# What would organisms be like in four physical dimensions?

In this question, I asked about a matter-filled 4-spatial-dimensional universe, with a gross structure defined by my own answer to that question.

Then, in this question, I asked about what the periodic table would look like, assuming that something like classical atoms could exist.

Now, assuming that there is a 4D Carbon-analog element that can form complex structures alone and in combination with other elements, what can we expect living organisms in such an environment to be like?

We can assume that there are generalities that apply in both 3D and 4D, such as autotrophy/heterotrophy and reproduction.

These 4D organisms do not need to interact with a 3D-universe — they exist and interact solely within their own 4D-universe.

Specifically:

1. What — aside from occupying an extra physical dimension — would need to be different between a 3D and a 4D organism? What is likely to remain the same?

2. What would a 4-dimensional lifeform be able to do that a 3D lifeform cannot? What new structures are possible and are any existing structures more efficient in 4D than in 3D?

3. What can a 3D lifeform do that a 4D lifeform cannot do? What 3D structures do not function and are any existing structures less efficient in 4D than in 3D?

4. Are any symmetries more likely in 4D than in 3D?

• Well, have a +1 from me for asking an awesome question that goes way over my head. Commented Oct 14, 2015 at 3:41
• maybe just maybe a 3 or higher dimensional(excluding time) entity will perceive and experience gravity differently, what I'm saying is that the entity might never know what a wall looks like let alone painting it ;) Commented Oct 14, 2015 at 5:11

Well, the first obvious difference would be that the square-cube law would basically be replaced by a cube-tesseract* law. Since 4/3 is closer to 1 than 3/2, this means that there could be greater variations in size. Another difference is that more of the (hyper-)volume is close to the (hyper-)surface. Since this means that important organs are closer to the surface, probably organisms would grow thicker protective skins. On the other hand, plants would probably profit from the increased surface/volume ratio, as they could keep relatively compact form and yet present a large surface to the light. Therefore I guess leaf structures would be less common, or otherwise, plants developing those would have more energy at their disposal, allowing them to evolve some more sophisticated abilities.

Since the floor would be three-dimensional, you'd need at least four legs on the floor for a stable standing (instead of three, as in our three-dimensional world). Therefore for an insect-like moving pattern, you'd need 8 legs. Coincidentally, if you extrapolate the four-legged pattern of the land vertebrates to four dimensions, you also get 8 legs. Therefore I conjecture that eight-legged animals would be common in four dimensions. Probably the typical four-legged animal would be long on one horizontal axis, and when looked on from above (so you get a 3D projection, similar to the 2D projection when looking at out animals from above), you'd basically see an ellipsoid with four left attached close to the front, and four close to the back, each arranged in a square. A stable standing would be achieved with two diagonally opposed front legs, and the two back legs on the other diagonal. From that fact, possible walking patterns could be derived.

When moving to an upright position (similar to humans), those creatures would walk on their four hind legs (interestingly, unlike our two foots in 3D, those four legs would still be enough to provide a stable stand without the help of specially formed feet, if positioned right), and have four arms available for manipulating things.

*) A tesseract is the four-dimensional equivalent to a cube.

• Have you heard of dynamic stability? That being the reason that humans don't generally fall down despite having only two legs in a 3d environment. Could a 4D creature have one, two or three legs? Commented Oct 18, 2015 at 22:22
• As I wrote, it's just a coincidence that you get four-legged "humans". What I did was to extrapolate the symmetry from 3 to 4 dimensions. However thinking about it, the extra axis would also allow for a rotational symmetry orthogonal to the forward-upward plane for the vertebrate animals, which would then lead to other multiples of two legs. I think it's still likely that you have the same number of front and back legs; the front legs would then turn into arms for upright beings. I don't think less than 6 limbs (three front, three back) would be reasonable, though. Commented Oct 19, 2015 at 20:32

Radial Symmetry in 4d would be like bilateral symmetry in 3d. A life form living in four spatial dimensions that has legs would likely have radial symmetry instead of bilateral symmetry. They would likely have at least five sides as five is generally the minimum number of sides for any type of radial symmetry on Earth. An organism in 4d could also rotate as it walks without turning around because it can rotate around its sides.

One thing that would be different is that a river would not be a natural boundary for none avian life forms in 4d meaning that rivers could not cause speciation events in the way they do in 3d. This means that sympatric speciation events would likely happen less often in 4d as natural boundaries would be less common in 4d. Speciation events can result from selective pressure as well though and I'm not sure how the rate of that would be effected by 4d. So one thing a 4d life form could do that a 3d life form could not is get around a river without going over or under it.

In 4d it would be possible for two organ systems to pass through each other without ripping each other apart. Hair getting tangled up would not be a problem so long as the hair consist of one dimensional strands because if it formed knots the knots could simply be untied through yanking.

In 4d the surface area to volume ratio is slightly larger than in 3d meaning that things would have slightly less volume per surface area than in 3d. This would barely effect life though because the difference in the surface area to volume between 3d and 4d is only 4/3 meaning that the difference in cell size between organisms living in 4d would likely be greater than the difference between cell size in 4d and 3d.

A brain would need to be larger and more complex in order to process information from it's environment.

• Hair would not be 1-dimensional, it would be 4-dimensional, but thin. Can a 4D object pass through itself? I.e. is it possible to tie a 4D object in a knot? Commented Oct 14, 2015 at 4:50
• @MontyWild I'm not sure if useable knots would exist in 4d. In 4d it would technically be possible to tie a string in a knot but the knot would be essentially useless as it could untie without having to go through the loop do to the extra dimension. I'm not sure whether it would be possible to tie a plain into a useful knot in 4d. Commented Oct 14, 2015 at 5:06
• Useable knots should be able to exist in 4D. To take a simple example, take an arbitrary knot in 3D (x,y,z), extrude it by some small amount into the 4th dimension w. Now, we know it can't move freely in the z dimension, or the 3D knot would be easily untied. So now let's take the 3D slice (x,y,w) for arbitrary z. Assuming I'm visualizing this right, we're now looking at a 3D knot just like the original, so moving in w shouldn't untie it easily either. Or to phrase it slightly differently, the 4D knot is a connected stack of identical 3D knots.
– Ray
Commented Oct 16, 2015 at 0:38
• A 2D knot would be impossible, since there wouldn't be a way to create the knot in the first place without one part of the string passing through another part.
– Ray
Commented Oct 16, 2015 at 0:40
• @MontyWild Strings cannot form knots in 4D, but sheets can. And that includes sheets that have been rolled up into tubes. So if you have tubular hairs / ropes, a 4D creature could tie them into knots. Commented Jan 13, 2018 at 2:11