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Imagine we are living in a parallel universe where the parallel lines can only exists as imaginary lines just not in the real physical world, how would road and skyscrapers work in such a universe?

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    $\begingroup$ They wouldn't, because life wouldn't exist. $\endgroup$ – Renan Feb 17 at 2:43
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    $\begingroup$ It all depends on what you mean by "parallel lines cannot exist in the real physical world". For that matter, how do straight lines (which are an abstract mathematical concept) exist in our universe? $\endgroup$ – AlexP Feb 17 at 3:00
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    $\begingroup$ depending on how you define "parallel lines", any curve could be fair game. So... are we talking about Non-Euclidean geometry? $\endgroup$ – Theraot Feb 17 at 3:03
  • $\begingroup$ 1) Look at a globe: Lines of longititude always intersect. 2) Roads & buildings don't have to have (and in practice don't have) lines that are perfectly parallel out to infinity, they just need to be approximately parallel for some distance, whether it's meters or hundreds of km. $\endgroup$ – jamesqf Feb 17 at 3:58
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    $\begingroup$ I love this question, but it needs a lot of clarification to be answerable. Specifically: What constitutes a line? Are we talking different geometry space (so just non-Euclidean parallel lines are banned) or are we talking Euclidean geometry where no two edges of any object can run parallel? If the latter, what constitutes an object? How does physics resolve this? Please fill in these details so the community can help you flesh out this interesting idea. $\endgroup$ – SRM Feb 17 at 14:16
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Almost nothing on Earth is perfectly parallel.

Let's explore why with a few thought experiments.

Two people start on Earth's equator, facing the North pole. They are separated by 90 degrees of longitude - perhaps one person is in a boat on the Atlantic, the other on the Pacific. Since North is exactly perpendicular to each person, their lines of sight are perfectly parallel. They continue forward until meeting at the North Pole. Fascinatingly, their parallel paths of travel converged - because spherical geometry is non-Euclidean. This is true of any two "parallel" roads on Earth: they are not parallel in the traditional sense because they will meet at a finite distance - yet they are still geometrically parallel. If you really want to keep roads equidistant, you have to curve them and cut out some distance, which is why lines of latitude can appear "parallel".

Two carbon-nanotube skyscrapers of infinite structural strength are built next to each other. Their bases are congruent and parallel. They are expanded upward, all the way to the moon's orbit. As height increases, the distance between them increases, because they are perpendicular to the Earth's surface - which is curved, not flat. Even the two sides of each individual structure would begin to diverge, if not artificially curved inward - and this is true on Earth.

As long as your universe has spherical planets, these phenomena will operate in the same way. This is one characteristic of manifolds: they appear Euclidean (flat) locally, so you can draw parallel lines that appear to stay equidistant from each other to infinity. That doesn't mean parallel lines act "parallel" forever in our universe.

There are countless more examples; consider that light bent around black holes travels in a "straight line" despite being guided along the curved path of space-time. Everywhere in our universe has some amount of gravity; no path, nor any set of two lines, can be straight or equidistant forever even here.

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  • $\begingroup$ "Almost nothing on Earth is perfectly parallel." Why this almost? If I undesrtand well enough the body of your answer, there are no structure on earth that isn't curved $\endgroup$ – Kepotx Feb 17 at 9:28
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    $\begingroup$ @Kepotx Although Earth is curved, the universe is not. Things not dependent on the surface of the earth for its structure (e.g. a book) could still have parallel lines. Likewise precise levels could be used to build a flat foundation so support beams in a single building can be parallel. This is why the example used two skyscrapers instead of the support beams in a single skyscraper. $\endgroup$ – Hink Feb 17 at 9:48
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Lots of Curves

Rather than assuming your parallel universe cannot have parallel lines, I'm going to assume it's in this universe, but parallel lines are a major cultural taboo and people go to great lengths to avoid them. (so physics works)

Architecture would likely use curves to avoid parallel lines. Skyscrapers could be built as cones or domes, and if necessary the floors could be slightly angled. Elevators would need to ascend on a curved path, possibly just attached to the outside of the building, with spiraled ramps leading up between levels.

Your city grid would likely be concentric circles with interconnecting lines. As long as the lines are (roughly) perpendicular when they connect to the rings, they will never be parallel no matter how many are added.

Polar Grid, from Pixabay https://pixabay.com/illustrations/polar-grid-circle-graphic-clipart-2187414/

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  • $\begingroup$ Well... a line is a circle with a radius of infinite... At some point, the circles are so large that for all means and purposes, the rings could be considered parallel rectangles. $\endgroup$ – Trish Feb 17 at 14:15
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There's a chance that our own Universe doesn't accept parallel lines

Lastest astronomical findings may point to the fact that our Universe is not flat, but actually curved on itself, and is a matter of heated debate. On a small, human scale, it means nothing, but in the big scheme of things it changes the development and eventually death of the Universe (a "Big Crunch" in this case). So, if the Universe is curved on itself, parallel lines are actually impossible, they will eventually collide on one point, they only appear parallel because the scales are short.

Your Universe can be also curved, maybe even more than ours making this effect measurable in relatively shorter distances.

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    $\begingroup$ I saw the articles about this. This is from the author's initial post - a discovery should reach an accuracy of at least five “sigmas” to be accepted by the community. Here we are slightly above three sigmas, so we are clearly below this acceptance level. - Keep in mind it is contradicting several studies that are above the 5 sigma level, so don't update your textbook yet. $\endgroup$ – Hink Feb 17 at 15:50

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