Thursday, 31 March 2011

Pi, Infinity and Zero

If you visualize pi as the connection between a circumference and diameter, and you release the circumference from the diameter, you have just created reality (male and female, yin and yang, north and south). Pi, as a potential point, and diameter as connection between two real points, creates a mandatory circumference, even though the entire circumference may not show up in one place at one time. This is a mouthful and it takes some time to ponder. But this is how we end up with movement (any entity, process, system). Movement is nothing more, or less, than the connection of two points, a general X and Y. If we weren’t moving, we would not perceive (call) movement as movement. But, for now, this is a diversion.

We can begin with the idea of a point, and assume, from point, that point must assuredly be circle, since point, if it is real, must have a diameter (even if this diameter is too small to discern). Therefore, there are no points that are not also circles, or, another statement that goes with this one is, there are no points, just circles. So no matter how we start, we are at (with) a (the) circle (and also of, to, and from it). If we are the starting point, we are a circle that connects to many other circles (which we may view as points or units or entities, processes or systems). It becomes apparent, quickly, that many words are used to describe the same entity (similar entities). (Also, fortunately and unfortunately, there is no starting point, and there are only starting points.)

But back to pi. Pi, if it connects a circumference and diameter, also connects a half circumference and diameter. Half of the circumference might be showing, and the other hidden. Or, the diameter might be hidden with both halves of the circumference disconnected. If we eliminate any idea of time and-or space, and start out, instead, with pi, as a connector of diameter and half-circumference, we can notice pi is a necessary infinite movement that starts at zero. This allows for any movement anywhere with no limit, in no particular order, and with no particular direction, destination, and-or relationship to the original point. Pi, in this sense, is invisible. (Zero is an invisible circle.) How do we get there?

Imagine pi as a point on an invisible circle. Whatever point you pick is connected in a straight line (diameter) to another point, creating a mandatory circumference or set of circumferences. Assuming we collapse all the circumferences into one, and, further, assuming we divide the circumference in half at the point where the diameter connects one point to another on the same circle, we have created a mandatory second diameter (and third) which will act the same (be connected to another two half-circumferences which also function as diameter connected to another two half-circumferences ad infinitum. There is no limit to this expansion. This is how we get to infinity. It is also how we get velocity and acceleration, because even though the diameter and half-circumference connect the same two points, they are mandatorily different sizes, and this is the origination of time and space.

A constant connects two points via two routes which end up an infinite number of routes, and also a finite number of routes (either you stay on the sphere which is the total set of circumferences of one diameter or you escape it). Zero, in this view, then, would have to be pi. That is, pi is playing the role of both zero and infinity, as it connects a stationary spot to two moving spots, where the two moving spots are not moving at the same rate (or in the same direction, necessarily). This is a very simple analysis that ends up being quite complex, once we recognize, we are certainly NOT limited by our conventional ideas of dimension (sphere). That is, sphere is actually line, because they are both a connection of two circles.

Line means one circle has escaped from the other. Sphere means one circle has trapped the other. In this way, our idea of dimension is flawed. Dimension is more easily thought of in circles (since points must be circles, and thus eventually spheres). You can never say (know) how many circles are within a sphere. As long as you have one circle, you necessarily have two, which may, or may not, appear as a sphere (depending on where YOU are in time and-or space). You cannot disconnect a sphere from a circle or a circle from a sphere, but you must if you want to know what is really going on. One sphere interacts with another as if it were two circles. It is unaware it is a sphere.

We have confused the notions of point, circle, line, and sphere; we can also throw in disk, if we want to be clear. In actuality, we are stuck with circle (of circles) because of how a circle operates. That is, it always begins with pi, at any location, in either time or space, which mandates a diameter and circumference (line and circle) connection somewhere. These can appear in many shapes. All of the shapes, however, must reduce to line, and then circle. Line and circle can also be viewed as line and curve. This is how we get parabola and any open shape. And, also, any closed shape, trapezoid, for instance. What we see around us, as reality, and in our symbolic universes, is a wild set of lines and curves, in no particular order, except, if you look very closely, you will always find a one-two-three relationship (somewhere).

This one-two-three comes from pi-diameter-circumference, but does not necessarily end up in sphere (sometimes, but never always, because diameter and half-circumference are not equal). What it boils down to, though, is line is zero not equal to infinity. Circle is zero equal to infinity. Pi makes the decision, not zero or infinity. Pi is the observer that determines where things start and where they end. There are no starts and ends. Just pi on a continual circle connecting many things to many things. We see this as reality. We name it cosmos, universe, solar system, star, planet, cell, dna, particle, force, speed, acceleration, dimension, and, most important, gravity.

We experience it as ‘I,’ where I is an observer. Pi is the only observer, because observer, once he-she-it makes an observation, must make a circle (between observer and observation). Thus, observer is (always) circle. And what we ‘see’ is a snapshot of something that can never be ‘seen.’ Only a circle can create a circle. However, circle can be (and is) known by many different names. Movement, in this view, is an alternate word for observation (and thus an alternate word for circle). And movement, in physics, and also biology, is quite a convenient unifier. If we notice one entity observes another via relative movement (determines a size and speed) to determine whether or not and how to interact with it, and all entities do this, we can use movement as the entity and process that joins any two entities (or processes) together. (Movement is another word for line is another word for circle.) (Pi is doing the analysis and comparison, not either of the entities.)

Any entity and-or process produces the survival and reproduction of movement, as movement produces the survival and reproduction of any entity and-or process. Movement then, originates at pi (acting as zero) and can go to infinity (depending on the observer). We begin and end nowhere (via mind). Via matter, we have a beginning and end, and this is tied to the beginning and end of our surroundings which also have beginnings and ends. All of us, however, because of pi, have no true beginning, nor end. This is the beginning (and perhaps, in a sense an ending) of an understanding of the mind matter relationship in (via) physics (and biology).

Written by Ilexa Yardley, Conservation of the Circle, and The Circular Theory

Saturday, 19 March 2011

The Way of the Tau

The circumference of a circle is 2π times its radius. This is a bit confusing, having to stare at a factor TWO in an expression that is supposed to produce the circumference of ONE whole circle. Also, if you rotate the circle radius ONE full turn (360 degrees), your radian expression will contain an irrelevant factor TWO. Inelegant, clumsy, unnerving.

But these (and plenty more) annoyances can be resolved, if you care to define a different circle constant, τ (Greek letter tau):

τ= circumference/radius

Now, using this new circle constant τ , one full rotation (360 degrees) of a unit radius becomes simply 1 x τ = τ radians. Similarly, turning the radius 60 degrees means turning it one sixth (1/6) of a full turn and hence τ/6 radians, turning it 90 degrees is equivalent to ¼ of a full turn and hence τ/4 radians, and so on. Immensely more intuitive and straightforward, when there is no fooling around with an inexplicable factor 2, isn't it?

The inspiration of using the circumference/radius definition as the proper (and the one-and-only) circle constant first occurred to mathematician Bob Palais (in the article "Pi is wrong!"), but the idea of using the Greek letter τ (Tau) to denote it is due to physicist Michael Hartl. Hartl declared June 28 2010 "Tau Day", and he will probably celebrate the first Tau Day anniversary on June 28, 2011.

2π occurs in a large number of important mathematical expressions (Gaussian probability distribution, Fourier transform, Cauchy's integral formula, etc.), so replacing 2π by τ makes things easier and simpler.

The price of this simplicity is that the expression for the area of a circle becomes A = ½τr2, which contains an inconvenient factor ½. But Michael Hartl maintains that there are a lot of quadratic forms of this type in mathematics and physics (distance fallen: ½gt2, spring energy: ½kx2, kinetic energy: ½mv2, etc.), so the area formula for a circle will be easy to remember.

So τ = 6.283185 ... is the only circle constant worth remembering. Forget all others – there can only be one winner! But victory may take some time, I guess.