How many 3D cubes can fit inside a hypercube in R^4?

[eljose]

let suppose we have an hypercube in R^4 then m y question is how many 3-dimensional cubes could we put inside our hypercube?...

 

[matt grime]

A rhetorical question for you to ponder: how many squares are there in a cube?

 

[damoclark]

let suppose we have an hypercube in R^4 then m y question is how many 3-dimensional cubes could we put inside our hypercube?...

If you mean "How many cube faces does a hyper-cube have"?
then 8, possibly

otherwise your question doesn't make sense, as there is no number of 3-dimensional cubes that we could put inside a hypercube.

 

[eljose]

the question is let,s suppose we have a four dimensional space,then could we put inside this four dimensional space our 3-dimensional space?,i think the question has been answered when considering a plane made by an infinite numer of curves or a line made by an infinite numer of points

 

[Data]

Is the statement $$\mathbb{R}^3 \subseteq \mathbb{R}^4$$ true? (Hint: NO!)

 

[HallsofIvy]

Good point. (But there is, of course, a subset of R⁴ that is diffeomorphic to R³.)

 

[Data]

True. And now that I think about it, my original post isn't anything close to a good answer to the original question at all~

 

[Aki]

A rhetorical question for you to ponder: how many squares are there in a cube?

there are infinite squares in a cube. so to draw a conclusion: there are infinite cubes in a hypercube.

 

[Icebreaker]

I don't really understand the erm "put inside". What if a squre is bigger than the face of a cube? Wouldn't the cube only be able to contain squares that are smaller than or equal to the size of its faces?

 

[Alkatran]

I don't really understand the erm "put inside". What if a squre is bigger than the face of a cube? Wouldn't the cube only be able to contain squares that are smaller than or equal to the size of its faces?

Only if they weren't on curved surfaces.