62

TMM;DR (Too Much Math, Didn't Read): For anyone who doesn't want to go through the derivations and calculations below, here are the important points from my answer: We're not working with the same space as normal, friendly, Euclidean space. This means that while we can still integrate and differentiate scalar functions defined on this space, we need to ...


55

You specify "creature", but would a fungus fit your requirements? Taking inspiration from Armillaria ostoyae, one example of which is the largest living organism, covering 3.4 square miles of the Malheur National Forest in Oregon (aka the Humongous Fungus). Despite its size, as it is a soil organism consisting largely of microscopic filaments interacting ...


51

The world would become progressively more alien as you moved across it. Unlike on Earth, where animals and humans can move across the entirety of the world given a few thousand/million/tens of millions of years, an infinite flat earth would always have somewhere that's far enough away that nothing from where you are has ever been there. This would probably ...


44

In four dimensions, you don't have a rotation axis (fixed straight line) but a rotation plane (fixed plane). However, not every 4D rotation has a rotation plane; there are rotations which have no fixed points (except for the origin). Indeed, rotations in 4 dimensions have six parameters instead of the three we know from 3D space. The corresponding rotation ...


31

It might be infinitely tall, just like a tree (or giraffe) that simply doesn't end. Ever. A snake-like being would work also, provided that it either flies/floats/burrows or is flat enough that it is not a barrier to other terrestrial creatures.


27

No The necessary condition for a liquid body to be in a gravitational equilibrium is that gravitational potential should be constant over its surface. For the cube with density $\rho$, by integrating $$ V(\mathbf r) = \int_{cube} \frac{G \rho}{\lVert \mathbf r - \mathbf r' \rVert_\lambda} d^3 \mathbf r' $$ for various locations $\mathbf r$ on its surface, ...


26

This would actually be really awesome. It wouldn't really be a planet anymore, but it could conveniently be called a plane or plane-et. We could find the epicenter of life. Presumably, abiogenesis is quite rare. On an infinite flat Earth it will certainly have occurred elsewhere, but not likely very near to us. Theoretically, we would eventually be able to ...


26

Israel has developed several ways to find tunnels made by Hamas. There most advanced one relies on a series of sensors and computer systems that detect slight vibrations in the ground and triangulate them. But simpler models have worked relatively well for decades. In WW1 where mines were common, soldiers developed a whole host of means to detect mines ...


22

There are some real-life species that do a decent impression of this already, namely fungal mycelium and aspen forests. Your criteria, as I understand them, are as follows: Infinite spatial extent Infinite age Does not blanket the entire surface of the infinite world Does not impede movement of humans over/under/through the space occupied by it too much Is ...


20

Make a Maze. A concept that comes up in a lot of fantasy novels is a sandbar maze outside of a harbor. This then lets you restrict access - outsiders don't know the patterns, and if they try to invade they ground their ships and you can then proceed to shoot them from a nice safe distance. Expanding space could be used in the same way to protect important ...


19

As Danijel has pointed out, the question can be reframed as "what is the shape of the equipotential surface under this force law?" There are further two different ways to view the results: the external perspective, where we see what shapes these "planets" map onto if we overlay them on our normal Euclidean space, and the internal perspective--what would ...


18

Could have a fractal outline and that would be an infinite surface within a finite volume. A Mandelbrot fractal looks as if it had a head (and a posterior as well) so that may serve your purpose. https://en.wikipedia.org/wiki/Mandelbrot_set Maybe this is not what you mean by infinite.


17

How do you stop enemies from digging under you country? You dig under your country first, small tunnels with seismic sensors (which isn't going to be so easy, since you're receiving a planetary surface's worth of seismic waves from above). Then deploy mines, and an underground force to intercept enemy excavations. With current technology it is relatively ...


16

Oh boy! A pocket dimension! What can't you use it for? Ok, you have a traversable wormhole to a big place. Call it a stellar-sized bag of holding. You can put all sorts of things inside it, perhaps even an entire solar system. The main problem with a solar system for an interstellar empire, especially one with enemies, is defending star systems. Put ...


15

My answer isn't going to be anywhere near as in-depth as your question but I'm happy to supply a few inputs. First, I'm going to take a look at your particles: 1. Particles can only move at one constant speed. 2. Particles can only move in six directions. (Or four in my 2D model.) 3. Particles have "gravity". I'm going to imagine two particles in a two-...


14

For now, I'll focus on calculating magic numbers and finding the island of stability, although I have several remarks to make later on the possibility of instability for atoms of all isotopes and atomic numbers. We need to use the nuclear shell model to find the magic numbers. This means that we have to treat the nucleus as an $N$-dimensional quantum ...


14

The problem can be restated as follows: find an infinite plane non-intersecting curve which does not partition the plane. That's easy. Two immediate examples, described in polar coordinates: $r = \exp(\frac{1}{0.01\theta + 1})$ (the red curve in the illustration), and $r = 1+\left(1-\exp(-0.05\theta)\right)$ (the green curve in the illustration). Two ...


13

Its similar to fighting in a 5ft wide corridor of infinite length. A defensive fighter would want to face opponent on a north south axis along the corridor. A large shield or two men standing abreast you block the entire world from getting past. Two attackers could charge forward with spears knowing their foes could never side step around them. In a ...


12

A lot of people ignored the question and delved into the rabbit hole of physical impossibilities given the physics of our known universe that were ruled out in the question. Given an infinite, planar type world with similar characteristics and physics to earth (though I am not sure how different temperate zones would come about) I agree that with others that ...


12

The Spherical World Society and Evil Twins There is a model of the projective plane known as the projective sphere. It has the following properties: Since it is based on a sphere, which is orientable, you can consistently assign objects orientations which do not change as you move along it. Unlike the hemisphere model, the sphere model is basically unique. ...


11

Some other place on Physics Stack Exchange we discussed the physics of gravitational attraction in the world of Asteroids. For the most part, it's not a problem. In order to predict gravity from an object (like a planet), you just sum up the field from an infinite lattice of such objects repeated infinitely. It's easy to see that with simple Newtonian ...


10

Life starting from a single point will create a so called Fairy Ring. Life will move outwards, expanding the ring seeking for a new resources. Many concentric ever expanding rings may be created as a new life evolves, that is able to use resources that the life in the previous ring was not able to use.


10

Everything dies. In fact, everything dies in a manner much more spectacular than if you had crashed the moon against Earth. Or Venus. Or even Saturn. The physics and mathematics behind this are enough to fill a few books, but the short version of it only needs you to take a couple things into account: Wormholes have mass. Yes, even if they are the Ellis/...


9

1 on 1, you are in a world of complexity. Martial artists have an infinite edge over non-martial artists as they can appear to kick on the right side and hit the opponent on the left! Also there are very high chances a short range projectile weapon (like a slingshot) would be used by every combatant. The most shocking battles would ensue when the opponents ...


9

The space we live in has to be complete, with the "outside" not simply another spatial direction of the same geometry. You might postulate another orthogonal $w$ axis that works the same as $x, y, z$, and we're not able to move that way because everything is lined up at $ w=0$ now so nothing can get get beside you to push in that direction. There might be ...


9

filling Look at space filling curves. You can apply the same idea to filling a 4th dimension with a 3d ribbon. You still need an infinite amount, though. You might also consider how 3d matter exists in the 4d world. Just like sheets of paper (or even ink on a sheet of paper) in our 3d world is not really 2 dimensional but meerly extremely thin in the ...


9

I can give you a Newtonian analysis of the situation, which may or may not be correct. If you want a Newtonian universe, then great. If you want a general relativistic one, you'll need something more complicated. Example 1: Square domain As a simple introductory example, let's say your universe is a rectangle with sides of length $l$. In other words, if ...


9

You would get tattoos which would be recognizable as mirror images of themselves. Rightie-pride people would get tattoos depicting their pride and belief. These would vary from person to person but have similar themes, and be publically viewable to greater or lesser extent. Likewise leftie-pride people (but different tattoos, of course!) A rightie who ...


9

Build up You would want to build up anyway since there is so much more room up there. But building up also allows you putting things like heavily reinforced concrete in the enemy's path. Of course, even the heaviest concrete can be breached, but it will take time and be very noticeable. One might think that building a concrete floor under your entire ...


9

Your creature is infinite and yet doesn't take up all the space in an infinite universe. Because you yourself specified that you want your creature to have a head, and therefore an end, I suggest that your particular creature is actually finite. Your creature is a mobius strip with some mathematical knots added for good measure. Being a mobius strip design ...


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