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

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Discussion Overview

The discussion revolves around the question of how many 3-dimensional cubes can fit inside a hypercube in R^4. Participants explore the implications of dimensionality and the relationships between different geometric shapes in higher dimensions.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants question the validity of the original question, suggesting that the concept of fitting 3D cubes inside a hypercube may not make sense.
  • One participant proposes that there are 8 cube faces in a hypercube, which could be interpreted as a response to the original question.
  • Another participant raises a rhetorical question about the number of squares in a cube, suggesting that there are infinitely many squares, and extrapolates this to imply there are infinitely many cubes in a hypercube.
  • Some participants express confusion over the phrase "put inside," questioning whether larger squares could fit within the faces of a cube.
  • A later reply mentions that while \(\mathbb{R}^3\) is not a subset of \(\mathbb{R}^4\), there exists a subset of \(\mathbb{R}^4\) that is diffeomorphic to \(\mathbb{R}^3\).
  • One participant reflects on their earlier response, acknowledging it may not have adequately addressed the original question.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the original question. There are multiple competing views regarding the interpretation of fitting cubes into a hypercube, and the discussion remains unresolved.

Contextual Notes

There are limitations in the assumptions made about dimensionality and the definitions of fitting shapes within one another. The discussion also touches on the nature of geometric relationships in higher dimensions, which may not be fully resolved.

eljose
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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?...
 
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A rhetorical question for you to ponder: how many squares are there in a cube?
 
eljose said:
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.
 
Last edited:
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
 
Is the statement [tex]\mathbb{R}^3 \subseteq \mathbb{R}^4[/tex] true? (Hint: NO!)
 
Good point. (But there is, of course, a subset of R4 that is diffeomorphic to R3.)
 
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~
 
matt grime said:
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.
 
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?
 
  • #10
Icebreaker said:
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.
 

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