On Dimensions

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

Examples:

Dim (Circle) in 2-dim plane = 1

image

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.
image

Note:
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.

image

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 !
image

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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