Can a finite creature understand infinity

Shapes for an Infinite Beast?

There are some real species that already make a decent impression of this, namely mushroom mycelium and aspen forests.

To my knowledge, your criteria are as follows:

  1. Infinite spatial expansion
  2. Infinite age
  3. Does not cover the entire surface of the infinite world
  4. Do not prevent too much the movement of people over / under / through the space they occupy
  5. Not particularly prone to being cut into pieces
  6. Has exactly one head

# 3 and # 4 can be addressed by having your creature exist mostly underground (or mostly in the air, but underground is easier), and # 5 by being in the form of a huge network. Like the mushrooms and aspens mentioned above. These organisms can survive if large parts of themselves are destroyed because there are many more redundant parts everywhere. And in your case, when # 1 is met, there is an infinite area that is invaded by this organism. A finite amount of destruction would hardly be perceptible to the entire organism.

# 6 is probably the hardest criterion to meet. Each of the functions typically attributed to animal heads would be better served by a collection of nodes distributed over the network than by a central node in one location. A unique mouth could never absorb the infinity of nutrients that the organism would need to feed itself. A singular brain could never respond to stimuli that are infinitely distant. If cutting off the head could kill the entire organism, it would never live to infinity and fulfill # 2. And if the world is flat and uniform, why should one place be upside down and not another?

Perhaps the organism has a fractal network of nodes, each of which has its own brain. The smallest nodes directly control the behavior of the organism in a certain area. Larger nodes regulate larger areas and delegate the micromanagement of these areas to the smaller nodes within. Each node communicates with its neighbors and reports to the next node of the next largest size. At each higher level in the hierarchy, the number of nodes decreases as the distance between them increases. In the limit, the most infinite level will contain a single node that is infinitely far from anything that could be considered a "head". But for all practical purposes it might as well not exist.

I should also point out that an infinite space can contain multiple infinite, non-intersecting volumes. In fact, your first paragraph contains two: the sky and the ground. Each is infinitely large, but only takes up half the space in your world. In fact, an infinitely long pipe running in a straight line across the ground has infinite volume but only covers an infinitesimal fraction of the surface of an infinite world. Just because your creature is infinite in volume doesn't mean it has to occupy all of the volume in the world. There can be a lot of space around it.

Update : If you want something more mobile than a plant or a mushroom, you can grow various sensory organs (e.g. eyes) and grasping elements (e.g. tentacles) that inform and are controlled by the next node. Which ... could give it a serious Cthulu-like look. If this is not quite the aesthetic you are aiming for and you want it to be easier to communicate with people, you can grow humanoid "avatar bodies" tied to the main nutrition and communication network with nearby nodes, but have their own brain and are able to act at least somewhat independently.

DLosc

+1 This answer addresses all of my concerns about the question. An additional point about heads: if the "main head" is lost, one of the "smaller heads" could be promoted to the new "main head". (Of course, this cannot happen very quickly due to the signal propagation time.)

Someone else 37

@ TylerSigi Mushroom mycelium and aspen forests linked a few paragraphs below. I think I could have made that clearer. Fixing.

Someone else 37

@DLosc The nice thing about the infinite fractal solution, however, is that the "main head" or other nodes in the infinitely high levels do not actually have to exist. No matter how high you go up the chain of command, you will never climb the finite steps. And if you happen to find yourself at a finite distance from a node with infinite levels, this is completely irrelevant to the story - provided it is not suitable for communicating with nodes that have infinitely many levels below, but only with nodes that have are infinitely far away, and therefore each message will take infinitely long to reach its destination.

Someone else 37

@DLosc "The Head" can therefore just as easily be viewed as a religion or folklore, rather than something that can actually have a direct impact on history.

PyRulez

What if the head is infinitely large and therefore absorbs an infinite number of nutrients?