On Dimensions

The dimension of a hypersphere inside a n-dimensional space = \boxed {n - 1}


Dim (Circle) in 2-dim plane = 1


As we approach near the neighborhood of the tangential point on the circle, the curvature of the circle disappears, there is no difference between the circle and the tangent line (dim = 1).

Hence, Dim (Circle) = 1

A point on a circle is determined by one independent variable only, which is the polar angle.

The dimension of the ambient space (2-dim plane) is not relevant to the dimension of the circle itself.

Dim (Sphere) in 3-dim Space = 2

The 2 variables (longitude, latitude) determine a position on the globe. Therefore dimension of a sphere is 2.


Interesting note:
Four Dimension Space (x, y, z, t): What will we get if the 4th dimension time t is fixed (frozen in time) ? We get a PICTURE !

Reference :
Love and Math by Edward Frenkel http://www.amazon.co.uk/dp/0465050743/ref=cm_sw_r_udp_awd_53swtb16779PY

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