Magic happens at the boundary.

It seems as though our physical senses cause a flattening of reality.

Immense, complex forms pass through our flattened world and we imagine we know them; We fool ourselves and mistake the glimpse of a shadow, for the entire picture.

Power of logic and dexterity of thought have become master in our film-like reality, but meaning remains elusive. We can only perceive the surface of things as they pierce our world. Their greater extension lies elsewhere.

Terence Mckenna’s notion of the transcendental object at the end of time, springs to mind.

A topsy turvy version of science’s ‘big bang’, beginning of time, singularity; The multi-dimensional singularity at the end of time!

A strange attractor drawing us ever towards itself, like a huge gravity sink, pulling us through the medium of, what we think of as, ‘time’.

Mckenna asserts that the piercing of this ‘object’ through our ‘flat’ reality cause the ‘ripples’ we call history, culture and civilization.

As the object breaks through the surface and comes into sight, time appears to speed up. Increasing novelty floods the system as we hurtle towards the singularity. The historical similarities of scale fold in upon one another and overlap. Echoes, resonances and patterns are repeated at an ever increasing rate.

In mathematics we have complex numbers. The combination of a ‘regular’ number with a strange extension into another dimension that drags it away from the well beaten path of the number line.

Such numbers have both a ‘real’ and an ‘imaginary’ component, for instance 2+3i

Essentially a new dimension has been added.

Instead of the one dimensional number line with zero at its centre and negative numbers running to infinity one way and positive numbers running to infinity the other, a new landscape has opened up; A fresh axis of imaginary numbers perpendicular to the existing line of numbers, stretching ‘up’ and ‘down’ to infinity creates a 2 dimensional plane, in which this ‘new’ species of numbers are able to inhabit.

Now the square root of -1 has an answer! It is i (and the square root of -4 is 2i)

It is easy to see pictorially. Suppose we multiply an arbitrary positive number, say 3, by i. The result is the imaginary number 3i. The number 3 has been rotated off the number line and is now 90 degrees out of phase. If we multiply by i again we get another 90 degree phase shift and we are back on the real number line. A flip of 180 degrees via an ‘imaginary’ realm.

Hence If we multiply a positive number by i squared , the number undergoes an 180 degree phase shift. 1 x i squared = -1

Disappearing through another dimension and into the ‘upside down’.

In the 70's, John Hubbard began looking at the dynamics of ‘Newton’s method’, which is a powerful algorithm for finding roots of equations on the complex 2-dimensional plane previously discussed.

The method takes a point on the complex plane as an estimate or approximation of the root solution. This is a starting point, upon which a certain computation is carried out, that improves the approximation. By doing this repeatedly, always using the previous point to generate a better one, the method bootstraps its way forward and rapidly homes in on a root.

Hubbard was interested in problems with multiple roots. In that case, which root would the method find? He proved that if there were just two roots, the closest root to the approximation would always win. But if there were three or more roots, he was baffled. His earlier proof no longer applied.

So Hubbard did an experiment. A numerical experiment.

He programmed a computer to run Newton’s method, and told it to color-code millions of different starting points according to which root they approached, and to shade them according to how fast they got there. (How many iterations it took to get to the solution)

he anticipated that the roots would most quickly attract the points that start nearby, and thus should appear as bright spots in a solid patch of color. But what about the boundaries between the patches?

The computer’s answer was astonishing.

The borderlands looked like psychedelic hallucinations. The colors intermingled there in an almost impossibly complex manner, bordering each other at infinitely many points, and always as a three-way. In other words, wherever two colors met, the third would always insert itself and join them.

Magnifying the boundaries revealed patterns within patterns.

At the heart of the enclosure, all is ‘as it should be’. It is at the boundary where magic happens!; The transition between the worlds.

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