# Unit ball in space of d dimension

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

• 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. – David Simmons Apr 13 '14 at 8:04
• 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? – thetatheta Apr 13 '14 at 8:09
• 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). – David Simmons Apr 13 '14 at 8:11
• You might also be interested in reading about manifolds. – David Simmons Apr 13 '14 at 8:16
• I correct my second comment. It should say "... curves through a second dimension ..." – David Simmons Apr 13 '14 at 8:51

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.