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I do know that the logarithmic spiral was not a mathematical invention and it has an occured behaviour. The logarithmic spiral is very often related with the circle and I'm curious which is the property that brings it so close to a circle.

Well, if we cut off the spiral with a line, which passes through the center os the spiral and then we draw tangents in the intersection obtained points, the angle made by the tangent with the radius is invariable. This property brings the spiral closer to a circle.

The Logarithmic Spiral has another name called Equiangular Spiral:

r=ke^(lt), where t is angle, r the radius vector, k and l are the parameter or constants.

The tangent ratio  of the angle  between  the radius vector and the tangent is r/(dr/dt= r/(kle^(lt)) = 1/l and is free from the angle t. Therefore the tangent and the radius vector make a constant angle. Hence the name Equiangular Spiral.

The radius vector , after a every peiodic t = 2pi, enlarges in an exponential way , by the power 2lpi.

If the  parameter l=0, r =ke^(0*t) becomes a constant and and the tangent ratio of the angle between the radius vector and the tangent becomes = 1/l= 1/0 =infinity, and resulting in a circle.

Therefore, the circle is a special case of Equiangular Spiral with l=0