Where is the actual space inside a tesseract?

[empleat]

Hey,
i can't see actual space in tesseract.
Like in 3 dimensional cube, you have squares as faces and between them there is a space.
But in tesseract where you have 8 cubes as faces, this leaves 0 space for actual space inside, i can see only 8 faces and nothing between them.
I first thought it has to do something perhaps with, that it is only 2d representation of tesseract i am looking at. But i can see space in 2d repesentation of a cube.
Someone can explain this to me ?

Also i find confusing that new axis, when you create tesseract, is penpendicular to all 3. True in 2d you can't create 3rd axis which is penpendicular to all 2 either without passing through itself. In 2d you wouldn't see where would third axis would go either, that would be impossible. But that's only inductive proof.
Still i am a little skeptical, i didn't find proof of 2d space and only hints of 4d.

 

[Buzz Bloom]

i can't see actual space in tesseract.

Hi empleat:

I am guessing that you are thinking that "space" is limited to being three dimensional. Since a tesseract is four dimensional, it's (hyper-) volume is four dimensional. Your confusion seems to be the same as looking for the area (2D) inside a pyramid (3D).

Regards,
Buzz

 

[empleat]

I don't think space is limited to 3 dimensions, because it wasn't proven yet, or that has 4 or more.It is tough to imagine tesseract when we are 3 dimensional beings.

Your confusion seems to be the same as looking for the area (2D) inside a pyramid (3D).

What do you mean ? I can see 2d slices in 3d pyramid.

When you want to know area of space inside a cube, that's called volume. So for cube if edge is 2, V = 8. I wanted to know where is space inside a tesseract. Because i saw just 8 cubes. Because cube consists of a 6 squares, or 6 faces. I didn't know faces are only 2d and tesseract's surface consist actually from 8 cubical cells. By this site: https://rechneronline.de/pi/tesseract.php hypervolume of tesseract - a4 is 16, when edge 2. So volume, inside a tesseract is actually these 2 cubes, one starting and one i push through, connected with 8 edges. It confused me that surface is at the same time volume, like in cube bottom square and top square and in between is space, because 4d space pass through itself. I have aphantasia i cannot really imagine anything in my minds eye, maybe something but it is like 99% opaque and have no continuity and i cannot manipulate it, i need to look at a picture and it is still tough as hell to track vertices etc, because i have bad visual memory.

Btw is it true that 4-dimensional beings could see through walls, like we could see 2d dimensional beings ?

 

[Buzz Bloom]

What do you mean ? I can see 2d slices in 3d pyramid.

Hi empleat:

I apologize that I cannot understand the meaning of what you write. What do you mean by a "2d slice"? What do you mean by a 2d slice being "in" a 3d pyramid? What do you means by "see" a 2d slice in a 3d pyramid?

Regards,
Buzz

 

[Drakkith]

So volume, inside a tesseract is actually these 2 cubes, one starting and one i push through, connected with 8 edges.

Volume is the 3-dimensional space enclosed by a 2-dimensional closed surface, just like area is the 2-dimensional space enclosed by a 1-dimensional closed boundary (a line).

It confused me that surface is at the same time volume, like in cube bottom square and top square and in between is space, because 4d space pass through itself.

If by 'surface' you mean 'surface area', or just 'area', then no, they are not the same thing, as I explained just above. Also, 4-D space does not pass through itself anymore than 3D, 2D, or 1D space does (they don't).

Btw is it true that 4-dimensional beings could see through walls, like we could see 2d dimensional beings ?

Yes it is. It's analogous to us being able to see every side of a 2D object at once, where a 2D person would only see a 1D slice corresponding to the sides facing them.

I wanted to know where is space inside a tesseract. Because i saw just 8 cubes. Because cube consists of a 6 squares, or 6 faces. I didn't know faces are only 2d and tesseract's surface consist actually from 8 cubical cells. By this site: https://rechneronline.de/pi/tesseract.php hypervolume of tesseract - a4 is 16, when edge 2. So volume, inside a tesseract is actually these 2 cubes, one starting and one i push through, connected with 8 edges.

Remember that squares are built from lines, which take up zero area. And cubes are built from squares, which take up no volume. So a tesseract would be built from cubes, which which don't take up 4D space.

 

[empleat]

Buzzbloom:
like 3d pyramid is created of 2d slices, so tesseract with 3d slices. I can see 4d space now. I had trouble seeing 4 dimensions in that 8 cube shenanigan. Because in 3d you can see clearly space, because we are 3d beings, but 4d pass it through itself in projection and that's what confused me.

If by 'surface' you mean 'surface area', or just 'area', then no, they are not the same thing, as I explained just above. Also, 4-D space does not pass through itself anymore than 3D, 2D, or 1D space does (they don't).

By that site surface area volume is 64 = 8 x a3. I was mistaken and thought - like for cube: top and bottom square is part of a volume of a cube. So 2 cubes are part of a volume of a tesseract.
Ye i knew it doesn't pass through itself, but in projection it does and that is confusing to me.

Remember that squares are built from lines, which take up zero area. And cubes are built from squares, which take up no volume. So a tesseract would be built from cubes, which which don't take up 4D space.

Again i didn't know that 3d cube doesn't take any volume inside a tesseract. Because if i have e.g: edge 2 times 2 is squared and times 2 is cubical, i took that square with surface 4 as a part of the volume. So if i multiply 2 by 0, i would have no volume just surface right ? That's why this have been confusing for me, thanks for clearing that up.
It still confusing thinking about it, because my 3d brain, but now i know everything i wanted to know and i don't have any more questions.
Btw brain makes shapes up to 11 dimensions, but that are still projections, because we are in 3d space.

 

[Buzz Bloom]

like 3d pyramid is created of 2d slices, so tesseract with 3d slices.

Hi empleat:

I think your use of "created" is misleading and confusing. The 2d "slices" are the boundaries of the pyramid, not part of the pyramid. They do not create the pyramid, they bound the pyramid.

Your use of volume is also confusing. A volume is some amount of 3d space. An amount of 4d or higher dimensional space is commonly called a hyper-volume in order to avoid confusing the concept with 3d space.

I hope this is helpful.

Regards,
Buzz

 

[scottdave]

All of this talk suddenly reminds me of the Menger Sponge, which is a 3-D fractal approaching unbounded surface area, while at the same time approaching zero volume capacity.
Now we're supposed to imagine how much volume a 4-D solid object "contains"?

 

[Drakkith]

So if i multiply 2 by 0, i would have no volume just surface right ?

No. You'd have nothing since 2x0=0. No volume, no area, no distance, nothing.

Btw brain makes shapes up to 11 dimensions

I don't know what this means.

 

[jim mcnamara]

@empleat. I am going to make analogy for you about assumptions or more correctly, definitions

For the record, definitions (stated assumptions) change everything in Mathematics. They are formalized as definitions to eliminate confusion. You appear to be making stuff up (assumptions) and not realizing it.

Lets do very simple geometry with a twist. All correct answers to same question, that are all different:

For 3 flavors for a simple triangle, here are 3 correct statements about triangles, they contradict each other without the definitions of their geometry "homeland" being specified!:

The sum of the 3 interior angles of any triangle is:
#1 exactly180 degrees
#2 always more than 180 degrees
#3 always less than 180 degrees

The answer is in this little problem:
A polar bear stands precisely on the North Pole.
She walks due South 1000 feet
She turns and walks due East 1000 feet
She turns and walks due North 1000 feet
How far is she from the North Pole?

This explains #1 versus #2 and why they can be correct:
https://www.quickanddirtytips.com/education/math/what-are-euclidean-and-non-euclidean-geometry

#3 above is yet another "homeland" of geometry. For example, it was used to navigate New Horizons to the planet Pluto:

https://www.nasa.gov/mission_pages/newhorizons/images/index.html

Point: watch your assumptions. They can confuse you mercilessly.

 

[andrewkirk]

The answer is in this little problem:
A polar bear stands precisely on the North Pole.
She walks due South 1000 feet
She turns and walks due East 1000 feet
She turns and walks due North 1000 feet
How far is she from the North Pole?

I am not sure that this distinguishes between geometries. It seems to me that, taking the most natural interpretations of 'South', 'East' and 'North', it may be the case that, for most reasonable geometries, the answer is that she is back at the North Pole.

Is the question that distinguishes geometries rather something like 'through how many degrees of angle has she turned immediately after she commences a second lap of the course' .

Of course, none of this detracts from your point that one needs to scrutinise one's assumptions very carefully, to avoid getting confused.

 

[andrewkirk]

Someone can explain this to me ?

The following visualisation works for me. It may work for you, or not.

Visualise by letting the extra dimension be time, and considering the appearance of the object over a period of time.

An example of doing this that is simpler than a tesseract is a solid hypersphere of radius $R$. If time is the fourth dimension, the object will appear to we 3D beings as:

- first there is nothing
- suddenly a point appears. Let us label the time at which it appears as time 0
- the point grows as a solid sphere until it reaches a maximum of radius $R$ at time $R$, then
- it shrinks until it is a single point at time $2R$, then
- there is nothing from then on

The solid sphere's radius at time $t$ with $0\le t\le 2R$ is $\sqrt{R^2 - (t-R)^2}$.

Now we'll work up to considering the tesseract.

First, consider how an empty square of side-length L looks to an inhabitant of a 1D world. The world will be a ring, in order to allow the inhabitant to inspect both sides of any solid object and thereby determine its length (since it can only see in 1D, just as we only see in 2D). The 2nd dimension of the square is time for the line-dweller. What they will see over time is:

- first there is nothing
- a line of length L appears for an instant, at time 0
- the line instantly disappears and is replaced by two points where the ends of the line were
- the two points remain in place for time L, then
- a line of length L appears again for an instant, then
- there is nothing from then on

The 'space inside' the square is the space between the two dots, that is there for time L.

Next, consider how a flatlander (2D dweller) sees an empty cube of length L, with time being the 3rd dimension.

- first there is nothing
- a solid square of edge length L appears for an instant, at time 0
- the solid square instantly disappears and is replaced by an empty square in the same place (ie the inside of the square vanishes)
- the empty square remains in place for time L, then
- a solid square of edge length L appears again for an instant, then
- there is nothing from then on

The 'space inside' the cube is the space inside the empty square, that is there for time L.

With that buildup, you can probably guess what a tesseract looks like to us 3D creatures.

- first there is nothing
- a solid cube of edge length L appears for an instant, at time 0
- the solid cube instantly disappears and is replaced by an empty cube in the same place (ie the inside of the cube vanishes)
- the empty cube remains in place for time L, then
- a solid cube of edge length L appears again for an instant, then
- there is nothing from then on

The 'space inside' the tesseract is the space inside the empty cube, that is there for time L.

I hope that helps.

PS:

But i can see space in 2d representation of a cube.

Actually, we can't see the space inside an empty cube because we only see in 2D. All we can see is its outside. We need to perform some sort of experiment to discover the space, such as insert a probe, cut it open or weigh it. Similarly, if confronted by a tesseract, we'd need to do that sort of thing to distinguish between the solid and the empty cube. We'd need to be very quick with the solid cube, since it's only there for zero seconds. But we could cheat and use a 'thick-walled empty tesseract', for which the walls of the empty-cube slices are k distance units thick, and the beginning and ending appearances of a solid cube last for k time units.

Even a 4D creature could only see in 3D and would need to perform experiments to determine whether the tesseract they saw was empty - like the one we are discussing - or solid.

BTW, for us, a solid tesseract would appear in the same way as an empty one, except that the cube slices would always be solid, rather than empty except at the beginning and end.

 

[Auto-Didact]

Btw brain makes shapes up to 11 dimensions,

I don't know what this means.

I'm assuming he is referring to this research.