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Menger’s Intricate Sponge by Juan Francisco Rodriguez and Carlos Peña

In the Advanced Calculus class of José Luis Zamora, we had an introduction to the limits of functions through arithmetic and geometric sequences or series. Within this process, we obtained the necessary knowledge to develop our fractal project. Juan Francisco Rodriguez and I, Carlos Peña, chose as a project the Menger’s Sponge. As a product of hard work and commitment, we produced a presentation that contained conclusions like how the Menger’s Sponge has an infinite surface, and at the same time, it traps a volume that tends to zero. We think of objects with three dimensions like a line, one square, or a cube. However, in fractal objects, there can be a fractional number of dimensions. The Menger’s Sponge is neither bidimensional nor tridimensional, given that its dimensions are 2.7268. It is more than a surface but less than a solid object due to its dimension being 2<d<3. Together with other conclusions, we formulated equations that described the behavior of the number of figures, volume, and surface area for any iteration of the sponge. 

The development of this work was beyond the expected by explaining how we formulated the equations. This project poses critical and advanced thinking since it requires additional knowledge of those acquired in class. The most challenging part of the project development process was manipulating the equation of the surface area to return the same as a non-recursive equation. To make up this non-recursive equation, we had to resort to the help of our classmates, who developed the same project. With the help of Gabriela Orjuela, we succeeded in solving this enigma.

Finally, we invite the community to access the QR code found in the evidence shared, which contains our project’s different products and processes. As progress, we would also like to share the third iteration of the fractal built at school with the 3D printer. We feel very proud of our quality work and hope the community enjoys it, as well. 

Jose Luis Zamora High School Mathematics Teacher

Juan Francisco Rodriguez and Carlos Peña Eleventh Grade Calculus Students


En la clase de Cálculo Avanzado de José Luis Zamora tuvimos una introducción a límites de funciones mediantes series y sucesiones aritmeticas y geometricas. Durante este proceso obtuvimos el conocimiento necesario para empezar el desarrollo de nuestro proyecto de fractales. Juan Francisco Rodriguez y yo, Carlos Peña elegimos como proyecto la esponja de Menger. El desarrollo de este trabajo fue más allá de lo esperado al explicar con minuciosidad el proceso con el que formulamos las ecuaciones. Este proyecto plantea un pensamiento crítico y avanzado en cuanto a la formulación de la ecuación del área superficial. Esto, ya que requiere de conocimientos adicionales de los adquiridos en clase. Invitamos a la comunidad a evidenciar nuestros logros ingresando a los archivos compartidos. ¡Nos sentimos orgullosos de nuestro trabajo de calidad y esperamos que la comunidad lo disfrute!

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