Introduction

Associating between Mathematics and Metaphysics

"a dialectical process in which each system is brought about to enlighten the other. through a translating metaphor, questions in one system are solved in the other than translated back. "

Using metaphor to connect perspectives is the theme of this site. This application can be represented as

Math::metaphor::Metaphysics

Unit circle: without and within.

From the math side of things, the unit circle is a natural representation of a bounded finite being in an unbounded space. The metaphor says that the within is like the subjective, the without like the "Communal CONsensual UNI-VERSE-ALL Reality" we share. The unit circle itself represents the boundary between the two.

Within reflects without: reciprical relation.

The metaphysical side informs us that within a holographic universe, everything without you is reflected within you.  This is stated directly by Rumi, and alluded to in other spiritual traditions. Mathematically, this connection between the within and the without is described by the reciprical relationship through which every point outside the circle is connected to a point within.  Here is the model, and a smattering of points without and their recipricals within in a matching color.

Giving the within and the without equal weight

In relating to the universe it is easy to look from the perspective that I'm small and the universe is big. As you can see above, the reciprical relationship makes the center of the circle dense with connections. So there is another way of seeing things in which you are as valuable as the universe, and your heart as large. This is represented mathematically by mapping the area of the interior of the circle to an area equal the mapping of exterior.

I devised the Tic Tac Toe Transformation which performs this balancing.

Read Webpages or a PDF, on the Tic Tac Toe Transformation and its implications.

Another mapping that performs this balancing is Stereographic Projection which I will address at a later date.

The exponential function in the complex plane applied to the unit circle and reciprical relationship.

This is a more complicated application.

It is shown that the Unit Circle is formed in the Complex Plane by applying the exponential function parametrically to the values -1 or +/- I. This means that the path followed by the complex value of a raised to t, as t changes from zero upwards. If a is 1, the point sits on the real value of 1.  It is clear that with a of -1, the value is real -1 when t is odd and real +1 if t is even (including zero).

The Unit Circle is traced out as the parametric path of the complex exponential when

a = -1 cycles every 2 powers

a = I cycles every 4 powers

a = -I cycles every 4 powers

as t goes from zero, where all are 1, to 4, where all are 1 again and the cycle repeats.

With the Unit Circle in place, the exponential function is applied to 2: +,2 -2, +2I, -2I; and the interior recipricals, Absolute Value = 1/2: +1/2, -1/2, 1/2 I, -1/2 I.  These values are selected because they are the second, reflected across zero into the negatives, and rotated orthogonally to the Imaginary axis. The reciprical to two, one-half, is also so reflected and turned.

Essentially, three dualities are being explored together:  Within/Without, Positive/Negative (180 degrees), and Real/Imaginary (90 degrees).

Read Webpages and see more visuals on the exponential function in the complex plane.