# Shapes for an infinite animal?

This creature exists in an infinite world, a flat landscape that extends ad infinitum, where light rains from the sky from infinity during recurrent day-night cycles, and, similarly, the ground goes down continuously.

All kinds of critters populate the cosmos including the skies and the underground realm. This bizarre world has existed for an infinite time and for this reason I want (at least) one creature to have inhabited it for an infinite time. And that's not all, this particular being is also physically infinite(its body has no end); obviously I don't want it to take up all the space, but only a part of it.

In the specific I was thinking of giving it a shape similar to something like Gabriel's Horn, so it would leave a large area available, however this would basically cut most of the surface in half, and on top of that most of his body is vulnerable to attack since it's so thin; for this reason I would like to avoid "infinitely small" bodies.

What infinite shape can I give my organism so that it wouldn't be too vulnerable but at the same time leave room for people to move about relatively easily, without cutting off parts of the world that are close by from each other? Also, while its body is infinite it should have only one head.

I am going to be a bit lax with the physics of this world to allow for some basic functionalities, but I want to put special attention to this aspect.

EDIT: Thanks a lot to everyone for the inputs, I went with the answer that better fits the requirements, but I will probably incorporate elements from other interesting answers

• A creature that is physically infinite in an infinite universe...but isn't actually infinite? I love it! Apr 24, 2018 at 14:55
• Reminder to close-voters: The problem cannot be fixed if the OP is not made aware of it. Apr 24, 2018 at 17:17
• That being said, I fail to see how one answer can be judged as "better," "more complete," or "more applicable" than another. A creature with long legs is just as viable as a creature that floats, for example. So either I fail to understand the question in its entirety, or it's primarily opinion-based. Apr 24, 2018 at 17:20
• While I see @Frostfyre's point, I disagree that this question deserved to be closed. The OP provided criteria (not too vulnerable, room for others, can't cut off parts of the world, one head) and the answers generally reflect this criteria rather than being all over the map. If an OP provides all the criteria to guarantee only one right answer, then 99% of the time the OP has the answer already. I'm voting to reopen.
– JBH
Apr 24, 2018 at 17:37
• @JBH I've opened a meta thread on this question. Apr 24, 2018 at 21:12

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:

1. $r = \exp(\frac{1}{0.01\theta + 1})$ (the red curve in the illustration), and

2. $r = 1+\left(1-\exp(-0.05\theta)\right)$ (the green curve in the illustration).

Two infinite yet bounded curves which do not partition the plane: r = exp (1/(0.01t + 1)) (show in red) and r = 1 + (1 - exp (-0.05t)) (show in green). Made with the graphing calculator at Desmos. Own work, available on Flickr under the CC-BY license.

Converting the infinitely thin lines into lines with finite widths is left as an exercise. (Hint: make the width inversely proportional with $\theta$.)

Note that the red curve grows inwards, and r will always be greater than 1, while the green curve grows outwards and r will always be less than 2. No matter how long the curves get they will never partition the plane in two disjunct regions.

Also note that I'm taking "an infinite time" to mean "a really long time". Things become truly weird when infinite values are actually allowed. In particular, any event which is not impossible becomes necessary...

• This is really cool, a spiral doesn't part the plane and it might have a "head" at the center Apr 24, 2018 at 16:28
• " In particular, any event which is not impossible becomes necessary...". I don't think it is true. Imagine that every second, you are picking one real number at random (from range 0 to 1 for simplicity). Picking 0.5 exactly is possible, but it is far from being sure, even taking infinite amount of time. I think that it has zero chances of happening in infinite amount of time, to be exact, despite being possible. Continuum >> aleph zero. Apr 25, 2018 at 11:58
• @ArturBiesiadowski: The probability of picking exactly 0.5 is zero, so the event is almost impossible. But in an infinite amount of time, any event which has finite non-zero probability, no matter how small, becomes certain. For example, the probability of a species evolving which looks just like humans, and for an individual to be born who looks just like Shakespeare, and who speaks early modern English, and for that individual to write a play which is word for word identical to Hamlet is vanishingly small, yet finite: so in an infinite amount of time it will certainly happen. Apr 25, 2018 at 12:25
• @ArturBiesiadowski I think you may be under appreciating the true scale of infinity. Apr 25, 2018 at 12:42
• @ArturBiesiadowski regarding 2: this isn't relevant in an infinite-time scenario, because the external factors are also (eventually) permutated such that the vanishingly improbable is allowed to becomes certain. If that's impossible, the dependent improbable occurrence is actually impossible, and we've just calculated the probabilities wrong. Apr 25, 2018 at 14:09

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 with plant roots it is mostly invisible, so even if it spread out over the whole world it would not get in the way. Its anatomy of largely self-sufficient parts would seem amenable to infinite scaling both in time and space. While this fungi is considered parasitic, it would be easy to imagine this being commensal or symbiotic with the plants of your world.

• Brilliant. A fungi is certainly closer to an animal than it is to a plant. This is certainly the most suitable answer thus far. Welcome to Worldbuilding! Apr 24, 2018 at 15:57
• That's a nice idea, but I was thinking more like an animal Apr 24, 2018 at 16:27
• Corals are animals. It could be a coral or something of that sort with some sort of feeding / digestive structure that expands infinitely in all directions through the ground. Posting that as a suggestion here rather than my own answer since I don't feel that my idea is significantly different from this answer, the only difference is the kingdom that the organism belongs to. Apr 24, 2018 at 16:39
• Although not an animal, Pando is similar. Apr 25, 2018 at 0:26
• a fungus is about the only thing that will work since and infinite animal would need infinite food which it could never eat with a finite mouth. Fungus can eat with its entire body so it does not have this issue.
– John
Apr 25, 2018 at 2:01

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.

• +1 For the snake-like creature. I picture a flying dragon-like snake that has a head but maybe no tail. Apr 24, 2018 at 19:39
• I instantly thought it must be Jörmungandr Apr 25, 2018 at 3:52
• I love the idea of a snake maybe a meter thick that has a head, but the other side goes forever. It's been around for ever and legend says it can answer any question. People go on pilgrimages to find the head. They follow the body, never knowing how close they are to the head, or if they are going endlessly in the wrong direction. Apr 25, 2018 at 8:37
• The creature can indeed answer any question but unfortunately it only speaks a language where any answer takes infinitely long to pronounce. Apr 25, 2018 at 13:23
• There are legends that it grows constantly and the world will end if it ever encounters it's tail… Snake, the mythos. Apr 25, 2018 at 18:25

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:

1. Infinite spatial extent
2. Infinite age
3. Does not blanket the entire surface of the infinite world
4. Does not impede movement of humans over/under/through the space occupied by it too much
5. Is not particularly vulnerable to being chopped 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 having it take the form of a vast network. Like the aforementioned fungi and aspens. These organisms can survive large chunks of themselves being destroyed by virtue of having many more redundant parts all around. And in your case, if #1 is fulfilled, there is an infinite area of land infested with this organism. Any finite amount of destruction would barely be noticeable to the organism as a whole.

#6 is probably the hardest criterion to fulfill. Any of the functions typically attributed to animal heads would be better served by a collection of nodes spread throughout the network, rather than a central node in one location. A singular mouth could never take in the infinitude of nutrients the organism would need to sustain itself. A singular brain could never respond to stimuli infinitely far away. If cutting off the head could kill the entire organism, it would never live to be infinitely old and fulfill #2. And if the world is flat and uniform, why should any one location have the head and not another?

Perhaps the organism has a fractal network of nodes, each with its own brain. The smallest nodes directly control the organism's behaviors in a certain area. Larger nodes govern larger areas, delegating micromanaging those areas to the smaller nodes therein. Each node communicates with its neighbors and reports to the nearest node of the next size up. At each higher tier in the hierarchy, the number of nodes decreases, while the distance between them increases. In the limit, the infinity-th tier will contain one single node, infinitely far away from everything, which could be considered the "head". But for all practical purposes, it may as well not exist.

Also, I should 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 infinite in extent, but only takes up half of 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 has an infinite volume doesn't mean it has to take up the entire volume of the world. There can be plenty of space around it.

Update: If you want something a bit more mobile than a plant or a fungus, you could have it grow various sensory organs (e.g. eyes) and prehensile limbs (e.g. tentacles) that respectively inform and are controlled by the nearest node. Which... could give it a seriously Cthulu-like appearance. If that's not quite the aesthetic you're going for, and you want it to be able to communicate with humans more easily, you could have it grow humanoid "avatar bodies" that are tethered to the main network for nutrition and communication with nearby nodes, but have their own brains and are able to act at least somewhat independently.

• +1 This answer addresses all of my concerns with the question. One added point about heads: if the "main head" is lost, one of the "lesser heads" could be promoted to be the new "main head." (This might not happen very quickly, of course, due to signal propagation time.) Apr 24, 2018 at 17:32
• @TylerSigi Fungal mycelium and aspen forests, which are linked a couple paragraphs down. I guess I could have made that more obvious. Fixing. Apr 25, 2018 at 2:54
• @DLosc But, the beauty of the infinite-fractal solution is that there's no need for the "main head" or any other nodes in the infinitely-high tiers to actually exist. No matter how high you go up the chain of command, you'll never get past the finite tiers. And if you happened to find yourself a finite distance away from an infinite-tier node, it will be entirely irrelevant to the story- assuming it doesn't deign to communicate with nodes infinitely many tiers below it, it will only interact with nodes infinitely far away, and thus any message will take infinite time to reach its destination. Apr 25, 2018 at 3:08
• @DLosc "The Head" thus may as well be considered religion or folklore, rather than something that can actually have a direct impact on the story. Apr 25, 2018 at 3:09
• What if the head is infinitely large, and therefore takes in infinite nutrients? Apr 25, 2018 at 18:54

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.

• The Mandelbrot set is definitely infinite and it has a "head" however since it goes infinitely small, it would look as a de facto finite thing right? Apr 24, 2018 at 14:06
• Could be an issue if this world has quantised space. In that scenario the fractal can only occupy finite space. Apr 24, 2018 at 14:33
• In Terry Pratchett's Discworld series, there's a butterfly that has ragged edges. Due to the fractal nature of the universe, this has become the "Quantum Weather Butterfly" (for the "a butterfly flaps its wings and creates a tornado." ) This is actually true of this tiny little butterfly. Perhaps ragged edges? Apr 24, 2018 at 16:41
• As you're saying: You can only fit an infinite object of N-1 dimensions in a finite space of N dimensions. Hence, the creature may be an infinitely long line (zero width) in a limited area of the 2D world, but it cannot be 2D. Although... there are fractal dimensions... Apr 24, 2018 at 19:07
• @SilverCookies The outline of a 2 dimensional fractal is infinitely long. One of the consequences of this is that if you treat coastlines as actual fractals, instead of close approximations, every coastline has the same length. Your creature could reproduce smaller infinite creatures with less area but the same infinite boundary (just like the Mandlebrot set has miniature Mandlebrot sets embedded in it). The skin can expand to encompass any area you wish as it eats and grows. Apr 25, 2018 at 5:53

The first thing that came to my mind was an infinitely long giant centipede. It can be tall enough to let other things pass underneath it, but they would have to be weary of it's constantly moving legs.

• You mean an infinipede? Apr 25, 2018 at 1:09
• Or a googolpede? A kajilliopede? Apr 26, 2018 at 16:14

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 it goes on infinitely in a finite space. Over the lifespan of the universe, your finite creature can grow infinitely. You can have any number of these ancient creatures in your universe.

Complexity to the creature design can be added with a combination of multiple knots. A mathematical knot is joined at both ends and cannot be undone. The simplest knot is a circle, or unknot. There are many different complex knots and new ones are still being calculated (similar to values of pi). I like to think of a Mobius strip as essentially a twisted unknot, although I do knot now if this would hold up to mathematical scrutiny. Some fractals can actually be considered "wild knots". A particularly complex knotted area can act as your creature's head, brain and any other limbs you may wish.

Your MobiusBody doesn't even have to be solid plane but can be holey, allowing other creatures to travel through without breaking the creature up into separate pieces. To allow the creature to defend itself, it can move the complex KnotHead around the MobiusBody. I've attached an image with the first number of knots, I leave the rest up to your imagination.

• The creature doesn't have to be finite; a line on a plane is infinite, and the plane itself is infinite, but the line doesn't take up the entire plane. Apr 24, 2018 at 15:42
• @Felthry, true in theory. However, the creature doesn't have to be finite...but should have a single head on one end? To me, that implies the OP does indeed intend to find or describe the finite end/head of this infinite creature. Hence why they are asking for shapes other than a 'line' that doesn't involve us playing an infinite game of 'snake' :) Apr 24, 2018 at 16:15
• How is this "infinite" while for example a torus isn't?
– pipe
Apr 24, 2018 at 17:34
• @pipe. as far as I am aware, a torus is defined as a tubular circle and has to maintain those circular dimensions. A Mobius strip and knots are circular but are not restricted to circle equations. They can be as squiggly as you wish. So while I specified that the Mobius strip was actually 'finite' it has the ability to have 'infinitely' more surface area than the torus. I think my edit explained my thought process.... Apr 24, 2018 at 18:11
• In mathematics, there are three things that match the everyday description of "line". An interval has a beginning and an end. A ray has a beginning but no end. It extends to infinity in one direction. A line has no beginning or end. It extends to infinity in both directions. The OP is asking for a ray-like creature with a head but no tail or an infinitely long tail. Apr 27, 2018 at 5:41

Immanent being.

https://en.wikipedia.org/wiki/Immanence

Immanence refers to those philosophical and metaphysical theories of divine presence in which the divine encompasses or is manifested in the material world.

This is a trippy concept. If you want a big snake that is infinitely long, or has infinite number of teeth this will not work. But your world already deals with the infinite. An infinite being would occupy the entirety of it. Immanence is a good companion concept for what is shaping up to be high concept fantasy.

Usually immanence is considered in the context of God - a conception of God is that God exists throughout the entirety of creation; not transcending creation but occupying creation. There is no reason your creature could not be like this. It exists throughout the entirety of what is. It is not the same as the world but it is inexorably intertwined with it.

You might or might not be able to sneak up on such a being. It is everywhere at once, but a cat can sneak up on my foot and I might not notice. You might or might not be able to attack such a being. An attack cannot drive it out of a place that it occupies because that is impossible; infinite-1 = infinite. But it might change the character of the being that occupies that place.

You might be able to kill such a being. The death of it will probably change the world it occupies. Remember: he's a god; it will take more than one shot.

• An infinite being does not necessarily take up the entirety of an infinite world... the two infinities are not necessarily equivalent. Otherwise, a really cool idea. Apr 25, 2018 at 16:57

This world could be filled with fractal critters. They can be as large or as small as you want, as they just have infinite detail: the closer you get to the critter, to more of it you can see.

Perhaps Menger sponges inhabit the oceans? They can be made into some cool carpets. Or perhaps mollusks with Apollonian gasket shells? An Ikeda map jellyfish? Mandelbrot manta rays?

I know you only asked for critters, but why stop there? An infinite world could also have fields of Mandelbulb flowers, crops of true Romanesco broccoli, Dragon island chains, caves with Sierpinski stalagmites, Pythagoras trees, and storms of Lichtenberg lightening and Koch snowflakes.

The options are, by defnition, endless!

• Infinite detail is an issue if space is quantised. It might not be in the OP’s world, but it’s worth considering. Apr 24, 2018 at 14:34
• @JoeBloggs: I assumed by the OP's Gabriel's Horn example an 'infinite surface, finite volume' critter was the goal, so infinite detail should be possible in this world. If not, then the Gabriel's Horn critter could not be infinite as the tail could only be as thin as the world's smallest quantifiable unit. Apr 24, 2018 at 14:51
• Good point! I didn’t think of that. Worth the OP realising though. Apr 24, 2018 at 15:48

A mist or vapor

Asking for the shape of an infinite body that isn't infinite is somewhat... definite. So we're trying to get as close to infinite as we can. The problem, of course, is that the body gets in the way of every other body in the world.

What would be the difference between such a creature and a ring of impassible mountains restricting access to the rest of the infinite world, making the world definite? After all, you're going to encounter this creature quite literally 99.99% of the time, so one hopes it has legs so you can walk beneath it... but then there'd be no sun...

Unless the creature is something more ethereal, something that can penetrate into every nook and cranny, extend from beneath the ground to the heights of the sky, something that people can walk through and still experience the entirety of this voluminous world.

Something that smacks just a bit of atmosphere...

Conclusion: the creature is a mist or vapor

Which also means you don't need to worry about where it's mouth is so that people can hear it roar from anywhere and everywhere.

• "you're going to encounter this creature quite literally 99.99% of the time" - Why? The world itself is infinite, it can therefore contain an infinite number of infinite beings and statistically you would never encounter a single one of them. Apr 24, 2018 at 15:54
• @Samuel, the problem with playing infinity games is that it generally comes down to philosophy. If two objects of infinite size exist in infinite space would they ever meet? The philosophy of mathematics says it's possible they never do. The philosophy of physics says they must be coincident and therefore always meet. You're welcome to choose your favorite philosophy, but in the end, it's the philosophy chosen by the OP that matters, which suggests he's looking for a being that simultaneously fills the world yet allows for the movement of others. My answer supported that apparent belief.
– JBH
Apr 24, 2018 at 16:19
• I don't think that's correct. In both mathematics and physics, infinity is boundless. You may be thinking of countable vs uncountable infinities. But that difference doesn't apply here. An infinitely large creature can occupy an infinitely large universe and still have room for infinite friends, but never meet one after wandering for infinite time. Apr 24, 2018 at 16:45
• @Samuel, If you don't like my answer, downvote it. If you have a better answer, post it. I'm not going to argue philosophy with you.
– JBH
Apr 24, 2018 at 16:47
• Your answer is fine, your assumption that the creature must fill all of space because all infinities are equal is what's wrong. It's possible the creature fills all of space, but it's far from required (or even likely). Apr 24, 2018 at 16:51

### Higher Dimensions

If you're amenable to more dimensions, then you could have something that exists infinitely, but takes up only a finite amount of our three dimensions.

It could be as simple as the 4th dimension being time, or perhaps something further; some other Nth dimension that I can't imagine, being bounded my my own existence in three.

• You could have it be a unit sphere in infinite dimensions. Infinite volume but everything is "close", so one head is feasible. Apr 25, 2018 at 1:09

An infinitely long serpent-like fish that flies through the air. The fish is inter-dimensional/inter-planar, so doesn't have to physically fit on whatever world it happens to be currently inhabiting.

I've seen this type of creature used in a campaign in the past (Party was in an airship swallowed by the fish, then the inside of the fish became the campaign setting and the characters had to escape).

• This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review Apr 24, 2018 at 17:28
• @Renan, while the answer could have been fleshed out more, it does technically answer the question. See the OP's 2nd to last paragraph.
– JBH
Apr 24, 2018 at 17:34

Do Superorganisms count? If so, you could have an infinitely large ant colony.

Since ant colonies can already grow arbitarily large, this isn't too far fetched. The fun part is the ant colonies could pull some Hilbert Hotel style stunts, if needed.

Since you specify only one head, we will say it has only one Queen, which is the "head" of an ant colony (head of state, that is).

This is very similar to a fun math book that was circulating on the web a couple of years ago: "Life on the Infinite Farm" https://www.math.brown.edu/~res/farm.pdf

It contains several thought experiments about what infinite animals might look like (as exercises in thinking about infinity).

• It probably does not answer the OP's question but +1 for the link to the book! :) Apr 25, 2018 at 7:16
• for more resources see: richardevanschwartz.com/farm.html May 8, 2018 at 20:56

According to myth, they were created when great trees displeased the Gods, and were cursed to lose everything above ground. And indeed, they do seem a little like plants - each one an immense tap root, wider than a house, drilling deep into the rock.

They branch. Branches snake off horizontally through the stone, beloved by miners because the bloody wood is easier to tunnel through than rock. The branch-roots always end somewhere interesting - porous stone saturated in oil, an underground river, or even a seam of precious metals that perhaps, eons ago, used to be a hot mineral spring.

But the main tap roots go only down. Perhaps they are nourished by magma. Perhaps they have found whatever lies beneath that. One thing we do know, is that they eat meat.

Near the surface, the root swells to the size of a sports ground, and ends in a flat top in the soil layer. Once every decade or so, a seemingly innocuous patch of grassland will collapse when a group of animals is walking across. Whatever triggered the attack - a herd of cattle, a pack of wolves, a legion of troops - all fall into an open maw. Huge tentacles sweep across the nearby ground, blindly grasping anything that escaped and hurling them into the corrosive mucous in the throat. Eventually they calm, and weave themselves again into a lid over the pit, to eventually be hidden beneath wind-blown soil.

Why do they do this? The amount they eat must be insignificant compared to the infinite body below. Perhaps it is indeed a curse. Perhaps they just like the taste.

For bonus weirdness, you could have a similar creature living vertically in the sky - like an infinitely tall sequoia whose trunk and leaves are held aloft by vast hydrogen balloons. Tentacles swoop down to pluck up their prey.

Male and female of one species?

What about a creature that has a limited height and width that is infinitely long and constantly traverses the landscape.

I'm imagining something like a giant centipede or worm where there's huge arches between each supporting foot of the creature. Large enough that if it happened to come across a building (or maybe even something larger like a city) it could just step over it. The legs could follow the exact same route through the air as they move into position. And each 'foot' would land in the exact same spot that the previous 'foot' occupied. Naturally the land underneath would become super-compressed over time. Perhaps people could find a way to divert the path of a section of this creature to use the land crushed by its feet as a really flat and solid foundation for building?

Anyways the creature would be infinitely long and each section would be supported by its own independent legs and feet.

I'm not sure about nourishment though. Maybe a symbiotic relationship with creatures that live on it. Fungi, animals and even plants could live on something this large that is always predictable. It follows the same path with the same timings. Perhaps it moves at the same rate as day and night? This really depends on the exact workings of the day/night system. But if some sections would be in day while other sections are in night, then you could maybe have portions of this creature that are constantly in daylight and portions that are constantly in darkness. This could lead to variations in different sections of the creature and the symbiotic creatures that live with it.

There's a lot of things you could do with this sort of creature as it's path could be very random meaning there'd be huge sections of land without any worm and huge sections of land that might be governed by the nearby worm paths. It'd be large enough in height that it's not insignificant and is difficulty to damage. How exactly would it recover from damage though? Perhaps some sort or merge with other pieces of the worm. That's up to you.

I just like that in an infinite world there's a creature equally unending that affects almost every part of this infinity, helping form the civilizations and creatures that live in all areas of that infinite world.

# Yes, a snail

Consider a snail: an initially tiny snail continuously grows new shell at the mouth, which grows wider and wider. The tiny central coils of the spiral are just the snail's shell from when it was younger. If you zoomed out, it would look the same as a younger snail.

# As long as we redefine matter

Snails in our universe had finite origins, and thus an initial finite size. Any real snail would have a minimum size as a result of the atomic structure of matter.

But in your abstract universe, given the other infinities, we can change the structure of matter to have no atoms, no indivisible pieces. Turtles all the way down. In that reality, the snail can have infinite age, whilst remaining a finite size; the spiral could go inward forever.

# Infinite animals cannot be made of atoms and live

One key issue with any infinitely large creature (in any dimension) is that any signal or biological process would be propagating forever from the head. So while it could start to move in some direction, it could never finish moving. And any nutrients consumed at the head would have to be only asymptotically absorbed in an ever diminishing tail (or the tail would starve). So I think any infinite animal would have to be asymptotically small in the older direction and thus require non-atomic matter.

# Snails live in the outer chambers

A snail neatly resolves the issue of moving an infinite animal; though infinitely long in the spiral, the finite size and mass of the animal permits it to still move, to still procreate, etc. You have a choice whether to have parts of the animal still living all the way down the spiral, or whether to have an infinite body containing a finite lifeform living in the outer chambers.

# Gabriel's shortened horn

We can make Gabriel's horn shorter too. Instead of the animal growing linearly along the x axis, we make it grow exponentially along the x axis. So regressing backward in time, every year it grew half as much as the previous year. This is the frog jumping half as much each time (albeit backwards) and has finite size.

# Meet the family

One issue with an infinitely old animal is reproduction. If I am infinitely old, then I wasn't born, so I cannot have parents in that sense. Nor can I have children that are born and are also infinitely old. However, if I was a division from another infinite animal, then I too could divide to yield offspring.

This would mean that as long as we match our infinitely-divisible matter with some mirroring phenomenon by which any animal could divide or replicate (down to all its infinite detail), then we have a way for a whole family of infinite snails to exist independently from each other yet still related. I can imagine some mirror or portal thing which duplicates an animal while halving its physical scale; the total finite amount of mass & volume is conserved, just the snail comes out half the size (as it is fractally self-similar).

To understand the process you need number theory and the infinite hotel. But the effect is similar to cell division.

# Two become one

On the flip side two animals could merge to become one twice as big; by a process reversing that of the division, the combined animal would be an infinite merge of the original two.

This gives as most of the components of reproduction as we know it, except one: a mechanism for variation.

# A finite change

Our snails grow in time, and as with real snails, turtle shells, tree rings, you can look back in time by looking at the historic growth of the organism. If a snail underwent some change at random intervals (voluntarily or not), then it would be visible somewhere down the spiral, even though the spiral goes on infinitely long. This would then make one snail distinct from another snail, even where they were divisions of the same animal in some earlier time.

This mechanism gives us a way to have individuals, and a reason for the merging process; the snails have a way to be different, and a reason to choose one snail to merge with over another. The differences may be random (analogous to mutation) but the selection need not be.

# One common ancestor

All snails can have derived from a common ancestor snail; in effect a division creates twins, so whilst they can ultimately yield lots of different individuals it would suggest that looking backward down any given animal's shell you could find an original snail where all snails are the same beyond this point.

The longer a snail has gone without dividing (compared to other snails), the larger it would be (given the division mechanism), so one half of the original snail could still exist alongside a larger number of smaller snails who had gone on to divide and sum separately.

# Voila a snail

An infinite animal, complete with reproduction, ancestry and family.

An infinite animal can be a snake, that starts somewhere, and then just extends, e.g., east, infinitely.

There are infinite problems with that animal. Not least of which, that (assuming a random placement of other creatures) infinitely many creatures won't encounter it, ever. If it has huge girth, there is the weird problem that inifninitely many people will be hindered in their mobility (because the snake lies around like a huge wall), while at the same time infinitely many people will never see the snake in their life (because they inhabit the infinitely big part of the world that the snake is not in life-time-wandering-distance of).

If it is coiled, or similarly distributed that its (basically) 1-D body is still all over the 2D-landscape (or even the 3D heavens) it may be encountered by everyone, and, given the interstices between coils are large enough) also avoided.

Fun fact: if the scales on the snake are randomly colored, there will exist a part of the snake that has a red arrow pointing exactly into the direction of the snake head, with an exact distance to the head in inches given, in orange, and small bold black print under it "[protagonists childhood nickname that just her siblings knew]: Go talk to the head".

• That last part could be an amazing plot device May 28, 2018 at 10:09

My own point of view of your problem:
Your creature should have infinite size. Either of surface area or volume, but it doesn't really matter to you. You just want it big. So immediately, peano curves come to mind. However, I don't think this quite fits with the spirit of your question. So first thing to consider: what does life need to live?

1. Life can reproduce itself,
2. Life can respond to stimuli, and
3. Life can grow.

There are other qualifications, but they are debatable, and it is easy to imagine a type of life without them. So tackling each one by one:

## Reproduction

Life needs to reproduce. How it does this is up to you, but it can't be a stand alone organism. Your organism probably needs to have various stages of life, early stages in the creature's life cycle involve moving as far away from the mother organism as possible. Middle stages involve growth and reproduction. Finally (this part is optional), towards the end of its life, it shrinks and prepares to die.

## Response

This one's probably harder. An infinitely large organism would certainly have issues with responding, especially with a centralized nervous system. It would either have to react by reflex, or have separate "brains" at different spots. Each "brain" could in turn connect to a larger "brain", which connects to larger "brains" ad nauseam. How you tackle this problem could have huge effects on the sentience of the creature. i.e. the fractal network of brains could make the creature incredibly intelligent (certainly smarter than most humans), whereas reflex would make it incredibly stupid (about on scale of an insect).

## Growth

This is the most difficult of your concerns. If I had to take a stab at suggesting a growth pattern, I'd suggest exponential growth where each step occurs in half the time of the last step so that it grows in a finite period of time, however how it gets the nutrients for this I don't know. Perhaps through sketchy, hand-wavy, use of light splitting into antiparticle pairs and then doing something with this, but I have no clue how you plan on discussing this if you even will. This part will take some serious ignoring of laws of physics to work.

## How I'd do it

Disclaimer: I don't know what your goal with this is, so this is just my own ideas put into an example.

So, when the creature is first born, it is a small single celled organism, it whips a little flagella attempting to get as far away from it's mother's main body as possible, as it travels along its mother's body, eating anything it can endophagize (I'll pretend that's a word). It keeps swimming until it reaches its next phase in life during which it undergoes rapid mitosis and begins to form a tendril system. These tendrils grasp at anything they can and secrete digestive fluids onto what they have, slurping the resulting fluids to the main body, which is beginning to form a central brain. The creature is about the size of a cat.
The creature grows at a super fast rate as the tendrils begin to stiffen at the older spots. At stiff spots, chloroplast like organelles form to collect sunlight. The tendrils at the end begin to branch forming more mobile tendrils at each branch, smaller brains are formed each connecting to the last. The creature grows like this for almost a century, at which point it is infinitely large, it's tendrils wrapping around others of its kind forming a sort of large octopus-y forest like formation. The creature is definitely sentient, but lacks much movement capability.
At some critical point, the main brain decides to die, and shuts down, killing its immediate surrounding parts. This releases new baby versions of itself which race down the tendrils at the same speed as it dies. Each brain of the original creature activates its area's death, and releases food and nutrients for the young.

It's infinite in higher dimensions. Any shape in higher dimensions would do. Perhaps infinite in all higher dimensions and not always in our lowly ones.

At any point it can project itself down into lower dimensions and take up that space in it's entirety.

This answer is inspired by Eldrazi, and although you don't need to follow this pattern this excerpt might be helpful:

Each Titan lives outside of the planes. When one wants to feed, it extends a part of its "body" into the plane, to create a physical manifestation of itself there, as well as an army of drones that are extensions of its body and will.

The Spirit Dragon Ugin compared this to a man sticking his hand into a pool of water; the man is the Eldrazi Titan, and the water is a plane. The fish--those who dwell on the plane--see only a part of the man--his hand. Likewise, the inhabitants of a plane can see only a part of each Titan. Even if the Titans appear to be independent beings, their physical forms are just part of a greater entity outside the plane. The same is true for every drone that the Eldrazi had created; they are all just part of the Titan that made them.

• This doesn't appear to answer the OP's question, "What infinite shape can I give my organism...?" Please do not assume people will follow links or that links will exist forever. Provide enough information in your answer to be self-contained. Thanks.
– JBH
Apr 24, 2018 at 16:50
• @JBH I could improve this by suggesting some higher dimensional shapes, but the answer is only inspired by the linked article, and isn't needed to fully understand the answer. I'll try to fish something out anyway Apr 25, 2018 at 9:49

Interesting/Horrific Idea:

Your creature is not one being but a branching collection of infinitely long snakes/dragons/centipedes. Each creature has finite width and is infinitely long but at any given time has only a finite length exposed. The remaining length is still coiled up inside the creature's 'mother' as it can never finish dragging its entire infinite length out. So travel long enough from the head you hit the mother creature. Travel long enough from the mother's head you eventually reach the grandmother and so ad infinitum. No member is infinitely old but the age of the members is unbounded and the entire family can be said to be infinitely old.

Further Idea: Travel deep enough down the generations and the last mother disappears into the ground. We are living on some 'layer' of this creature. The day/night cycles are caused by it moving.

Three ideas:

1) A dome creature that messes up spacetime. Imagine this: if you are a mile away, it looks like a fairly ordinary dome, in the distance, or what have you. But the closer you get, the bigger it appears. It occupies infinite space in every direction, but is contained in compactified spacetime. It's just like how 1/x approaches infinity at 0. Right up against it, if you put your eye on this beast, it would look as though it never ends. For this to make sense, as you got close, the appearance of the rest of the world would have to bend backwards, so even what was in front of you would now be behind in some direction. This beast could easily have a well defined head at one particular location. There could be as many of these creatures as you like, so that they're relatively popular on the land. (it doesn't have to be a dome. Any creature shape could theoretically do this)

2) An infinite collection of somehow-connected finite things. I imagine an infinity of spheres. It could be that they all are managed like limbs from one head sphere. They, together, are 1 entity. Or maybe they all have a head in their centers, that share consciousness, but if N of them are destroyed concurrently, they all "die". This would necessitate they should be able to regenerate over time, if one is destroyed in some way.

3) An ethereal being that is omnipresent, but is physically accessible by some imaginative way.

If we borrow from mythology...https://en.wikipedia.org/wiki/J%C3%B6rmungandr

...then your creature could be a large serpent. It has a head but is still infinite because it has swallowed its own tail.

But is that satisfactory as "infinite" by definition? Sure, if inside it's mouth is a wormhole that extends the creatures length indefinitely.

Or, what if your creature does NOT have a head, at least not in the traditional sense. Yours is a circular cylindrical creature that has eyes and mouths and ears all over its body. At any given point of its infinite length it can see, smell, hear, and bite.

How was a creature like that born? Which end came out first? Well, you could leave up to myth and the readers imagination, OR the whole creature came out of its parent all at once, already circular in its shape. That's how its cells formed when it was in the womb of whatever birthed it.

Or if this is fantasy setting then just magic it.

As for it not being vulnerable... the creature could be so large that inhabitants of your world think its a mountain range. Any "earthquake" would destroy even an army of attackers.

Well, since your world is also infinite, the shape of the creature doesn't really matter. There is enough room in your world for infinitely many of said creatures. So the answer should be: It looks however you want it to look.

It could be shaped like a cat with an infinitely long tail, infinite hind legs on an infinitely long body, infinite front legs to support its infinitely wide shoulders, or infinite teeth in its infinitely huge mouth. Whatever the case, it just doesn't end in at least one dimension, same as your world. Since it has to be a world where our physics don't apply, our understanding of biology doesn't really matter. It could look as if made by Picasso or imagined by H. P. Lovecraft.

You can imagine all sorts of creatures, as long as they live underground. People and everyone else will be able to walk all over it without even noticing what lies beneath their feet. Give the body enough holes (or even make it web-like) and the underground denizens won't be bothered by it either. Since the ground extends down infinitely, the creature can also grow very big in all 3 dimensions. You can hide the head wherever you like. :)

Bonus story opportunities:

• The head could be yearning to experience the sky and flight, thus trying to move its body out (with all sorts of disastrous story-fuelling side-effects).
• Since it's impossible to see any sizable portion of the creature at once, nobody (including the reader) could realize that it's the same creature until a big "revelation" later. Until then - well, you do come across these "big burrowed snake" creatures every now and then, but they're few and far between so nobody has made the connection that they're not actually distinct entities.

## Infinitely Large Head

If the animal is infinitely large, it will need an infinitely large head with which to gain infinite nutrients. To do this, simply take an regular animal, and make it infinitely wide (and also tall enough to allow things to move underneath it when needed). So it has infinite teeth, infinite stomachs, infinite eyes, an infinitely large brain (with many redundant parts), etc...

Part of the problem is that infinities are somewhat counterintuitive. The infinite hotel is a classic example. (*) You want an infinite being that doesn't take up everywhere and lets people wander round, and doesn't get too thin? Nothing could be simpler!

Your being is a conical being, which slithers along the ground, and is obligingly flexible so it can slither over other infinite beings. Its head is at the bluntly rounded apex of the cone, and cones of course have infinite volume, if they aren't truncated. There is no "thin" point, and it takes up an infinitesimal part of the world.

(* a hotel has an infinite number of rooms, and is 100% booked. If one guest turns up, move everyone into the next higher room, and the new guest can stay in room 1. If a (countable) infinite number of guests turn up, move every guest into a room of double their present room number, which frees an infinity of odd numbered rooms for all the new guests. If an uncountable number of guests turn up, buy stocks in Cantor's AirBNB business and quit the hotel world!)

The Haze

The universe is filled with a mist like substance extending into infinity, this mist is in fact the diffuse body of a super-organism, able to form denser pockets to serve its various biological purposes, filled with flora and fauna which fuel its biological processes in a similar fashion to the gut-flora of more conventional creatures.

The Head is a single location in the universe where The Haze started, a literal Genius-Loci. You can find the location of the Head by following an infinitesimal gradient of density in the Haze, or more conveniently by following a faint flow of energy and material which fuels The Head.

Think Steven King rather than Lovecraft, The Mist, but the mist itself is a monster as much as the creatures within it.

The answer is coral. It is an animal, exists and uninhibited could grow to an indefinite size as an animal. Unlike a tree it can be cut to make more separate animals or reproduce to have larvae.

## A Hive Mind

As hive minds work, it can expand infinitely, just spreading from one single mind to another.
To really be infinite, it should be able to assimilate any kind of living species.

Let's call it:

## The Knowledge

It is actually built from every living species on your infinite world and is actually looking forward to assimilating samples of every living thing.

Since the power of an infinite mind is monstruous it wants to acquire the most knowledge from the whole universe.

It has no real reason to assimilate every single mind since their evolution is a source of infinite knowledge for it.

A hive mind actually has only one "head", aka mind but it cannot be cut out so it's really infinite in space in time.

It is seen by other living organisms not as a predator but more like a god of knowledge which makes them the greatest gift, by assimilating them from time to time (they gift their minds to live forever in the hive mind).

The hive mind is not blocking any dimension on earth or in the sky since it's almost everywhere and nowhere at the same time.

There are no real animal instincts like on Earth, where little turtles know from the beginning they should run to the sea. The watcher actually teaches the young with the single beings he assimilated from those species to let them know the basics of their species.

When the species grows communals the humanity dies, it lets them teach each other (as we do) and give them great lectures from those that have been assimilated (even eons after their physical death). This can even give the communal species a boost in innovation and actually secure all knowledge that has been assimilated until the end of time.