### THE PYTHAGOREAN THEOREM

##### Main MenuFitzgerald, Lorentz, Einstein, and Relativity

The Pythagorean theorem is the fancy name for a something a man in Ancient Greece learned from someone else, supposedly the Egyptians. It was known before by the Sumerians who probably gave it to the Egyptians, and no one knows its true origin because it goes all the way back to pre-history.

It is a statement relating to all right triangles. If the sides of a right triangle are labeled "a" and "b",
and the hypotenuse is labeled "c", then a2 + b2 = c2.

This theorem and its many variations are found in many parts of the mathematics of the universe. The Lorentz factor found in the Lorentz transforms used by Einstein and in nether theory are based upon the Pythagorean theorem. This is illustrated in the parts of this website called Lorentz Factor Derivation and Time. The Pythagorean theorem was also used in the derivation for the construction of the pentagram and the old "life series" which Fibonacci discovered only recently and is called the "Fibonacci series" in modern times.

The reason that the Pythagorean theorem and its variations are so ubiquitous has to do with what the theorem actually is. The law of conservation of energy is a fundamental rule of the universe. Vectors are also rather fundamental. The three spatial dimensions of our universe are always at ninety degrees to one another because one dimension may be considered a line that is made of a series of points which are all equidistant from any two points, along the line of another dimension, that are equidistant from the point where the two lines cross. But it is much easier to say that the two lines are at right angles or ninety degrees.

A vector has length that is meant to illustrate its magnitude and a direction. It is a line with an arrowhead on one end. When two vectors are at ninety degrees to one another, they can show two velocities. If an object in space is moving with one velocity shown by one vector, and another velocity is applied at right angles to its direction of movement, the object then moves at a resultant vector that is the vector sum of the original vector and the one that was applied to it. The angle of ninety degrees is the only angle where the maximum magnitude of the applied vector can act directly sideways on the original direction of movement.

The original vector may be shown as one side of a right triangle, and the applied vector shown as the other side. The resultant or sum of the two vectors becomes the hypotenuse of the right triangle. The magnitude of the sum of the two vectors that are the sides of the triangle is found by using the Pythagorean theorem. This means that the square root of the sum of the squares of the magnitudes of the two vectors that are the sides is the magnitude of the resultant.

Kinetic energy is found by the formula mv2/2 where "m" is mass and "v" is velocity. This means that it is found by the using the square of the velocity. The magnitudes, of the squares of the vectors which form the sides of a right triangle, are actually parts of the kinetic energies of any mass moving in the prescribed manner. Since "m/2" remains the same before and after one vector is applied to the other, it is the squares of the velocities which must be added to find the resultant.

Therefore, the right triangle with the Pythagorean theorem is a means of showing vector addition of kinetic energies. The law of conservation of energy means that the resultant kinetic energy must be precisely the sum of the component energies. So the Pythagorean theorem is a statement of the law of conservation of energy and the law of dimensions. Looking at it another way, it can be used to define the nature of dimensions. Either way, it is at a level very close to the top of our universal fundamentals.