
What Is the Geometry of the Universe? - bryanrasmussen
https://www.quantamagazine.org/what-is-the-geometry-of-the-universe-20200316/
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zomglings
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.

~~~
uoaei
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.

~~~
airstrike
+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.

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russellbeattie
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.

~~~
perl4ever
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")

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

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marcus_holmes
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"

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virtuous_signal
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).

~~~
earenndil
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/](https://roguetemple.com/z/hyper/)

2:
[https://github.com/zenorogue/hyperrogue](https://github.com/zenorogue/hyperrogue)

~~~
abnry
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/](http://hypernom.com/)

~~~
eru
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.

~~~
zenorogue
Here is my blogpost on various weird geometries used in games:
[https://medium.com/@ZenoRogue/non-euclidean-geometry-and-
gam...](https://medium.com/@ZenoRogue/non-euclidean-geometry-and-games-
fb46989320d4)

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montalbano
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...](https://www.goodreads.com/book/show/599877.The_Shape_of_Space)

~~~
option
thanks - seems like a book worth reading

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joan_kode
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](http://www.geometrygames.org/CurvedSpaces/index.html.en)

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daxfohl
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.

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Giho
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](https://www.cadforum.cz/img/petals.gif)

~~~
somewhereoutth
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.

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hirundo
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.

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

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joe_the_user
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.

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

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

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m_j_g
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.

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hi41
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.

~~~
m_j_g
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...](https://www.reddit.com/r/askscience/comments/2pu2o0/is_there_any_time_dilation_between_us_and_the/cn03iuf/))
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."

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

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mijkal
Dodecahedron :-)

~~~
wolfgke
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...](https://en.wikipedia.org/w/index.php?title=Homology_sphere&oldid=911674219#Poincar%C3%A9_homology_sphere)

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...](https://en.wikipedia.org/w/index.php?title=Binary_icosahedral_group&oldid=940730315)),
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."

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

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unnouinceput
tldr; We don't know.

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option
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?

~~~
strategarius
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)

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johndoe42377
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.

~~~
Strilanc
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.

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johndoe42377
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).

~~~
antepodius
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.

