1
$\begingroup$

If I have a unit ball in space $R^d$ then in how many dimension space its surface will be represented. I know the answer is d-1 but i am unable to convince myself. can anybody give me some intuition. I also did'nt got one line from wikipedia

In mathematics, a distinction is made between the sphere (a two-dimensional closed surface embedded in three-dimensional Euclidean space) and the ball (a three-dimensional shape that includes the interior of a sphere).

and can you give me some reference so that i can understand this topic well means what are the property of ball in $R^d$ why people use them in proof,etc.

I know this is very broad but you can narrow down to any level. I want to collect as much info as i can

$\endgroup$
  • $\begingroup$ Consider a circle in two dimensions. By marking, arbitrarily, a point on that circle as a point of origin; we can represent any point on the circle using a one-dimensional coordinate system. I.e. any point on the circle can be represented by a single number $x\in [0,2\pi R)$ ($R$ is the radius of the circle), where $x$ represents the clockwise (or anticlockwise) distance from the arbitrarily defined origin. $\endgroup$ – David Simmons Apr 13 '14 at 8:04
  • $\begingroup$ are you saying that if I cut the circle I will get a line. So if I tore the ball it will work in same manner because tearing it will destroy one axis.. am I on the right path? $\endgroup$ – thetatheta Apr 13 '14 at 8:09
  • 1
    $\begingroup$ Exactly. The circle is just a line (which is one dimensional) that curves through a third dimension (the curvature happens to be constant $\rightarrow$ the ends of the line are connected). $\endgroup$ – David Simmons Apr 13 '14 at 8:11
  • $\begingroup$ You might also be interested in reading about manifolds. $\endgroup$ – David Simmons Apr 13 '14 at 8:16
  • $\begingroup$ I correct my second comment. It should say "... curves through a second dimension ..." $\endgroup$ – David Simmons Apr 13 '14 at 8:51
0
$\begingroup$

Consider a circle in two dimensions. By marking (arbitrarily) a point on the edge of that circle as a point of origin (to be used to define a coordinate system), we can represent any point on the circle using a one-dimensional coordinate system. I.e. any point on the circle can be represented by a single number $x\in [0,2\pi R)$ ($R$ is the radius of the circle), where $x$ represents the clockwise (or anticlockwise) distance from the arbitrarily defined origin.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.