PM [B08] The Gate-crasher

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GATE-CRASHERPM [B08] The unexpected


The unexpected guest revealed

We have been working busily on 𝑖𝑖and 𝑒𝑒, and so far are happy with them. But while they are on their way to a union, there is an unexpected guest knocking at the door – the angle 𝜃𝜃 (theta).𝜃𝜃 has been a common enough object. It is the familiar symbol for an angle and has been sitting around nice and quiet. Looks like having no cause for suspicion.Unfortunately, it is found to have hidden a lot of secrets behind more than that just those meeting the eye – it is suspected to be the culprit that have cause science so much ‘success’.


Angles in Radians

In angular measurement, the angle (𝜃𝜃𝑐𝑐) is defined by the ratio of the arc (𝑠𝑠) to the radius (𝑟𝑟):

𝜃𝜃𝑐𝑐 =𝑠𝑠(𝑎𝑎𝑟𝑟𝑎𝑎)𝑟𝑟

A radian unit ( �𝜃𝜃𝑐𝑐) is the angle subtended at the center of a circle by an arc equal to the radius of the circle:

�𝜃𝜃𝑐𝑐 =𝑟𝑟(𝑎𝑎𝑟𝑟𝑎𝑎)𝑟𝑟 = 1 𝑟𝑟𝑎𝑎𝑟𝑟𝑖𝑖𝑎𝑎𝑟𝑟

�̂�𝜃𝑐𝑐

𝑠𝑠 = 𝑟𝑟

𝑟𝑟


Why specifically 𝜃𝜃𝑐𝑐

Here the angle is specifically written as 𝜃𝜃𝑐𝑐 instead of 𝜃𝜃because:

❶ It is a circular measure, not an angular one;

❷ It is a special quantity which is not entirely circular or angular.

That is:

𝜃𝜃𝑐𝑐 𝑖𝑖𝑠𝑠 𝑟𝑟𝑛𝑛𝑛𝑛 𝑛𝑛𝑡𝑒𝑒 𝑠𝑠𝑎𝑎𝑠𝑠𝑒𝑒 𝑎𝑎𝑠𝑠 𝜃𝜃

𝜃𝜃𝑐𝑐 ≠ 𝜃𝜃

≠


Linear ↔ Circular

The equality process is actually a transformation (artificial one) changing a linear quality 𝑟𝑟 to a circular quantity 𝑠𝑠, or vice versa:

𝑟𝑟 ↔ 𝑠𝑠

Linear ↔ Circular

Therefore 𝜃𝜃𝑐𝑐 is not a normal quantity but a ratio of mixed nature between the arc and the radius. It is a twister.

𝜃𝜃

𝑟𝑟


Radian Analysis

Here is the evidence.The angle in radian is measured by:

𝜃𝜃𝑐𝑐 =𝑠𝑠(𝑎𝑎𝑟𝑟𝑎𝑎)𝑟𝑟

Multiply both levels by 2𝜋𝜋:

𝜃𝜃𝑐𝑐 =2𝜋𝜋 𝑠𝑠2𝜋𝜋 𝑟𝑟

2𝜋𝜋𝑟𝑟 is the circumference (𝑎𝑎) of the circle, so:

𝜃𝜃𝑐𝑐 = 2𝜋𝜋𝑠𝑠𝑎𝑎

𝜃𝜃𝑐𝑐 = 2𝜋𝜋𝑠𝑠𝑎𝑎

𝜃𝜃𝑐𝑐

𝑠𝑠

𝑎𝑎


𝜋𝜋 the culprit

Here we see what has not been unusual.

𝑠𝑠/𝑎𝑎 is a defined quantity. 𝑠𝑠varies but c remains constant. So in 𝜃𝜃𝑐𝑐 = 2𝜋𝜋𝑠𝑠/𝑎𝑎 the size of 𝜃𝜃𝑐𝑐depends on the size of 𝑠𝑠modified by the accompanying constant 2𝜋𝜋.

The key now lies with the constant 𝜋𝜋.

What kind of a constant is 𝜋𝜋?

𝜃𝜃𝑐𝑐 = 2𝑠𝑠𝑎𝑎


Radius & Circumference

In nature, the straight line and the circle are two different entities which are absolutely independent of each other. On the straight line, there is no point where the geometry is circular. On the circle, there is no point where the geometry is straight. The only thing that keeps the circumference of the circle (𝑎𝑎) related is its radius (𝑟𝑟 or diameter) is:

𝑎𝑎 = 2𝜋𝜋𝑟𝑟

Even this is not perfect. It is only an artificial relationship simply because 𝜋𝜋 is irrational.

𝑎𝑎 = 2𝜋𝜋𝑟𝑟

Circumference = 2𝜋𝜋 × Radius


𝜋𝜋 irrational

𝜋𝜋 is an irrational number meaning that it cannot be expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are commonly used to approximate 𝜋𝜋); consequently, its decimal representation never ends and never repeats:𝜋𝜋 =3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 . . .

Wikipedia

Get real! Be rational!


𝜋𝜋 transcendental

Moreover, 𝜋𝜋 is a transcendental number – a number that is not the root of any nonzero polynomial having rational coefficients. The transcendence of 𝜋𝜋 implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge. The digits in the decimal representation of 𝜋𝜋 appear to be random, although no proof of this supposed randomness has yet been discovered.Wikipedia


𝜃𝜃 is also irrational

So 𝜃𝜃 is also irrational because of its relationship with 𝜋𝜋 in:

𝜃𝜃𝑐𝑐 = 2𝜋𝜋 𝑠𝑠𝑐𝑐

That is why 𝜃𝜃𝑐𝑐 has also been so successful in correlating the arc and the radius. In this universe, if there is anything that can links up a linear and circular quantity, this must be 𝜋𝜋 and 𝜃𝜃𝑐𝑐 .It is an artificial compromise. But it is done so immaculately that the inventors himself must have thought that it is what is intended by nature as well.

𝜋𝜋


The Four Stooges

So far we have found the three weirdos: 𝑖𝑖, 𝑒𝑒, and 𝜃𝜃, four if 𝜋𝜋 is also included. They are the most beautiful and important symbols of mathematics. They are our hope of understanding the weird microscopic world.

𝜋𝜋 𝑖𝑖𝜃𝜃

𝑒𝑒


THE HAPPY UNIONTo be continued on PM [B09]:

Date post:	21-Mar-2017
Category:	Science
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PM [B08] The Gate-crasher

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