# what alternative topological shapes could alien life take on?

I've always been intrigued by the idea that alien life would be completely different from life on this planet. recently, I've been doing some research and found out that humans, along with other vertebrates and even some invertebrates are topologically shaped like toruses. but there are also some invertebrates that are topologically shaped like spheres. and it made me wonder, what other possible shapes could life on other planets appear life?

• There are several books or tv series produced on just alien life on a single planet. Please narrow this down a lot, if you want it to stay open
– L.Dutch
Dec 17, 2021 at 9:05
• basically: every shape. or even shapeless - imagine a sentient heap of goo. Dec 17, 2021 at 9:11
• @L.Dutch: The question is explicitly about topology, not about the geometry of the alien beings. A learned discussion of whether is conceivable to have a living entity topologically equivalent to a thickened tame non-trivial knot would be quite interesting. Dec 17, 2021 at 16:27

# Not Much

An animal in three dimensions must occupy all 3 of these dimensions

Given this, it seems like the only topological shapes possible for life would be the ball, or some variation on the solid torus

However, there are still a few things you can play with on the topological level. For example, you could give your aliens an even genus (where there are an even number of holes through the body, which could be an interesting constraint to work in

• I think there are many. Dimensions: trees are fractal, which is a broken dimension. Snake and worm actually cover no more than 2 dimensions. In 3d, you have cylinder (another way of looking at the snake)... you can have tetrahedron.. there are many basic geometrical shapes that can be built on that. A variation in limb count could be regarded as "topological variation". What about head count.. if you take function into account, not only topology, it becomes huge. Dec 17, 2021 at 12:48
• @Goodies A snake or multilimbed creature would still be a multi-holed torus if the topology is based on the digestive tract. A tree being fractal is interesting but it may topologically be a sphere or an highly multi-holes torus. Dec 17, 2021 at 13:22
• This touches the subject of topology. Mathematically, a tree is definitely not a sphere or torus. But nearly all lifeforms on Earth are based on the cylinder, even the octopus could be regarded as a cylinder: mouth, anus, space in between.. the limbs are extensions sprouting from that basic shape. If you go look at alien life, I don't think it is so limited. An alien lifeform could be a cloud-like shape shifter with no fixed topology at all. I don't think shape is really limited, when you propose any alien life with gravity X, evolution Y under atmospheric circumstances Z, all unknowns. Dec 17, 2021 at 13:44
• It depends on why the OP has the definition of a sphere or torus, I assume a sphere is a cell so it can absorb nutrients across its surface, similar to a tree as it draws from its leaves and roots. A cylinder is just a stretched out torus and it is too simple to say life is either shape as animals have more than just 2 holes, there are 9 or more. Dec 17, 2021 at 13:51
• @Goodies: In topology, there are no tetrahedrons or cylinders; those things are balls (aka spheres). Topology is not geometry, and the question is about topology. Dec 17, 2021 at 16:20

Handlebodies

From a topological point of view, the only possibilities for an animal (or any solid body) are:

Solid Ball:

Solid Torus:

Solid Double Torus:

Solid Triple torus:

and so on.

Of course two topological types might look different. For example the torus is topologically equivalent to a coffee mug:

and the triple torus is topologically the same as the 3-handlebody:

So your creature is always a ball with some tubes removed. The number and arrangement of the tubes determines with topological type you get.

A simplified model of an animal with one tube removed going from the mouth to anus is indeed a topological torus.

But a real animal has more than one tube missing. For example the urethra adds a second tube connecting to the first tube near the end; the nostrils add two more tubes that connect to the first tube near the start (hence you can spit water or hot coffee out your nose) and the ears add two more tubes that connect to the start (hence you can swallow to relieve pressure on your ears).

The upshot is that topological type is not an interesting way to describe animal shapes. For example an octopus with a single digestive tube and no ear canals etc is the same topologically as a cow with no ear canals etc.

The extra limbs (protrusions) do not make a difference to the topological type because you can always stretch a rubber shape without tearing to make extra limbs.

Likewise intrusions like the uterus and lungs don't change the topology since they go inwards but not out the other side.

That means a stomach that takes in food, digests and absorbs nutrients, then spits waste back out the mouth hole does not change the topology either. If there are no other tubes then the animal is topologically a sphere.

In fact topology cannot even distinguish between an (imaginary) animal with two unconnected digestive tubes -- perhaps one for food and one for water -- and an animal with a single large tube (digestive tract) that goes all the way through, and a second smaller tube (urethra) that connects the outside to the first tube!

This is what you should play with to make your aliens truly alien. Having a body plan where some of the tubes do not connect to each other. This is not a topological notion but should be easy to understand. Formally you could describe it by designating an "inside" and "outside" of the animal and only allowing us to stretch and squeeze the inside and outside separately.

I think the obvious outlier is the octopus; they are extremely intelligent problem solvers, invertebrates, eight arms, three separate hearts, five lobed ring brains, distributed brains (they have neurons in their arms) and they can solve complex puzzles human six year olds cannot solve, and they can use tools, even by example: A care giver for an octopus accidentally dropped a scraper into the tank; and the octopus grabbed it and began using it, as intended, to clean the inside glass of its own tank.

I don't think being water animals should restrict them; in fact it means they are already adapted to a near weightless environment; like outer space. No artificial gravity needed.

And just like we have to carry our air with us in space, they'd have to carry their water with them, but that is not a show stopper. Just a heavier lift.

Octopi are actually much more adept and dexterous than humans at manipulating objects; their suckers mean they can easily grip an object firmly without any handles -- We are restricted by what we can wrap a hand around; they are not restricted to such small sizes.

If octopi had human levels of intelligence, I think they have all the physical attributes needed to create a fairly sophisticated undersea civilization, with science and advanced mechanical technology.

Note that octopi can also survive for up to minute out of water. They cannot breathe so they are "holding their breath" in a way, depending on oxygen already in their blood. But they can come out, manipulate or grab things, and even navigate "walking" to other tidal pools, crossing a few yards.

Kind of like us being able to navigate under water for a short time.

No reason a human-intelligence octopus could not invent a mechanical pump and a "breathing" apparatus that circulates fresh water in a "land suit" that leaves their arms free and allows them to work on land for extended periods of time. Much like our dive suits.

I would not discount octopi-like aliens.

• Topology is not geometry, and the question is about topology. An octopus is topological torus just like a human. Dec 17, 2021 at 16:23