[Harri (OP)]

Can anyone help me visualise a flat universe that is expanding in every direction? It would seem to me that anything that is flat could not retain its flat shape if it were expanding in every direction.

[Halc]

Can anyone help me visualise a flat universe that is expanding in every direction? It would seem to me that anything that is flat could not retain its flat shape if it were expanding in every direction.

Don't understand your inability to visualize it.
Consider any shape, like a heart of size 1.  Now double every dimension in it, so it is twice as wide, tall, and thick, but size 2 now.  It is the exact same shape, just expanded.  Ditto for the flat universe.  If it was flat before, doubling its space in all dimensions does not affect the shape at all, so it's still flat.  Expansion does not inherently change the shape of something.

[RobC]

What will be the ratio of its thickness to its length?

[Halc]

What will be the ratio of its thickness to its length?

The universe has neither thickness nor length.

[pzkpfw]

Flat, here, does not mean like a pancake or CD.
It's about whether it's curved.
https://en.wikipedia.org/wiki/Shape_of_the_universe#Curvature

[Harri (OP)]

If I travel along the surface of the heart I will come around to where I started as it is curved. If I travel along a flat object I either fall off the edge or go on forever. When I think of something flat I automatically think of something with angles and edges. What does 'flat' actually refer to when discussing the expanding universe?

[evan_au]

What does 'flat' actually refer to when discussing the expanding universe?

"Flat" means that it follows Euclid's geometry that you learned at school (on large scales, like bigger than galaxy clusters):
- If you have a line, and a point not on that line, there is only 1 parallel line you can draw through that point
- The sum of the angles of a triangle add up to 180°
- The circumference of a circle is π times the diameter

There are also varieties of non-Euclidean geometry where:
- There is more than 1 parallel line through the point (or none at all)
- The sum of the angles of a triangle add up to more than 180° (or less than 180°)
- The circumference of a circle is more than π times the diameter (or less than π times the diameter)

These ideas seem strange, until you realize that the surface of the Earth is a non-Euclidean surface.
- There are no parallel lines of longitude (they all intersect at the poles)
- You can draw a right triangle with internal angles adding up to 270° (one side is on the equator, and one vertex is at the North pole)
- You can draw a circle around the equator whose circumference is just twice the diameter

See: https://en.wikipedia.org/wiki/Non-Euclidean_geometry

In the vicinity of a black hole there are also small-scale distortions, in addition to any large-scale distortions.

[RobC]

How does string theory describe the shape of the universe?

[Harri (OP)]

School? I well remember the day my parents were asked to take me out of school! The alternative was that the school would throw me out! So even the most basic mathematics is beyond my grasp. But, here's what I'm seeing. I get the Euclidean non Euclidean thing after reading up on it. The universe is flat and I'm seeing the parallel lines. The universe is expanding in all directions and I'm seeing those parallel lines expanding into a trumpet shape. So where am I going wrong? 

[evan_au]

I'm seeing those parallel lines expanding into a trumpet shape

You could imagine the universe as a sheet of rubber. Draw two parallel lines on it while it is relaxed.
 - If you grab the rubber sheet by 2 edges and pull, it will stretch on one direction, and shrink in the other direction. This will make the lines bend; closer to each other in some parts, farther away in other parts. They are no longer straight lines.
- However, if you hold the rubber sheet all along all 4 edges and pull equally outwards, the two lines will remain straight, and will be the same distance apart all the way along (just further apart than they were originally).
 

[Halc]

I get the Euclidean non Euclidean thing after reading up on it. The universe is flat and I'm seeing the parallel lines. The universe is expanding in all directions and I'm seeing those parallel lines expanding into a trumpet shape. So where am I going wrong?

The trumpet shape is probably due to something graphing space vs time.  Flat space should depict parallel lines in a graph of x vs y (both space), not x vs t.  If you expand that graph, the lines remain parallel.

The surface of Earth is a nice example of non-Euclidean 2D space.  It is difficult to draw it on a flat paper map, and any attempt to do so ends up distorting continents and such.  Parallel lines drawn on the ground always meet.  Triangles have their angles add up to more than 180°.  That's a great way to show that the surface of Earth is not flat.
They've actually attempted to do the triangle thing to measure curvature of space.  It does actually curve locally due to gravity, but on a large scale it seems to be flat.

[Harri (OP)]

The trumpet shaped universe I envisioned doesn't make sense now in light of what I've read in the replies! Discovering just what is going on in our universe bit by bit is quite enlightening so thanks guys.

[Bill S]

The trumpet shape has been around for a while.  Eg.  https://www.independent.co.uk/news/science/the-universe-is-not-round-say-scientists-it-is-shaped-like-a-trumpet-560046.html. 

I have always assumed it was a “mathematical” model, intended as an aid to understand; rather than something that was meant to be taken too literally. 
No one will be surprised that I have grave reservations about the physical demonstration of:-
 
“The thin end would be infinitely long - but so narrow that it would have a finite volume.”

[Halc]

The trumpet shape has been around for a while.  Eg.  https://www.independent.co.uk/news/science/the-universe-is-not-round-say-scientists-it-is-shaped-like-a-trumpet-560046.html. 

I have always assumed it was a “mathematical” model, intended as an aid to understand; rather than something that was meant to be taken too literally. 
No one will be surprised that I have grave reservations about the physical demonstration of:-
 
“The thin end would be infinitely long - but so narrow that it would have a finite volume.”

OK, Mr Arthur gets a good prize for bad reporting of science.  No scientist claims the universe is shaped like a trumpet, an object with edges.  The bit about finite volume comes probably from Gabriel's horn ( https://en.wikipedia.org/wiki/Gabriel%27s_Horn#:~:text=Gabriel's%20horn%20(also%20called%20Torricelli's,surface%20area%20but%20finite%20volume.&text=The%20properties%20of%20this%20figure,Torricelli%20in%20the%2017th%20century. ) a 3D shape which has zero to do with the shape of the universe (a 4D thing).

What the scientist are doing is taking a circle of space and plotting just that circle as it expands over time.  The circle can be any size, and nobody suggests that the universe is bounded at that circle, so the actual shape is something 3D (not a 2D circle) without bounds, expanding in a non-linear curve over the 4th dimension, time.  The 'trumpet' shape is just what you get if you graph the expansion rate (the scalefactor to name it more correctly) and twirl that graph on its time axis.  It is the shape of expansion at best, not the shape of the universe at all.

Oh, and this news is decades old.  I knew this expansion curve 30 years ago, even though it definitely has been tweaked since then. I think it was merely published as a way to get hits to their hostile site, which I found impossible to read without ad-blocker on.

[Harri (OP)]

And that's where the trumpet shape came from in my head. It is something I've seen in books and on the internet. It is basically a diagram to illustrate the expansion of the universe, not the actual shape of the universe

[Bill S]

I was just looking back over some of the posts in this thread and thinking what a lot of information can emerge from discussing things about which, fundamentally, people agree.  Then I followed Halc’s link and found something different, again.  “….associating the divine, or infinite, with the finite.”  It was the “divine, or infinite” bit that stood out.  Even if it doesn’t actually equate the two; the suggestion is there and may, albeit subconsciously, contribute to the reticence of scientists and mathematicians where considering anything other than “mathematical” infinities is concerned.  Just a thought.