Polyhedra and Euler's Formula (11/5-11/8)
We continued our exploration of volume ratios by looking at a hexagonal prism coffin box created by two sisters (3rd grade and K) that was five times larger than the original template. Not only did they bring in their prism but also the template they created. The template was pasted together with many sheets of paper. The linear ratio of the four coffins we explored was 2:3:5:10, so they would be able to hold exactly
8 : 27 : 125 : 1,000 pieces of candy, respectively. These volume numbers are the cubes of each of the linear numbers. Believe it or not, the 10x10x10 calculation was the easiest of the four volume conversions. Too bad we did not have 1,000 pieces of candy to test it.
We then explored hexagonal prisms found in nature as well as those that are man-made. See the attached pdf with pictures of the New York Supreme Courthouse in lower Manhattan, an extremely large telescope in Germany, Fort Jefferson in Dry Tortugas National Park, and honeycombs created by bees, a cloud over the north pole of Saturn, and Hanksite crystal. Why bees chose a hexagonal prism for the shape of their home is the subject of a future lesson (conceptually, bees want to do the least amount of work to contain the largest volume of honey; regular hexagons, triangles, and rectangles tessellate [fit together without spaces] but hexagons have the largest volume to perimeter ratio).
We then began to look at other POLYHEDRA. “Poly” means many, and “hedra” means faces. The children recognized the word polygon where “gon” means sides. The triangular pyramid, otherwise known as a tetrahedron (“tetra” means four) is the polyhedron with the fewest faces. The children created their own creations of polyhedra and were asked to name them and then count their faces, vertices, and edges.
V = Number of Vertices (these are the corners or points)
F = Number of Faces (these are the surfaces of the solid)
E = Number of Edges (these are the segments where the faces intersect)
Above the kindergarten level, we studied an amazing formula developed by mathematician Leonard Euler (pronounced “oiler”), where he proved that for any convex polyhedron, the number of vertices plus the number of faces, less the number of edges will always equal 2.
V + F – E = 2 (it works for some concave polyhedron as well). “Convex” polyhedron means a solid that is not concave (in other words, if you were to connect any vertex to another vertex on the solid, that diagonal would be on or inside the solid). If the polyhedron is “concave,” some of the diagonals will be outside the solid.
We looked at some of the wonderful polyhedra created by my students. One was an icosahedron (20 triangular faces). We then “truncated” each of the twelve vertices which were pentagonal pyramids (this means that we cut off the pyramids) to form a truncated icosahedron (better known as a soccer ball; it is also the shape of certain forms of the element carbon).
Another shape was a compound polyhedral of three dodecahedrons (12 pentagonal faces) connected by two pentagonal prisms. We showed that even that concave solid followed Euler’s formula. It had 60 vertices, 42 faces, and exactly 100 edges (60 + 42 – 100 = 2).
I had a cylinder mixed in with other polyhedra and waited to see if students would challenge Euler’s formula given that cylinders have no vertices, two edges and three faces (but can we call the lateral area a face since it is curved?). Here, the formula would yield an answer of 1 (0 + 3 – 2 = 1). Sure enough, cylinders are not polyhedra because their faces have to be flat and all polyhedral have vertices. A few students looked at a sphere and noticed that it would have an Euler number of 1 as well (0 vertices, 1 face and 0 edges).
Homework:
1. Finish building and decorating your hexagonal prisms (coffin boxes) and decorate them with number sequences and shapes (use color as well).
2. Create your own polyhedra out of material of your choice:
a. Use straws and pipe cleaners to connect your edges.
b. Use popsicle sticks and glue them together.
c. Use paper rolled up around a pencil (put a small piece of tape in the middle to make a very strong straw or polyhedral edge).
d. There are countless other ways to create a polyhedron.
3. If you create your own polyhedra, count the vertices, edges and faces to see if Euler’s formula works.
4. If you do not have the time to create your own polyhedra, find polyhedra around your house and count the same. The most common polyhedron will be a rectangular prism with 8 vertices, 6 faces and 12 edges.
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The attached pdf entitled “Euler Polyhedron Formula Blank” is where they should record the vertices, faces, and edges of the polyhedra they create. If they know the name of the polyhedron, list it; if not, name it after themselves (for example, the “Alexhedron”). If bringing in their polyhedra is too cumbersome, take a picture of it to show in class.
I have also attached the pdf of the Hexagonal Prism Coffin Box.
| Attachment | Size |
|---|---|
| Hexagonal_Prisms_Man-Made_and_Natural.pdf | 4.08 MB |
| Euler_Polyhedron_Formula_blank.pdf | 253.32 KB |
| Hexagonal_Prism_Coffin_Box_Small.pdf | 42.85 KB |
