Was Aristotle right?

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Aristotle claimed that among the Platonic solids, only regular cubes or regular tetrahedra can fill three-dimensional space perfectly. In this multivariable calculus project, students investigate whether or not Aristotle was right! Students discovered that a combination of regular tetrahedra and octahedra can indeed fill three dimensions, and they found something surprising about tetrahedra in the process. To demonstrate their findings, students designed some 3D models: a collection of tetrahedra and octahedra, and tetrahedra "dipped" into a sphere to illustrate their approach. This project was all about exploring the properties of regular Platonic solids and using mathematical tools like multivariable calculus to understand how they fit together. How I Designed This Students had to create a 3D model to help explain the results of their project. We used Mathematica to design and export the files. For the tetrahedra and octahedra, we needed identical side lengths and orientation so that one face would lie flat on the printing bed. Therefore, we designed the models from scratch, finding coordinates for each vertex and using the DelaunayMesh function. For the sphere with the wedge cut out, students had to find the equations for the three planes of the wedges. We then used Mathematica's RegionPlot3D function to produce the solid figure. Project: Was Aristotle right? Overview & Background The premise of this project is Aristotle's claim that regular tetrahedra can fill three dimensions perfectly. Students are to investigate this claim using multivariable calculus and design 3D models to illustrate their conclusions. Objectives In this project, students should practice surface area calculations and gain experience with spatial reasoning. In designing the figures, students use skills related to intersecting lines, planes, and angles. Finally, students got to try using Mathematica and 3D printing software. Audience I assigned this project to my multivariable calculus class. The calculations are pretty complex, so I would suggest using this project with advanced students. Lesson This project took place outside of class. Groups of four each chose a project from a list of possibilities, and this was one of those options. I assigned the following write-up as a project description: In the 300s BCE, Aristotle claimed that three-dimensional space could be filled completely by cubes or regular tetrahedra, but by no other Platonic solid. This is obvious with cubes, but was Aristotle right about the other Platonic solids? What about combinations of these regular Platonic solids? So, the questions for this problem are already stated above. We know that you can fill space with regular cubes. Can you do it with any other regular Platonic solid? Are there combinations that will work? I'll leave it up to you to answer this however you want, but one idea is to imagine one vertex of the solid at the origin and a small sphere centered at the origin; what percentage of the surface area of the sphere is inside the solid? If it's not 1/n for some integer n, then what does that tell you? A complete answer to the questions above is not necessary to receive full credit. I do want you to address the question for tetrahedra, cubes (although this is sort of easy), and octahedra. Students must prepare a write-up, designs for 3D models, and a presentation. Students asked if they could use Girard's Theorem, and we decided they should try to avoid using it for the cases of tetrahedra and cubes. Duration Students had three weeks to complete this project outside of class. Their presentation was to last 15 minutes. Rubric & Assessment The project was worth 5% of the students' overall grade. Grading was done holistically, with half of the credit for the write-up and half for the presentation. Student work Students used the approach described in the "Lesson" section. They parametrized the sphere in the usual way and found the surface area subtended by a tetrahedron one of whose vertices is at the center of the cube. Finding this surface area gives enough evidence to say whether Aristotle's claim is correct or not.