The significance of the dimensional expression 2*i*pi is highly elusive. In fact there are direct connections to the mystical notion of "bindu" or point centredness or even more recently the physical notion of a singularity.
The way I look at it is as the centre point of the "imaginary" circle.
We can approach this initially with reference to the "real" circle.
Now the centre of this circle is equally the centre point of its line diameter. So in this sense it is at this point that both linear and circular notions are reconciled as identical. Put another way we could say that the quantitative and qualitative notions of interpretation (characterising Type 1 and Type 2 Mathematics respectively) are reconciled.
However the problem with the real circle is that it cannot be conceived in the absence of linear notions (relating to the size its radius). So though with the central line diameter we can depict the middle point as 0 with the right hand side in positive units and the corresponding left hand side in negative units (with both equal) in actual fact we represent both positively geometrically as lines (with equal extension). Now if we could imagine the circle in more dynamic terms where positive and negative aspects cancel each other out the circle would shrink to a point (with no linear extension). Likewise of course the line diameter would shrink to the same point. So at this point the line and the circle would be identical. Now inherent in the qualitative notion of the imaginary is this complementarity of positive and negative polarities. So the imaginary unit circle which is represented by 2*i*pi is therefore identical with this point (where in a sense the circle and line have collapsed to a point where they are indistinguishable from each other).
Now quantitative interpretation is of a linear rational nature based on finite notions of form; qualitative interpretation is directly of a circular intuitive nature based on infinite notions of emptiness.
So in fact we can have 3 different expressions for this non-dimensional point.
First we can have the rational interpretation where it is approached from the perspective of actual form i.e. as being represented by a quantity - rather like the infinitesimal notion - that is in the process of becoming nothing i.e. 0.
Secondly we can have the intuitive interpretation where it approached from the perspective of emptiness (through negation of the quantitative aspect). So from this perspective when we have surrendered any rational notion of nothingness we are intuitively left with the experiential awareness of its true qualitative nature.
Thirdly we can have the combined appreciation of both quantitative (rational) and qualitative (intuitive) notions.
As I say this area is in fact dealt with at length in the mystical contemplative literature where the three notions I have mentioned represent three stages in the specialised growth of pure spiritual understanding.
The first is often referred to a the arrival at a pure state of concentrated transcendent awareness i.e. as spirit as the centre of one's being. Here spirit is understood as beyond all phenomenal form.
However because initially there is lack of corresponding immanent awareness i.e. where spirit is seen as equally inherent in all form, this leads to a lingering phenomenal attachment to the very notion of this point. Put another way though maintaining the absolute (separate) nature of ineffable spirit one thereby still understands the notion in an unduly phenomenal manner.
So the second stage requires the negation of this lingering - merely - phenomenal understanding a spiritual point. This - when successful - leads in turn to corresponding free intuitive awareness of its qualitative nature.
However because both rational and intuitive are themselves dynamically complementary in experience, ultimately both refined rational and intuitive interpretation must be combined in the realisation of creation as both form and emptiness. This thereby entails both the immanent aspect (where spirit is inherent in matter) and the transcendent aspect (where spirit is beyond all form).
And by this third stage both the quantitative and qualitative aspects of understanding are coherently related.
We will see how these notions are related to the mathematical treatment of 2*i*pi in the next contribution.