miércoles, 14 de septiembre de 2011

WALTER MEYER’S

GEOMETRIC STUDIES


Presentation


These pages are mainly aimedathigh schoolteachers and students having a critical spirit.  Traditional teaching systems inhibit the critical spirit of both teachers and students.  It is paradoxical, for example, thatthe teaching of mathematicshas not questioned the measuring of the areaenclosed by a curvedfigure with square units of measurement. Wouldn’t it be more logical to measure that area in circular units of measurement?

We hereby present, in the first place, an ingeniousgeometric method that allows us to calculate the value of “pi” by means of the Pythagorean theorem (to see a article in blog: http://www.curiosidadesgeometricas.blogspot.com/2011/09/calculo-del-valor-de-pi-con-teorema-de.html).  Based ona theorem of the equality of perimeters of circumferences,varioustheorems for circumferences and volumes arepresented.  As a consequence of these results the concept of circular geometry is proposed.  This concepttakes us to a geometry not requiring the value of “pi”.


CALCULATION OF THE PERIMETEROF A CIRCUMFERENCE WITH SMALLER CIRCUMFERENCES


Suppose you have any given circumference of diameter D (linear units).





Now suppose you divide the diameter intoany “n” parts.



If you use each of those parts as the diameter of a circumference.



          Then the sum of the perimeters of all those circumferences is equal to the perimeterof the circumference of diameter D.


          Notice that the resultdoes notdepend on the number “n” oflengths in which you divide diameter D or on the length of eachone. This result is remarkableboth for its simplicity and its usefulness.  Based on it we maydefine a diametricalunit of measurement, which would allow us to “measure” the perimeterof a circumference without using the value of “pi”, thus avoiding the aforementioned inexactitude in the result. For example, we could define 1 diametricalcentimeter (1 dcm) as the “perimeter” of a circumference whose diameter is 1 centimeter.

Thus, 1 dcm + 2 dcm = 3 dcm.  In other words, when we add a 1-diametrical-centimeter circumference with a 2-diametrical-centimeter circumference we obtain 3 diametrical centimeters.




CALCULATIONOF THE AREA OF A CIRCLEWITH SMALLER CIRCLES

Suppose you have a square and you inscribe a circle in it.

Now suppose you divide it into n x n smaller squares.


If you inscribe a circle in each small square.


            Then thesumof the areas of those n x n circles is equal to the area of the circle inscribed in the large square.


           Then it is possible to define a circular unit of measurement that also does not depend on the value of “pi”.  We could define 1 circular centimeter (1 ccm) as the “area” of a circle whose diameter is 1 centimeter.  Thus, the area of one circle of 5 centimeters in diameter is equal to (5 x 5) times 1 ccm (circular centimeter)i.e. 25 ccm.

CALCULATION OF THE VOLUME OF A SPHERE WITH SMALLER SPHERES


Similarly, now suppose that you have a cube and you inscribe a sphere in it.




Now suppose that you divide it into n x n x n smaller cubes.



If you inscribe a small sphere inside each small cube.


click here to maximize


         Then the sum of the volumes of those n x n x n small spheres is equal to the volume of the sphere inscribed in the large cube. In this way we may define a spherical unit of measurement that does not depend on the value of “pi”.

         We could define 1 spherical centimeter (1 scm) as the “volume” of a small sphere whose diameter is 1 centimeter. Thus, the volume of a sphere of 4 centimeters in diameter would be equal to (4x4x4) times 1 scm, i.e. 64 scm.




OPEN PROBLEM

The previous analysis leads us to the possibility of a circular system of measurement that would eliminate the inexactitude introduced by “pi”.
More explicitly, we refer to a circular geometry. The logic question is if these units of measurement will be useful to measure perimeters, areas, and volumes of any curved figure. You could maybe attempt to answer this question by repeating the previous reasoning for the case of a known figure, such as the ellipse. If you need some ideas on how to proceed, e-mail me a message with your inquietudes to curiosidadesgeometricas@gmail.com .