If-we-have-an-n-dimensional-cube-How-do-we-find-its-volume-

Question Number 5398 by FilupSmith last updated on 13/May/16

If we have an n - dimensional cube .

How do we find its^{'} {v o l u m e}^{'} ?

Commented by Rasheed Soomro last updated on 13/May/16

Suppose s is measure of side of

n - dimensional cube .

volume (v) will be

v = s^{n}

\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots . .

0 - dimensional cube (point) :

^{'} {volume}^{'} (?) = s^{0} = 1

1 - dimensional cube (line segment) :

^{'} {volume}^{'} (length) = s = s^{1}

2 - dimensional cube (square) :

^{'} {volume}^{'} (area) = s . s = s^{2}

3 - dimensional cube (cube) :

volume = s . s . s = s^{3}

4 - dimensional cube (hyper cube) :

^{'} {volume}^{'} = s . s . s . s = s^{4}

\dots \dots \dots \dots \dots \dots . .

\dots \dots \dots \dots . .

\dots \dots

n - dimensional cube (hyper cube) :

^{'} {volume}^{'} = s . s \dots . s (n factors) = s^{n}

Commented by Rasheed Soomro last updated on 13/May/16

Can we think of - 1 - dimensional cube

^{'} {volume}^{'} = s^{- 1} = \frac{1}{s}

And so on .

Commented by prakash jain last updated on 13/May/16

There was an earlier question about surface

area and valume of n - sphere .

Commented by Yozzii last updated on 14/May/16

W h a t a b o u t o t h e r f i g u r e s i n d i m e n s i o n s

h i g h e r t h a n 3, l i k e c u b o i d s, s p h e r e s,

c y l i n d e r s, t o r u s, f r u s t r u m, e t c . ?

I s t h e r e a n y g e n e r a l t h e o r e m w i t h

r e g a r d t o t h i s ?

Commented by Rasheed Soomro last updated on 14/May/16

The anologues of the mentioned figures

(in various dimensional spaces / plane)

should be considered indidually .

For example cuboid has anologues

p o i n t, l i n e - s e g m e n t, r e c t a n g l e, c u b o i d,

\dots i n 0, 1, 2, 3 \dots . d i m e n s i o n a l s p a c e s r e s p e c t i v e l y .

If l_{1}, l_{2}, l_{3}, \dots, l_{n} a r e d m e n s i o n a l m e a s u r e s o f

n - d i m e n s i o n a l c u b o i d, t h e n v o l u m e w i l l b e

l_{1} l_{2} l_{3} \dots . l_{n} . O f c o u r s e a n o l o g u e s o f s o m e f i g u r e s

a n d t h e i r c h a r a c t e r i s t i c s (v o l u m e, s u r f a c e - a r e a

e t c) a r e s o m e w h a t d i f f i c u l t t o c o n s i d e r .

Commented by Yozzii last updated on 14/May/16

I n t e r e s t i n g \dots

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