#equations

New Post has been published on http://dancecompreview.com/the-5-equations-of-partnership/

The 5 Equations of Partnership

If ballroom dancing is the journey, then partnership is the ride. And for a greater journey, we need an equally great ride. Days will otherwise fade in lame resemblance, as dancers flounder on their path towards seemingly unobtainable goals. And performances will faint in the shadow of a lost potential. Of two lost potentials, to be exact.

But at the other end of the spectrum, where the dance floor blazes with fire, a couple dances and grabs the attention. Dancers share much more than beautiful choreographies and sharp technique. They exude with energy, harmony and with their own unique identity. This is only possible when dancers embrace the art of synergy.

So, how does that happen?

It all starts with Michael and Elsa – two random dancers. Each one of them has a unique dance imprint that reveals individual power. Let’s call these “Power (M)” for Michael’s and “Power (E)” for Elsa’s.

The two dancers are partners. How they perform together can make the dullness or the greatness of the dance.
And the possible results of that performance can be mathematically modelled and explained – yes, sometimes, (basic) math actually makes sense.

Result #1 = The Negative Sum of Both

Equation = 0 – [Power (M) + Power (E)]

Michael and Elsa are in a state of war on the battlefield of dance. Each one tries to bring out the best of his/her individual power, at the expense of the other. Michael wants to show how big and strong his moves are but Elsa seeks to reveal the sharpness of her technique. He puts her off balance and she blocks him. In the end, they both end up losing their energy in the fight, instead of joining forces for the dance.

Result #2 = The Gap Between the Two

Equation = Power (M) – Power (E) OR Power (E) – Power (M)

Elsa and Michael are both on the dance floor. And they are not fighting! But it feels that only one of them is dancing, because someone is outshining the other who is lagging behind. What people see from the outside is the empty space between two powers, as if each dancer lived on a different planet. The unbalanced nature of the performance stings the eye. And the result is a failure to communicate the spirit of harmony that should emanate from the dancing.

Result #3 = Their Average

Equation = [Power (M) + Power (E)] / 2

Michael and Elsa perform the dance together, but they still lack some harmony. Although no one is outshining or fighting with the other, they are both so focused on themselves that they forget that the dance is not just about each dancer independently, rather about the couple as an entity. Their sense of independence, although admirable, reveals a lack of communication. The initial success is that each one is seeking to master one’s part without hindering the role of the partner. But this is just half a success, because the absence of interdependence takes away the magic of the dance.

Result #4 = The Sum of Both

Equation = Power (M) + Power (E)

The couple finally understood that the two need to work together to harvest the energy of the partnership. They have also started to understand each other’s needs and seek to fill the gap, like in a puzzle. Instead of just doing their part, they support each other. Their dancing shows some cohesion. And the interaction between the two feels like the tide, a gentle flow that fills the movement with content. This stems from a collaboration where partners join forces.

Yet, there’s still one more step.

Result #5 > The Sum of Both Squared

Equation = [Power (M) + Power (E)]2

Not only are Michael and Elsa complementing each other’s movements, but they are also transcending them. They embody the dance, claim it as their own and inhabit every part of it, together. The performance is no longer a puzzle, but a seamless perfect picture of who they are and what they do. As for the interaction between the dancers, it is no longer limited to the tide; it is the entire ocean in action. The dancing finally distinguishes itself with an identity that is larger than the sum of the parties.

This… is Synergy!

So, what kind of dance partnership are you living today?
And what should you do next?

Author:Alexandra Kodjabachi
Photography:Egorich.ca DanceSport
Exclusively for Dance Comp Review

Beyond the Enlightenment Rationalists:
From imaginary to probable numbers - I

Imaginary numbers arose in the history of mathematics as a result of misunderstanding the dimensional character of numbers.  There was a failure to acknowledge that numbers exist in a context of dimension. This has earlier been addressed at length.^[1]  Simply put, numbers exist always in a particular dimensional context.  Square numbers pertain to a context of two dimensions and therefore to a plane,  not a line.  Square roots then ought justly reference a two-dimensional geometrical context rather than the linear one mathematics has maintained ever since mathematicians of the Age of Enlightenment decreed it so.  Square roots contrary to the way mathematics would have it can neither exist in nor be found in any single line segment,  because they do not originate in the number line but in the two-dimensional square.

Algebra, not geometry, provided the breeding ground for imaginary numbers.  They were given a geometric interpretation as an afterthought only, long after the fact of their invention. Rationalist algebraists, feeling compelled to give meaning to equations of the form b² = -4 came up with the fantastic notion of imaginary numbers. Only indirectly did these grow out of nature, by way of minds of men obsessed with reason.^[2]

Descartes knew of the recently introduced square roots of negative numbers. He thought them preposterous and was first to refer to the new numbers by the mocking name imaginary, a label which stuck and which continues to inform posterity of the exact manner in which he viewed the oddities.  It is one of the ironies of history that when at last a geometrical interpretation of square root of negative numbers was offered it involved swallowing up Descartes’ own y-axis. Poetic justice? Or ultimate folly?

Had the essential dimensional nature of numbers been recognized there would have been no need to inquire what the square root of -1 was. It would have been clear that there was no square root of -1 nor any need for such as +1 also has no square root.  As linear numbers,  neither -1 nor +1 can legitimately be said to have a square root.  Both, though, have two-dimensional analogues and these do have square roots, not recognized as such unfortunately by the mathematics hegemony.^[3]

In the next post we will look at a comparison between imaginary numbers,  which were formulated in accordance with this misconstrual about how numbers relate to dimensions,  and probable numbers which grow organically out of a consideration of how numbers and dimensions actually relate to one another in nature.^[4]  The first of these approaches can be thought of as rational planning by a central authority; the second, as the holistic manner in which nature attends to everything, all at once, and without rational forethought.

(continuedhere)

Image: A drawing of the first four dimensions. On the left is zero dimensions (a point) and on the right is four dimensions  (A tesseract).  There is an axis and labels on the right and which level of dimensions it is on the bottom. The arrows alongside the shapes indicate the direction of extrusion. By NerdBoy1392 (Own work) [CC BY-SA 3.0orGFDL],via Wikimedia Commons

Notes

[1] See the series of about nine posts that begins here.

[2] The Rationalists missed here a golden opportunity to relate number and dimension by defining square root much too narrowly. They seem to have been so mesmerized by their algebraic equations that they failed to pursue the search into deeper significance pertaining to essential linkages between dimension and number that intuition and imagination might have bestowed.

[3] As Shakespeare correctly pointed out, a rose by any name would smell as sweet. Plus one times plus one certainly equals plus one but that has nothing to do with actual square root really, just with algebraic linear multiplication.  Note has often been made in these pages of the difference between mathematical truth and scientific truth. Whereas mathematics demands only adherence to its axioms and consistency,  science requires empirical proof.  Mathematics defined square root in a certain manner centuries ago, and has since been devoutly consistent in its adherence to that definition.  In so doing it has preserved a cherished doctrine of mathematical truth, as though in formaldehyde.  It has also for many centuries contrived to be consistently scientifically incorrect.  The problem lies in the fact it has converted physicists and near everyone else to its own insular worldview.

[4] For an early discussion of the probable plane, potential dimensions, and probable numbers see here.

IN SCHOOL:

IN THE REAL WORLD:

if x^2+y^2=20 is a circle at which point will the abscissa will be 2?

If (b+3)x^2+(a-1)y^2+(a+b-10)xy+14x-21y-1=0 is an equation of  a circle what is the value of a and b?

I run this blog partly as a Terra Nova public outreach project, and partly as a commonplace book for my graphic novel. This post is squarely of the latter species.

For the Boston Globe, accompanying a review of Andy Weir’s new novel, Project Hail Mary. Happy I got paid to draw an astronaut.