pretty popular, Size wise I'd suggest 6" or larger. I'd also suggest F6 even if F4 is available. These will be on a German style mount if equatorial. Bit of a problem. The telescope is balanced with a counterweight. That makes the kit heavier and people may be surprised by how much the image bounces about when the scope is touched even just to focus it. This sort of thing pushes people to the heavier more expensive mounts.
Meade and Celestron. I feel these come into being a decent option with fork mounted Schmidt Cassegrains. The telescope balances itself meaning it's pretty easy to produce fairly stable mount at a lower weight than a German type. Up to 8" weight wise things are fairly manageable but some might want to check total weight before buying. There are other compound arrangements about as well even Newtonian variants. Maybe 6" for either. A Newtonian can give a perfect on axis image. Variants have a bit of a problem in this area. An old saying is that a 4" refractor can match a 6" newtonian. The main reason for this is the central obstruction given the same optical quality levels. There are no free lunches in this area as far as telescopes are concerned.
Dobsonians. The whole idea of these was to allow people to have much larger diameter scopes without the associated mount problems. I didn't find pointing one much fun.
Refractors. None APO's forget it. Tried and the optical quality was awful so colour fringing was irrelevant. A lot more money might fix that but it's doubtful that a maker would bother doing the work needed. APO's. Something like a moderate 4" may be a decent option price wise especially if F6 is available. Even F8. F4 would need more optical correction for decent results and costs for that can rocket. The old 4" matching 6" comments were based on an F15 none apo refractors. F8 newtonian.
GoTo. Make sure there is an option to correct alignment errors. You set up as instructed. You track to something and it isn't on centre. There needs to be an option to recentre and press a button to correct the alignment error. If interested in photography that may not be good enough. A slower F ratio scope needs longer exposures. Noise builds up while an image is being captured. The whole area is a money pit but within reason it is possible to cope with lower cost cameras and slower scope. It just means more images are needed. DSLR's can be an option. There is usually stuff around to reduce the focal length of all types of scope for photography. I've stuck away from astro photography but these are solid comments. I did intend to go in this direction but local street light changes make things even more unfeasable. Prior to that I aimed at having a permanent aligned mount in the garden attaching the scope when in use. An alternative is full computer control and a display in the house but I like looking. For dark skies all I did need was a sodium light rejection filter, That would also apply to photography. Things are way more complicated now and not much use for viewing.
Finders if wanted are interesting. Many may decide a right angle one is needed. Scopes often don't come with these.
Buy used. Some would say no as there will be something wrong with it. I've sold several that were fine. They may need realignment. A scope can be messed up even more if that is done incorrectly. Probably more of a problem on Newtonians as the first step is to make sure the focus tube is well pointed at the mirror also at the 2ndry mirror. There have been designs where the 2ndry mirror wont be truly aligned. Best left as is and try an align the main mirror entirely into view. A laser item can be bought to to help but if this initial centring is wrong things will go wrong. The mirror may need cleaning. An SC has 3 screws to align the 2ndry mirror. This may need doing even on a brand new scope after some period of use. If someone is prepared to demonstrate their used scope in use I'd say that is a very safe option. They may well be just upgrading. It's all save money with some risk really.
Also in winter a dew heater of some sort is likely to be essential. Even the main mirror on a newtonian cam mist up. It's one of the ways these get dirty.
LOL All based on actual experience. Rather a lot of it.
Song & Dong actually use scalar curvature which, in two dimensions, is exactly twice the Gaussian curvature for an imbedded structure in Euclidean space.
If you don't believe me, ask Wikipedia! 😀
The Universe has all sorts of deformations in space-time where it varies from being perfectly flat, but Song & Dong seem to be saying it is as good as flat on the smallest of scales.
I have no idea of the significance of that statement - perhaps Song & Dong would be better off forming a pop duo!
Steve may explain how a 3D construction can be flat, but should be careful to distinguish between geometry and topology.
While the geometry of the universe is flat, the topology is not!
Do you mean to tell me, TNT, that after following all my years of googling you have learned nothing about the geometry of space?
It reminds me of what my maths 'teacher' said the day I laughed out loud at a classmate's joke during one of her lengthy monologues.
Suffice it to say that it involved pearls and a certain breed of farmyard animals! 😍
It reminds me of what my maths 'teacher' said the day I laughed out loud at a classmate's joke during one of her lengthy monologues.
Suffice it to say that it involved pearls and a certain breed of farmyard animals! 😍
My most embarrassing math thing was in my 2nd yr at high school. I was off with the fairies 10 000 miles away when the teacher noticed and, pointing to me, said 'how do you treat x in these equations?' He had written about 5 or 6 up on the board. I duly told him how to treat x in each case. Complete silence from everybody. He looked me in the eye and said 'Well done. You got every single one wrong!'
^
'Well done. You got every single one wrong!'
OTOH, I was in High School in Oahu, Hawai'i. In the 70s.
We inhaled then.
Geometry was after lunch, sometimes, if the teacher was bored and we had to go through "last night's" homework he'd pass me the chalk and in a single stream of consciousness, while David Bowie was playing in my head, I'd do the entire thing... starting from the left blackboard and ending at the right most blackboard.
The rest of the class enjoyed it because they too had David Bowie in their head ( hint! ) and the teacher didn't call them to do anything.
Needless to say I broke the curve and I got a major in Math in college... ;-)
BTW, I learned how to double clutch an Alfa Romeo during a year of Electro Magnetism.... while sitting on a chair, looking out the dreary PNW mornings, I learned how to late brake, hit the apex and exit on full power.... this was done like 8 years before I bought an Alfa... amazing stories!
Gravity... well, I'm sure you will appreciate the gravity of the situation where in an expensive private college course I learned both EM Physics and The Art Of Driving An Alfa Romeo On Mountain Roads.
'Well done. You got every single one wrong!'
OTOH, I was in High School in Oahu, Hawai'i. In the 70s.
We inhaled then.
Geometry was after lunch, sometimes, if the teacher was bored and we had to go through "last night's" homework he'd pass me the chalk and in a single stream of consciousness, while David Bowie was playing in my head, I'd do the entire thing... starting from the left blackboard and ending at the right most blackboard.
The rest of the class enjoyed it because they too had David Bowie in their head ( hint! ) and the teacher didn't call them to do anything.
Needless to say I broke the curve and I got a major in Math in college... ;-)
BTW, I learned how to double clutch an Alfa Romeo during a year of Electro Magnetism.... while sitting on a chair, looking out the dreary PNW mornings, I learned how to late brake, hit the apex and exit on full power.... this was done like 8 years before I bought an Alfa... amazing stories!
Gravity... well, I'm sure you will appreciate the gravity of the situation where in an expensive private college course I learned both EM Physics and The Art Of Driving An Alfa Romeo On Mountain Roads.
I read Lord of the Rings in HS math. One teacher asked why I never looked at her? It was all in the book. Silver Reed, an excellent series of texts.
My most embarassing math semester was quantum mechanics in college. I would grind out 25 page treatments of the homework, which would earn me D- on tests. I asked the proff the solution to one Schroedinger problem; he said "it is obvious that" blah blah blah and something about euler equations. I looked at a book on euler equations in the library, it was full of equations and the numeric values of them for various values of x. What am I supposed to with that? Turns out the 11 guys making A's were attending homework sessions with a classmate after dinner in one of the dorms. I heard this years later from a fellow Rice reject, at a sales meet for electronic technicians. I couldn't afford a dorm, I was paying cash for my college & lived at home. I hated MaryJane smoke too, which the dorms reeked of. The 3 of us making D- were trying to learn from the proff. Stupid us. I paid Rice $3500 cash, they documented how stupid I was. Should have taken Engineering at UHouston in the first place and blown off the 4 year scholarship in Physics. My HS grades & SAT could have gotten me in MIT or CalTech, but I could not afford room & board nor airline tickets. My father thought his income & net worth were none of the school's business.
I learned to double clutch in the 6th grade from a school bus driver who could operate those old International 5 speeds as silently as a bunny. The other drivers, like my Algebra teacher, crunch grind BUZZ!!!
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I think you mean: Flat, because one can make it flat; Because you can make à cylinder from à flat sheet you bend.
In that meaning, a cone is flat, a sphere is not flat.
In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion
In geometry, 'flatness' refers to the behaviour of parallel lines as they go out to infinity.
Think of a flat sheet of paper: Lines that start out parallel will remain parallel as they extend along the paper.
Think of the Earth: Lines of longitude begin perfectly parallel to each other at the equator but eventually converge at the poles - the Earth is not flat.
Ignoring small-scale deflections from galaxies and black holes, rays of light left over from the Big Bang, which started out parallel, have been shown to have remained parallel - confirming that the overall geometry of the universe is flat.
https://www.livescience.com/what-is-shape-of-universe
Yet the universe is clearly 3d and light travels in a 3d fashion from source. The rays or photons what ever you want to call them do not travel in a parallel fashion. This is our normal view of dimensions that we are aware of all of the time. The flatness of the universe is something else. Some theories may mention extra dimensions that are curled up so small that we are unaware of them. The flatness of the universe is a similar type of dimension. We are physically unaware of it so infer it's flatness from what we are aware of.
Who knows. It's now all about empirically derived factors of dark energy and matter to make thing fit current thoughts. Interests in various ideas still remain. Then comes mainstream. 😉 One critisism of others Einstein made was that they were over educated - you might say stuck in a box.
In n dimensional geometry terms there is another dimension we are perfectly aware of. Often called the arrow of time. All lovely and straightforward until Einstein cropped up. While bending space time is a nice cool term bear in mind we are still really talking n dimensional geometry. His idea predicts things that have found to be correct. Some one does some sums and up pop black holes. Also singularities - crazy things really. There is nothing new about super dense matter, Seems there is no limit.