Curiosidades geométricas: Study of Areas of Circles and Squares

PREFACE

Dear readers:

Some ideas have arisen from the program “Analysis of the Inequality of the Curve and the Straight Line,” activities and ideas that are detailed below:

1 Diffusion

· Conferences ate several universities in Mexico and Chile

· Exhibitions comprising 37 shapes that didactically explain the steps followed to get to the aforementioned conclusions.

Due to the conclusions arrived, and looking for opinions on the subjects, exhibitions and talks have been carried out in several educational institutions: Universidad panamericana, Universidad del Valle de México, Universidad Autónoma UNAM, Instituto Politécnico nacional Plantel ESIME – Ticoman de Ingeniería, all in México. In Chile, I can mention Universidad de Santiago, Universidad de Playa Ancha, Universidad Católica de Valparaíso, Universidad Federico Santa María, Universidad del Mar, etc.

I would like to mention that as for the diffusion, the admiration of the public was explicit in the visitor comments book that we had during the exhibitions. For example: “I believe that the success was precisely due to the modeling of the axioms exposed.”

This modeling, which started in 1985, was of great clarity for the general public. Nevertheless, these new ideas did not resonate with the relevant authorities.

As an anecdote: A teacher drew a duck on the blackboard and one of the students (as children have a clean mind), said: “Miss, the duck has one eye!” The teacher answered: “No, sweetie, it has two eyes,” The student was confused and kept on looking at the drawing and asked the teacher: “Miss, where is the other eye?” Baffled, the teacher thought she could not tell the child that the other eye was behind the blackboard.

That is why it is better to show shapes to children instead of drawing them (shapes, models, etc.). This method is widely used in other countries.

2^nd Mathematics Conventions (papers sent and accepted) I took part in:

Argentine:

Santa Fe, UMA (Mathematics Union), Universidad Nacional del Litoral. LII (XXV Annual Meeting of Scientific Communications).

Presentation: “The Inequality of the Curve and the Straight Line.” October 16-20, 2002

Salta city, Universidad Nacional de Salta, III CAREM.

Department of Mathematics, Faculty of Exact Sciences.

Presentation: “Analysis of the Inequality of the Curve and the Straight Line”, October 9 and 11, 2003.

Brazil:

Blumenau: Universidad Regional de Blumenau, ULBRA (Lutheran University of Brazil).

Presentation: “Geometric Curiosities”.

XI Inter-American Conference of Mathematical Education. July 13 to 17, 2003.

Porto Alegre: “Analysis of the Inequality of the Curve and the Straight Line”, July 2003.

Cuba:

Congress at the University of Holguín, Matinfo.

Presentation: “Analysis of the Inequality of the Curve and the Straight Line”, October 2003

Congress at the University of Holguín, “Oscar Lucero Moya”,

Presentation as an author: “Why not the sphere as a new geometric pattern?” April 2007.

Chile:

VI Journeys on Innovation of Mathematics Learning. Universidad de Viña del Mar.

Presentation: “Geometric Curiosities”, November 2001.

Congress at the Universidad Católica de Valparaíso, SOCHIEM (Chilean Society of Mathematical Education), “Teaching Mathematics.” Special guest Dr. Jeanne Bolon PhD, Professor of the IUFM in Versailles, France. December 27, 2002.

Seventeenth Latin American Meeting on Educational Mathematics, Universidad Católica de Santiago, “CLAME RELME 17” (Latin American Committee of Educational Mathematics - Latin American Meeting on Educational Mathematics).

Presentation: “Analysis of the Inequality of the Curve and the Straight Line, Structural Foundations for the Extension of the Measurement System.” July 25, 2003

Spain:

Universidad Complutense de Madrid.

Presentation on the research within the scope of PRONOR-SIGCO about: “Method and System for the Calculation of Circumferentials, Circles and/or Spheres” (exploration model for teaching), November 2008.

3rd Stage:

We have formed a small group of academics. We have been able to advance in the research of different problems to demonstrate that it is more effective that students take part in the research of problems instead of going on learning formulas in integral and derivative calculus, etc., and having no idea of their development, thus creating mathematical automatons. This becomes errors that at large have produced increases in costs in different applications.

I have done nothing but applying knowledge that already existed to solve existing problems, realizing that there are substantial differences with the current system.

For just one subject we may have several calculation systems, because here in Chile we have cultivated the concept that mathematics are a rigid science instead of dynamic knowledge that will allow us to get to a solution through more than one path.

The Japanese education system focuses in that the student should develop his/her own system to solve mathematical problems.

I put forward these ideas so that we may advance in the didactics of mathematics.

Atte.

Walter Meyer Vergara

ANALYSIS OF SURFACE AREAS OF CIRCLES AND SQUARES

WITH AN EXTENSION FOR VOLUMES OF SPHERES AND CUBES

First, we present as a whole, perimeters of circumferences and squares, areas of circles and squares, and volume of spheres and cubes on Page 1.

Then, length of the arc of a circumference and length of the side of a square, area of a circular sector and area of the sector of a square, volume of a spherical wedge and volume of a cubic wedge on Page 2.

We divided this study into three stages, each one starting with a short review till we come up with the conclusion of the new ideas presented together: perimeters of circumferences and squares, area of circles and squares, volume of spheres and cubes.

a) Calculation of perimeters of circumferences and squares: PART A, Equality of perimeters of squares, Page 43, taken from the first text “The Inequality of the Curve and the Straight Line.”

b) Area of circles and squares: PART B, starting from the idea taken from the third book “The Relationship between the Circumference and the Square, “ pages 1-3.

c) Volumes of Spheres and cubes: Short review with calculation of volumes of spheres and volumes of pyramids in order to demonstrate that the result of the study of the area of circles and squares is also true for cubes and spheres.

Calculation of Perimeters

Of Circumferences

And Squares

II) Equality of Circumference Perimeters

Theorem The perimeter of a circumference is equal to the sum of the perimeters of n tangent circumferences aligned along its diameter.

Illustration for n = 4

Demonstration

Let d₁, d₂, … d_n be the diameters of the n circumferences and let P₁ + P₂ + ... + Pn be the perimeters of those circumferences. Now, the perimeter P of the circumference of diameter D is:

P = p D

= p (d₁+ d₂+ … + d_n)

= pd₁ + pd₂ + … + pd_n

= P₁ + P₂ + … + P_n

EXTENSION OF THE THEOREM OF EQUALITIES OF PERIMETERS OF CIRCUMFERENCES

Going on with this study, we will see how the theorem of the “Equality of the Perimeter of Circumferences” is true for the main geometric shapes, provided that they are inscribed in the circumference.

Note: We must take care with the value of n, because in some cases, in order to simplify, I have taken it as the diameter and in some other cases as the radius, so the diameter is expressed as 2n.

Some examples:

SQUARE

Theorem:

The sum of the perimeters of any “n” squares is equal to the perimeter of a square whose side “S” is equal to the sum of the sides of the “n” squares.

Fig. 32

Illustration

For n = 4

Demonstration:

Let S₁, S₂, S₃, … S_n be the sides of the “n” squares.

Let P₁, P₂, P₃, … P_n be the perimeters of those squares.

Then, the perimeter P of the square of side S is:

P = 4S

P = 4(S₁ + S₂ + S₃ + … + S_n)

P = 4S₁ + 4S₂ + 4S₃ + … + 4S_n

P = P₁ + P₂ + P₃ + … + P_n