Thanks for the invitation. Because we, humans, are limited to a world with three spatial dimensions. As for the rest, 'theories do their best'...

Thanks for the invitation. Because we, humans, are limited to a world with three spatial dimensions. As for the rest, 'theories do their best'...

Dear Saeed:

Since our perceptive and cognitive schemes have been shaped by interacting with a spatial world of three dimensions, our mental images can hardly go beyond that.

However, we can represent an indefinite number of dimensions mathematically. What we cannot do is to represent them intuitively.  

There are however some "tricks" that can approach us to an intuitive representation of more than three dimensions. In the movie Interstellar, for example, the main character navigates a structure (built by humans of the distant future) in which time has been turned into a spatial succession of "rooms". In each room the navigator is able to see events that have occurred at different points in time. 

The Fourth Dimension

Have you ever considered how a four-dimensional being would appear to you if it suddenly came into your room?

I think they would resemble flesh-colored balloons constantly changing in size. We would see disconnected blobs of flesh, some attached to claws, teeth, lips, eyes, or hair. Let me tell you why.

Think of it this way. Consider a two-dimensional world resembling a sheet of paper. How would you appear to the inhabitants of such a world if you tried to interact with them? The 2-D creatures would only see cross-sections of you as you intersected their universe. Your finger would look like a flat disc that grew in size as you pushed it through their world. Your five fingers might looked like five separate circles. They would just see irregular shapes with skin boundaries as you entered their world. Similarly, a hyperbeing who lived in the fourth dimension would have a cross-section in our space that looked liked a bunch of skin blobs.

A 4-D being would be a god to us. It would see everything in our world. It could even look inside your stomach and remove your breakfast without cutting through your skin, just like you could remove a dot inside a circle by moving it up into the third dimension, perpendicular to the circle, without breaking the circle.

A hyperbeing can effortlessly remove things before your very eyes, giving you the impression that the objects simply disappeared. The hyperbeing can also see inside any 3-D object or life form, and if necessary remove anything from inside. The being can look inside our intestines, or remove a tumor from our brain without ever cutting through the skin. A pair of gloves can be easily transformed into two left or two right gloves. And 3-D knots fall apart in the hands of a hyperbeing, much as a 2-D knot (a loop of string lying on a plane) can easily be undone by a 3-D being simply by lifting the end of the loop up into the third dimension.

What Do You Think?

Find out all about the fourth dimension in my book Surfing Through Hyperspace: Understanding Higher Universes in Six Easy Lessons (Oxford University Press).

Click here for more information on this book. In the book, I answer such questions as:

1. What would a being who could move in the fourth spatial dimension look like to us? Would these beings have teeth? If these beings evolved just like us, how would they look to us? Would they have internal organs like our own?

2. Would the beings look liked bags of skin or be more exotic?

3. What are the chances that we could learn to communicate at all with a 4-D extraterrestrial?

4. As a consequence of their access to a fourth dimension, what powers would seemingly amazing powers would they have? What would their sex acts be like? How might they think and act? How might we search for them?

5. If we encountered them, we would find they can perform levitation, bloodless surgery, disappear in front of our eyes, walk through walls. Nothing would be hidden from them, no matter how hard we tried to hide. Certainly if such a being was observed in biblical times, He would be considered a God with many characteristics of omniscience, omnipresence, omnipotence. How would an encounter today affect our world and religion? What is the religious significance?

6. Could God be a four-dimensional being?

7. Would we be able to hear the speech of a 4-D being inches away in the fourth dimension? How would sound work? What would the being sound like using 4-D vocal cords? What happens to photons in four dimesions? For example, how do 4-D beings see? We don't seem to be "losing" photons into the fourth dimension, so I'm not clear how 4-D creatures would use them to see us or themselves...

Some Random Thoughts

Is it hard to imagine a universe with four dimensions in which we are constrained to 3 dimensions? Could there be some mechanism constraining us -- such as being stuck to a surface via adsorption? If we were stuck to some surface in this manner, then the way free 4-D beings could interact with us would be constrained by that surface.

In our 3-D world, evolutionary mechanisms made us evolve into tubes. The earliest multi-celled sea creatures were tubes that could pump water through and pick up nutrients from the water as it went by. As life evolved, the primary structural differences involved the complexity of organs attached to the tube. We are still just tubes, but we carry a lot of that ancient sea around with us in the "inside" of a bag.

In higher dimensions, the fraction of volume near the surface of a baglike form increases dramatically compared to 3-D forms. Heat dissipation and movement of nutrients and oxygen (most of the blood vessels uniformly distributed through the bag will be close to the surface too) may be strongly affected by the proximity of the volume to surface. How would this affect the evolution of life? Would primitive 4-D beings may evolve so that they digest on the outer surface, with other organs, like brains and hearts not involved with nutrient an oxygen acquisition, deep inside? The outer surface would be digestive and pulmonary, and contain sense organs and orifices for excretion and sex. Is this tendency more likely as the dimension of the space increased?

Here is why the volume of a 4-D animal is more concentrated near the surface than 3-D animal volumes. Consider a D-dimensional sphere. The volume is

V(R) = S(D)*R**D / D.

where S is the solid angle in dimension D, and R the radius.

If you compare two spheres of radius 1 and 1 - a, where a is very small, the difference is

(V(1) - V(1 - a)) / V(1) = 1 - (1 - a)**D.

This is essentially the fraction of the volume within 1-a of the surface. Particularly, if you have a 10-dimensional sphere, then about 40% of the volume is within 0.05 R of the surface (a = 0.05). In a 4 dimensional sphere, about 34% of the volume is within 0.1R of the surface. In a 3D sphere, this is 27%. For that D=10 sphere, the fraction within 0.1R is 65%.

Here are some of my own images dealing with higher dimensions and spatial warps. Enjoy.

Klein Bottle

Banchoff Klein Bottle

Black Hole Embedding Diagram Showing Space Warpage

Gravitational Ripples Produced by Two Black Holes

Quantum Foam Embedding Diagram

Worm Hole Embedding Diagram

Explore the fourth dimension on your iPad or iPhone! Visit FourDimensionalApp for a cool iPad app for manipulating 4-D objects.

Return to Cliff Pickover's home page which includes the Wishing Project cataloging wishes from various cultures, computer art, educational puzzles, fractals, virtual caverns, JAVA/VRML, alien creatures, black hole artwork, and animations.

http://sprott.physics.wisc.edu/Pickover../fourth.html

My dear Saeed

I think, the human brain has evolved in a 3D world and natural selection has equipped us with software that can model and understand only 3 spatial dimensions. So, ingrained is this in the brain that when we see a 2D painting with perspective we cannot force ourselves to see it as a 2D painting.

Thank you

The theoretical physicist Sean Carroll wrote a blog post entitled "Why Can’t We Visualize More Than Three Dimensions?" (http://www.preposterousuniverse.com/blog/2009/03/30/why-cant-we-visualize-more-than-three-dimensions/) There he explained as follows:

"[…] why can’t we visualize things in more than three dimensions of space? Could a Flatlander, living in a world with only two spatial dimensions, learn to visualize our three-dimensional world? Could we somehow, through practice or direct intervention in the brain, train ourselves to truly visualize more dimensions?

I can think of a couple of explanations why it’s so hard, with different ramifications. One would be simply that our imaginations aren’t good enough to project our consciousness into a constructed world so very different from our own. Could you, for example, really imagine what it’s like to live in two dimensions? Sure, you can visualize Flatland from the outside, but what about asking what it’s like to really be a Flatlander? The best I can do is to imagine a line, flickering with colors, surrounded by darkness on either side. But the darkness is still there, in my imagination.

The other possible explanation is that the process of visualization takes up a three-dimensional space in our actual brain, preventing us from “tuning a dimensionality knob” on our imaginations. The truth is certainly more complicated than that (and I’m not experts, so anyone who is should chime in); the visual cortex itself is effectively two-dimensional, but somehow our brain reconstructs a three-dimensional image of the space around us."

Dear Saeed Al Rashid,

We used for our universe three dimensions - length, width and depth. We can imagine how would look in truncated dimensions - a plane in 2D or 1D along the line - but imagine how things would look if a greater number of measurements, it is difficult (if not impossible). We just can not imagine how something is moving in the direction, which is as it is not part of our concept of space. Our universe is inherent in the fourth dimension (time), but it also has only three spatial.

For someone in a three-dimensional world, you could do incredible things - in many respects - would have made you a god-like:

you could teleport from one place to another, disappearing in one place and appearing somewhere else;

you could rearrange or delete other people's internal organs, performing surgery without the need to open someone's body;

you could just remove someone from a three-dimensional universe in which he lives by putting it through some time in another location of your choice.

In the context of general relativity is very simple to build spatial and temporal framework within which the number of "large" (ie, macroscopic) measurements have changed over time. You could not only have a large number of measurements in the past but in the future you will be the

It may fall a chance; you even could build a space-time in which the number will fluctuate, changing into a large and downwards over time, again and again.

If we reduce or increase the number of measurements, we would change the way, for example, extend the line of the force field.

If it affected electromagnetism and the nuclear forces, to this disaster.

Therefore, to dream about the appearance of the fourth dimension is not exactly worth it. However, there is a positive note. We did not have to worry about global warming, as the increase in distance from the sun would greatly dampen our world faster than the rising atmospheric carbon dioxide heats it.

Regards, Shafagat

Algebraically, it is generated by applying the rules of vectors and coordinate geometry to a space with four dimensions. In particular, a vector with four components (a 4D vector or 4-tuple) can be used to represent a position in four-dimensional space. The space is a Euclidean space, so has a metric and norm, and so all directions are treated as the same: the additional dimension is indistinguishable from the other three.

In modern physics, space and time are unified in a four-dimensional Minkowski continuum called spacetime, whose metric treats the time dimension differently from the three spatial dimensions (see below for the definition of the Minkowski metric/pairing). Spacetime is not a Euclidean space.

https://en.wikipedia.org/wiki/Four-dimensional_space

Dear dr. Al Rashid,

I find this a very interesting question. In my opinion, experience, representation and imagination are different approaches to space. Thus, I would call “external world” the extension we experience around us. As for the representations of space, there are many options. For example, besides the representations relating to art, there are analytical and synthetic geometries. Finally, imagination is only bound by our incapability of abstraction from daily experience. Undeniably, the mentioned approaches to space are interconnected. But their connection is not a straightforward one. In this sense, I like dr. Mustafa’s answer.

In addition, when facing the external world with the aim of representing it, there is no objective need to link it to three Cartesian coordinates. In fact, only if the space is linked to three coordinates orthogonal to each other, it becomes impossible to represent a four dimensional space in the usual analytical way.

Geometric image of four dimensions is not easy, but geometric image of three dimensions and the time as fourth dimension is possible to imagine.

Thanks for the invitation. Our brain simplifies our world to save energy. In evolution this is a very important factor to survive. The three 'visible' dimensions are sufficient to survive, so it would be a waste of energy to perceive more. Nevertheless, because of decreasing restriction of energy it could be possible that further generations can see more dimensions.

One can imagine the effects of higher dimensions indirectly: Imagine you project a cube made of wires onto a sheet of paper. Rotation of the cube will lead to 'strange' behavior of the visible lines. If we have a 4-D cube and project it onto our 3-D world, the effects would be similar (and really psychedelic).

So, please, don't be jealous if someone tells you that he/she can imagine some 4-D-effects ;)

Best Regards

Andreas

PS: It is even more hard to imagine non-integer dimensions (e.g. fractals)... but even this is sometimes calculable. The Volume of a n-dimensional cube is a^n, where a is the edge length and n the number of dimensions. Well, if n=pi... it is calculable as well...

What we all perceive is the space-time that is a reality in four dimensions. All our actions and our thoughts are always placed in space and time, ie in four dimensions. We are so accustomed to act and think in four dimensions that often we do not realize it. However we can not draw on a two-dimensional sheet, items in four dimensions. Maybe we need a sheet with three dimensions!

 Dear Hussien and all,

Good explanation, however the creatures in  2-D surfaces or flat landers could not communicate with us in any way possible. I have similar thread about the search for the hidden dimensions of space, as  solutions of differential equations of nature tell us that reality is just 3-dimensional.  https://www.researchgate.net/post/Where_are_the_hidden_dimensions_of_space

Saeed, I'm tempted to answer in two ways. One, we can imagine 4 or way more than 4 dimensions. Two, we can't even imagine tomorrow, why would anyone expect us to imagine higher dimensional spaces?

Four dimensions are not that difficult, in truth. Our 3D space would be frozen in time, and we would be unable to perceive it at all, were it not for the fact that our 3D space is moving along the 4th dimensional axis. So at least that much we do perceive.

But look up "linear programming." It is an optimization technique that involves finding the optimal solution for a multi-variable objective function, given a set of constraint inequalities.

The graphical explanation for the LP technique is to make a "hyperplane," or multi-dimensional plane, out of the objective function, and then a hypersphere, or hyper polyhedron may be a better term, meaning a convex shape in multiple dimensions, out of the constraint set.

The optimization technique involves moving the hyperplane through the hyperpolyhedron, until the hyperplane reaches a vertex or an edge of the hyperpolyhedron. That will be the single or the multiple optimal solution(s). (Multiple optimal solutions if the hyperplane intersects along an edge.)

So, I submit, this is not so hard to picture. There's no reason to assume that only 3 or 4 dimensions exist, if math can create so many more dimensions with such ease.

How would a flat person perceive a 3D object? If the 3D object is moving, the flat person might just see a flash of something, as the 3D object traverses the plane of the flat person's world. Maybe we only need to detect unexplainable, brief glimpses of "things," and wonder if we've just perceived some hyper-object whizzing by!

What we see / perceive depends a lot on the medium and the environment. Our eyes are adapted for viewing in air.  If, we are in the water then light rays bend and it functions as a lens. Water has approximately the same refractive index as the cornea, effectively eliminating the cornea's focusing properties. Instead of focusing images on the retina, we now focus them far behind the retina, resulting in an extremely blurred image.  So, in the water we see only barely shapes.

The crystalline lenses of fishes' eyes are extremely convex, almost spherical, and their refractive indices are the highest of all the animals. These properties enable proper focusing of the light rays and in turn proper image formation on the retina. So, might be that under another environment human would be able to perceive 9D.

In fact brain is an open system that reflects the environment in which was established and evolves.  It functions and evolves in parallel and in equal dimensions with the environment on the basis of an exchange of energy and matter. This suggests that our vision/perception should reflect the relative dimensions and properties of the 'matter in motion as a whole’.  

Without being an expert I read the USpatent US2006/0071122 A1, “A pulsed gravitational wave wormhole generator system that teleports a human being through hyperspace from one location to another”, I assume that by creating a gravitational wave at the end the conclusion is the same, in air the 3D remains our reference field. 

Dear colleague

I agree with all the views forward by colleagues

Thank you

Well, I have a small remark here. We can imagine the fourth dimension - even the fifth, and the sixth dimension, f.i. That is after all what mathematicians do. What we cannot achieve is a visual perception of it.

This is fascinating. Spearhead science is exactly about this: the realities we take about cannot be seen in the natural sense of the word. We conceive them we imagine them.

Some examples of these realities are: mind, the brain, information, RNAm, DNA, global warming, extinction, etc., etc.

Human limited by 3 D

The main reason for our limited ability to see multidimensional objects is that, on the level of direct sensory experiences, our actual vision is inherently two dimensional. Planar projections are further processed to create impressions of a 3D space. The detailed discussion of how we create the perception of depth from 2-D images is presented at

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC21082/pdf/pq006517.pdf

For the spaces with dimensions higher than three, we have to represent shapes by the algebraic structures. From the purely mathematical point of view, we can easily develop a set of rules to manipulate geometric objects in the n-dimensional Euclidean space. The best example is the 4-D Euclidean cube which is known since antiquity.

The analogous methods can be applied for higher dimensions. One of the most elegant discussions of the subject was presented by Coxeter in the book on multidimensional polytopes

https://en.m.wikipedia.org/wiki/Regular_Polytopes_(book)

The situation is more complex for the multidimensional curved manifolds where we have to develop more complicated techniques to assist our imagination. The corresponding methods are emerging in many areas of the applied sciences, including the multidimensional data analysis.

Why can't we imagine four dimensions?

Because we can't fully answer some questions on 4th dimension i.e. space-time e.g.  can we go back / forward of time with empirical evidence?  What are the implications when we can do that?  How large is our universe?  How we know how large is the universe? etc.

Time is not a spatial dimension. It enters as an imaginary quantity into 4D calculations.

We can imagine 4 dimensions when change is occurring.

Dear colleagues, thank you very much