3-D Tessellations and the Platonic Solids Objectives: • The student will see how three dimensional shapes such as the Platonic solids fit together to form larger solids. Use this exploration to enter into a discussion of how regular bonds between atoms can form crystal latices and ultimately macroscopic crystals. Grade Levels: Tenth through twelfth grade Subject Areas: Mathematics (especially Geometry), Chemistry Timeline: One/Two 55 minute class periods National Science Standards: NS.9-12.1 Science as Inquiry National Education Standards: Geometry Standard for Grades 9-12 Connection Standard for Grades 9-12 Background: The platonic solids are the 5 three dimensional shapes that can be created by connecting together exact copies of regular polygons. Four equilateral triangles will produce a tetrahedron. Six squares will produce a hexahedron (commonly called a cube). Eight equilateral triangles will produce an octahedron. Twelve regular pentagons will produce a dodecahedron. Finally twenty equilateral triangles will produce an icosahedron. Plato demonstrated 2400 years ago that these were the only possibilities for regular three dimensional solids. In this activity students will produce models of each of these shapes and through discovery determine which of them can be repeated in a lattice. Materials: • Copies of the Platonic Solid Fold-up Patterns from the Space Foundation Biology and Physical Research Summer Institute Course (Attached to this document) for each student. • Card stock for the number of solids that your class will be making.

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White Glue Scissors Optional: Models of the Platonic solids to use as references of what the result should look like.

Lesson: Begin with a discussion of the two dimensional case. Discuss the properties of regular polygons and what a tessellation is. Have students conjecture about which two dimensional shapes will tessellate. Discuss the measure of the interior angles for each shape and show that the only three regular polygons that will tessellate are the equilateral triangle, the square, and the regular hexagon. Discuss what other shapes might tessellate. You should now be ready to move up to three dimensions. Open with a discussion of Plato and his interest in Geometry. Then discuss how Plato was the earliest writer to discuss three-dimensional geometry and was the first to identify what we now know as the Platonic solids. While regular polygons consist of a given number of equal line segments meeting at regular angles, the Platonic solids consist of specific number of regular polygons meeting at regular angles. Discuss what types of polygons can be put together to form regular spacial symmetry (equilateral triangle, the square, and the regular pentagon) and what number of each of these shapes can meet at one vertex. (3, 4, or 5 in the case of equilateral triangles and 3 in the case of squares and regular pentagons.) Try to get students to appreciate why it is the case that there are only 5 possible Platonic solids even though there are and infinite number of regular polygons. Decide if you are going to have students work individually on this or in groups. If groups are working together make sure groups are of about 5 individuals of mixed abilities. Pass out the Platonic Solid Fold-up Patterns, card stock, glue, and scissors either one set of patterns for each individual or one set per group. If you have them, pass around examples of what the finished models will look like. Get students to glue the patterns to the card stock. If working in groups see that the tetrahedron and cube patter going to the students who struggle more with geometric constructions while the octahedron should go a middle ability individual and the dodecahedron and icosahedron go to the stronger students in the group. If time allows every student to try making their own set, make sure that they start with the simpler shapes and work towards the more complicated ones. If students are working in groups the constructions should take about 40 minutes, if students are working individually allow a couple of hours of class time, knowing that some students may not complete all 5 shapes. After enough shapes have been constructed have multiple individuals or groups work together to see what shapes can be combined into tessellate space, and that are fill space without any gaps. The cube should be obvious. Are any other shapes capable of filling space without gaps? Why do some work while others do not? Have students

write down their explorations in there notes and what they have discovered. Can you imagine other 3D shapes besides the Platonic solids that might tessellate? Describe them. Extensions: If you have some really advanced students and you want to really set them to thinking, ask them what a four dimensional tessellation might be. What would a four dimensional shape look like? How would you know if four dimensional regular shapes can tessellate? Another topic that this activity can motivate is the discussion of crystal lattices. Discuss what are the possible configurations of atoms in a crystal and what the resulting crystal will be shaped like. Evaluations: There are two products from this activity that need to be evaluated. First is the 3-D models of the platonic solids. Sloppy construction could result in shapes that are not useful for the 3D tessellation exploration. It is important that these regular solids are indeed regular. The other product is the student’s written evaluation of the activity and what they discovered about the space filling properties of a variety of 3D shapes. Did they correctly identify which shapes will form a 3D tessellation of space? Do they understand why some shapes work while others do not? Resources: Good description of the Platonic Solids from MathWorld: http://mathworld.wolfram.com/PlatonicSolid.html Description of the Platonic Solids that you can refer your students to: http://www.mathsisfun.com/platonic_solids.html The original description of the five regular solids that connects them to the Greek theory of the four elements (the fifth element being what celestial bodies consist of): Plato, Timaeus. Platonic Solid Fold-up Patterns Addendum:

All graphics on this page are from Sacred Geometry Design Sourcebook