The dimension of a hypersphere inside a n-dimensional space =
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
The dimension of the ambient space (2-dim plane) is not relevant to the dimension of the circle itself.
Love and Math by Edward Frenkel http://www.amazon.co.uk/dp/0465050743/ref=cm_sw_r_udp_awd_53swtb16779PY