What Is the Geometry of the Universe? | Hacker News Reader

What Is the Geometry of the Universe?

quantamagazine.org

289 pointsbryanrasmussen69 comments

Quanta is such a top notch publication. I am consistently impressed with the quality of their writing.

Great article on geometry, particularly a nice explanation of the flat torus, and of whether things can or cannot be measured locally.

Quanta is really good, I agree. It seems to me like a reincarnation of Popular Science at its peak. If you like similar work, check out Nautilus. For more philosophical tangents, Aeon is great as well.

+1. Seems Firefox (or Pocket) is always suggesting articles from those 3 places to me in the new tab, and I'm happy for that.

I had the exact same thought while reading the article. Not the first I've read by far, but perhaps one of the few publications that consistently puts out short-form masterpieces.

Agreed - it's unusually well written. The author is a topologist as well as a science writer.

https://en.wikipedia.org/wiki/Erica_Klarreich

Quanta is a project launched and funded by the Simons Foundation, a nonprofit that does absolutely incredible work to support ongoing international scientific collaboration in a wide array of disciplines. Quanta is independently edited and not a direct organ of the Foundation (at least, afaik), but I've always felt like Quanta's stellar quality is a direct reflection of how seriously they take their mission.

You know how Saturn's North Pole is a hexagon? It wouldn't surprise me if someday we find some shape to the universe that isn't smoothly circular because of some fundamental force that we don't understand yet. I'd vote for a hexagon, since if you stack a bunch of circles, that's the shape you get (the other circles being other universes). Plus bees are probably interdimensional beings.

Aren't those Saturnian hexagons blue? Hyperintelligent shades of the colour blue, I reckon.

It's been shown that spherical cow, not bees, is homeomorphic to the Universe [1]

[1] http://en.wikipedia.org/wiki/Spherical_cow#/media/File:Spot_...

A cow is homeomorphic to a torus due to the digestive system.

People too. I'm a donut.

You're probably more a triple torus.

Way more than triple, considering things like nephrons in kidneys

We can limit our precision here for more fun :)

Seven holed donut due to extra holes of nostrils and tearducts.

https://m.youtube.com/watch?v=egEraZP9yXQ

How many of those go all the way through your body?

Not all of them but they don't need to; they just need to be topgraphically holey, which is why a teacup and a donut share the same number of holes for example.

Right, and the reason a teacup and a donut have the same number of holes is because the former has a handle. Holes that don’t go all the way through don’t count.

There is plenty of holes in the human body that go all the way through (for example the ducts in your eyes going towards your nasal cavity as well as various parts of said nasal cavity).

Fair enough, I guess we’re holey toruses ;)

Your vote is correct. You're not the only one voting hexagonal: https://maskofreason.files.wordpress.com/2011/02/the-library...

Every now and then I read some far-out scientific paper, and I'm pretty sure it's real, but I get fascinated with the question of how I really know that it's not pseudo-science by some crank.

Example:

"In this section, we investigate the band structures of plasmonic ribbons with four types of boundary conditions, i.e. zigzag, bearded zigzag, armchair, and bearded armchair"

(from "The existence of topological edge states in honeycomb plasmonic lattices")

A real crank would try to use professional-sounding language, only someone completely secure in their field would call sometime a "bearded armchair." ;)

naming things is hard.

I now have this mental image of the author sat there for three hours trying to come up with a name, and then just exploding with "fuckit! I don't care what it's called! It doesn't matter. I was sat in an armchair when I thought of it, so armchair it is! If they can have strange quarks, I can have armchairs"

A real smart crank is aware of this and because he only spends time on cranking this was the riddle he made. Not many academics would be confident enough to call this man on his bullshit.

Physicists are amazing in coming up with names, from WIMPs (Weakly interacting massive particles) to something like Nuclear Pasta[1], with all sorts of culinary phases.

[1] https://en.wikipedia.org/wiki/Nuclear_pasta#Phases

My personal favourite is the 'barn', an area approximately the size of the face of a uranium nucleus, invented to covertly describe a large-ish target size to hit (the 'broad side of a barn') with a particle beam.

Maybe, I'm guessing, when you're at such a niche area of science like topological edge states in honeycomb plasmonic lattices you've got to double back to calling things simple shit in name-pairs just to keep everyone onboard.

> Plus bees are probably interdimensional beings.

You misspelled "mice".

> I'd vote for a hexagon, since if you stack a bunch of circles, that's the shape you get

I can't imagine that, do you have a link to explain it or is it a joke?

Here you go! This explains it well.

https://en.wikipedia.org/wiki/Close-packing_of_equal_spheres

Imagine stacking marbles. They'll naturally fall into a hexagonal pattern as the most likely way of naturally stacking.

"pack" might have been a more appropriate term than "stack". Tightly packed circles (in 2d) form a hex grid.

More relevant, though, might be the shapes you get when packing spheres: https://en.m.wikipedia.org/wiki/Rhombic_dodecahedral_honeyco...

I think anyone who has experience playing old video games can intuit the first 1/3 of the article from this experience. Think of one of those spaceship games played on a 2D screen, where you can exit out the right side and re-enter through the left, at the same vertical coordinate. That's a 2D torus, or R^2/Z^2. And one can easily extend this by one dimension to generate a possible physical universe. The generalization is that now you are in a 3D room shaped like a cube, where you can fly into the ceiling and come out the floor (same with the left/right, front/back walls). This is R^3/Z^3, the simplest space like ours that isn't "infinite" (while finite, it might still be very big, and that's why we shouldn't expect to be able to detect this situation by simply shining a light and waiting to see it from behind).

https://en.wikipedia.org/wiki/Asteroids_(video_game)

We'd expect the size of the universe to to be, at minimum, (oldest visible star's age * c * 2 * star's distance from us) - right - unless we just haven't looked in the other direction and seen the image of the other side of that star.

That doesn’t take into account cosmic inflation in the very early universe. This is believed to have smeared out the universe into a mind bogglingly huge expanse, such that our observable universe is a mere speck in the grand scheme of things.

But if the star's visible on one side, it's close enough that it hasn't been redshifted into oblivion, right? The effects of cosmic inflation would only raise that minimum.

Yes.

The scientific consensus is that the observable universe is vastly bigger than (oldest visible object's age * c * 2 * star's distance from us).

See eg "We know that the Universe has been around for 13.8 billion years, but we also know we can see for 46 billion light-years. How is this possible?" at https://medium.com/starts-with-a-bang/how-is-the-universe-bi...

(And plenty of similar article when you Google 'size vs age of universe.)

Inflation isn't even necessary for this effect, but might amplify it.

Inflation is not necessary, because the cosmic background radiation is the furthest thing we can 'see'. And that background radiation stems from when the universe was 379,000 years old. Wikipedia says that inflation lasted at most until 10^−32 seconds after the big bang.

Yet, the background radiation seems to come from much further away than your simple calculation would suggest.

See https://en.wikipedia.org/wiki/Cosmic_microwave_background

Size of observable universe is 93.016 billion light years. The age of the universe is estimated at 13.8 billion years.

Observing gravitational waves might let us peek behind the curtain of the cosmic background radiation. Very exciting.

Hyperrogue[1] takes place in hyperbolic space. Definitely a good way to gain an intuition for it. It offers several different projections, so you can try them out. It's also open source[2].

1: https://roguetemple.com/z/hyper/

2: https://github.com/zenorogue/hyperrogue

Another game to check out in this vein is hypernom, developed by henry segerman, vi hart, and some other people that I don't remember: http://hypernom.com/

In a less principled way, there's also 'Antichamber' and 'Tea for God'.

Both use Escher-like spaces that are locally Euclidean but don't connect in proper ways. Eg if you go 360 degrees around a column, you might arrive at a different place in the level from where you started.

In Antichamber it's an interesting gimmick. In 'Tea For God' the mechanic is actually useful, because it's a way to fold a big level into the small boundaries of the VR space you defined on the Oculus Quest in your living room.

Here is my blogpost on various weird geometries used in games: https://medium.com/@ZenoRogue/non-euclidean-geometry-and-gam...

I probably spent more on Asteroids than on cocaine ;)

This is great. For anyone who wants to dive a bit deeper I would highly recommend the book The Shape of Space by Jeffrey Weeks. It's a wonderful insight into topology that can be enjoyed by anyone with high school math or above. I've even spotted it in the small library for graduate students at the Cambridge University maths department.

Definitely my favourite maths book.

https://www.goodreads.com/book/show/599877.The_Shape_of_Spac...

thanks - seems like a book worth reading

If you want to get an intuition for the weirder curvatures of 3D space, there's a playable "curved space simulator" created by topologist Jeff Weeks called Curved Spaces. I remember it from a while back, but apparently it recently got an update! Some combinations of curvatures and tilings get really funky to navigate.

http://www.geometrygames.org/CurvedSpaces/index.html.en

The article mentions topology but I feel like most everything in physics is geometrically oriented and makes no particular topology rules, just normal continuity. I feel like the next discovery will be topological in nature. Some relativity of topology that fixes locality.

I don't understand how they do the comparison between the torus and sphere. First to take a rectangular paper as an example for a torus and then say some parts would stretch doesn't make a scientific comparison to a sphere. As I understand the only difference between a sphere and a torus geometrical is that as a flat piece the sharp tops are cut of a flat piece of sphere. See link for picture of flat sphere: https://www.cadforum.cz/img/petals.gif

One way to see the difference is to consider a loop on the sphere or torus. On a sphere, any loop can always be drawn tighter until it is a point. On a torus, there will be some loops that cannot be drawn tight, as they go 'through the hole'. Another way is to imagine the sphere and the torus are hairy. You can comb a torus such that all hairs lie in more or less the same direction as their neighbours - if you try this with a sphere, there will be at least one point where the hairs completely diverge.

When we are whole brain emulations in the cloud, we can experiment with upgrades to our hippocampus to work with alternate geometries. There's no particular reason to assume that our current one is optimal for thinking. Maybe different geometries are better for different tasks. We could match the dimensionality of the workspace to the problem space. Want to visualize 7 dimensional data? Project it in 7 dimensional space with your brain adjusted to make it feel natural.

Characters in Greg Egan's hard sci-fi book Diaspora do this

Alternate geometries are interesting but it seems pretty clear that a flat universe is the simplest conclude when what we observe in the large scale is a flat universe (this is the case in most sophisticated observations despite general relativity demonstrating local curvature).

I think that the motivation for a universe that's bounded in some fashion comes from the philosophical paradoxes that come from an infinite universe. But I don't, maybe just deal with them.

This article helped me to understand a great talk by Lawrence Krauss which then became the book A Universe from Nothing.

I remember asking someone maybe a physicist since space is curved why do we locally see it has relatively flat.

It is like the surface of the Earth in an additional dimension. Looks flat locally but is actually curved.

The radius of curvature is much bigger than distances we daily measure, for a similar reason when dealing with small distances on the surface of the earth you can use flat, Euclidean geometry. We can measure the curvature of space(time), for example, Gravity Probe A managed to measure the curvature of space-time in the vicinity of earth.

I followed the gravity probe b experiment when it was happening. It tried to measure two effects of Einstein’s theory. In layman terms how far should one be to see the curvature of spacetime? My son was asking and I couldn’t offer an explanation as to how much the earth curves the spacetime around it. How could one use gravitational force value of 9.8 m/s2 help in grasping it the curvature of spacetime, since that is something we are aware of.

As curvature of space is affecting measurments of distances in angles, the curvature of spacetime is affecting both measurements of time and distance.

Concerning the curvature of space: If you measure sum of angles of triangle on the surface of earth, it will be grater than 180 deagrees thanks to positive curvature of earth surface. For triangle in your backyard the diference will be immesurable, but if your measure traingle with segments spanning for 6 thousends kilometers sum of the angles will be closer to 270 deg than 180 deg.

For the spacetime in the vicinity of earth you don't have to necessarly think about large distances, here (https://www.reddit.com/r/askscience/comments/2pu2o0/is_there...) someone calculated diference for two clocks so you can se scale :

"If so the clocks on Earth's surface and 7km down would see a relative drift of 0.001ns per second, so the lower clock would tick that much slower as viewed by someone on the surface. At that rate over the course of a billion years the surface clock would pull ahead by 525.6 minutes."

Haven’t read this yet, but another good question is why can’t we know the topology of the universe.

Dodecahedron :-)

Your snarky remark carries much more mathematical depth than it appears on the surface.

There exists a 3-manifold that is homologous to a 3-sphere but not homeomomorphic to it; the so-called Poincaré homology sphere:

> https://en.wikipedia.org/w/index.php?title=Homology_sphere&o...

Among the homology 3-spheres (besides the 3-sphere itself), the Poincaré homology sphere is the only one with a finite fundamental group.

Now for the plot twists:

1. A possible construction of the Poincaré homology sphere starts with a dodecahedron.

2. The fundamental group of the Poincaré homology sphere is the binary icosahedral group (https://en.wikipedia.org/w/index.php?title=Binary_icosahedra...), which is an extension of the symmetry group of the dodecahedron/icosahedron.

3. The huge plot twist: What does this all have to do with the question "What Is the Geometry of the Universe?"? Let me quote the Wikipedia article:

"In 2003, lack of structure on the largest scales (above 60 degrees) in the cosmic microwave background as observed for one year by the WMAP spacecraft led to the suggestion, by Jean-Pierre Luminet of the Observatoire de Paris and colleagues, that the shape of the universe is a Poincaré sphere. In 2008, astronomers found the best orientation on the sky for the model and confirmed some of the predictions of the model, using three years of observations by the WMAP spacecraft. As of 2016, the publication of data analysis from the Planck spacecraft suggests that there is no observable non-trivial topology to the universe."

The universe is a spheroid region, 705m in diameter.

tldr; We don't know.

When I was in school (not a physics major) I imagined universe as a 3 dimensional sphere, S3, which is a surface of a 4-dimensional ball that constantly inflates since Big Bang - hence the flow of time.

I wonder if that model can be easily disproven given what we know now?

I imagined exactly the same. However, Planck experiments show that the universe is flat. It means, our theory was wrong - or the sphere is so big that we can't notice its curvature, thinking we're living on a flat 4-dimentional surface (almost like with Earth in ancient times)

According to a proper philosophy there is no way to know. Only to have socially constructed consensus (hello, mr.Khun)

Let me remind you that abstract concepts like dimensions, does not exist outside human minds.

The question itself contain a type error: the concept is not applicable.

I don't see how the question contains a type error, any more than the question of the earth being round contains a type error.

Philosophy of science is an easy subject and a person of reasonable intelligence could grasp the principles without any difficulty.

The current contest is not of experimentally verified facts, but of fancy abstract models. For funding and the high social status of an abstract "researcher".

The vastly complex and expensive devices for "experiments" are made according to the current model - a layer upon layers of socially constructed abstractions. This is no different from theology.

You could also read about positivism and acceptance of the limits of what could be known (not accepted or believed).

Yes, the difficulty of making machines to test these ideas makes their results less trustworthy. But you're applying am uneven amount of ontological skepticism here. You talk about the 'limits of what could be known', but this applies just as much to your own senses or the existence of society as it does to the validity of scientific models.

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