
#1
Dec1609, 04:45 PM

P: 464

Let's say I have a four dimensional cube. Would it have a true surface area? I'm wondering if maybe it would have a surface volume rather than a surface area.




#2
Dec1609, 05:07 PM

P: 707

its boundary is not a surface but does have a 3d volume




#3
Dec1609, 05:59 PM

P: 1,400

Would this n1 dimensional boundary be a hypersurface?




#4
Dec1609, 06:13 PM

P: 707

Does an NCube have Surface Area? 



#5
Dec1709, 06:22 AM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,900

In n dimensional geometry, a "hypersurface" is the n1 dimensional boundary of a bounded ndimensional region.
As for dimensionless's original question, its really a matter of convention whether you call the 3 measure of the boundary of a 4 dimensional region "area" or "volume". That's why most people just talk about n or n1 dimensional "measure". 



#6
Dec1709, 07:13 AM

P: 464





#7
Dec1709, 07:30 AM

P: 707





#8
Dec1709, 08:34 AM

P: 608

Solution of the wave equation is quite different in even dimensions vs. odd dimensions.




#9
Dec1709, 10:46 AM

P: 464





#10
Dec1709, 11:09 AM

P: 1,400





#11
Dec1709, 03:28 PM

P: 707

A submanifold of dimension n1 is a called a hypersurface. You may be aware that you can have submanifolds of lower dimension as well. For instance in 4 space the Klein bottle can be embedded as 2 dimensional surface. 


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