Another interesting family is the Hill tetrahedra and their generalizations, which can also tile the space.Ī 3-dimensional uniform honeycomb is a honeycomb in 3-space composed of uniform polyhedral cells, and having all vertices the same (i.e., the group of is transitive on vertices). In particular, for every parallelepiped, copies can fill space, with the cubic honeycomb being special because it is the only regular honeycomb in ordinary (Euclidean) space. The simplest honeycombs to build are formed from stacked layers or slabs of prisms based on some tessellations of the plane. The more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered. There are infinitely many honeycombs, which have only been partially classified. However, not all geometers accept such hexagons. Interpreting each brick face as a hexagon having two interior angles of 180 degrees allows the pattern to be considered as a proper tiling. Similarly, in a proper honeycomb, there must be no edges or vertices lying part way along the face of a neighbouring cell. It is possible to fill the plane with polygons which do not meet at their corners, for example using rectangles, as in a brick wall pattern: this is not a proper tiling because corners lie part way along the edge of a neighbouring polygon. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space. They may also be constructed in non-Euclidean spaces, such as hyperbolic honeycombs. Honeycombs are usually constructed in ordinary Euclidean ("flat") space. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. For other uses, see Honeycomb (disambiguation). Here's a nice article that may give some ideas that students could look into to understand the purpose of tessellations in our natural world. As for the honey bees an interesting thing to look into is why do honey bees use regular hexagons rather than other regular polygon that tessellates- it has to do with optimizing the amount of honey a regular hexagon stores. I'm still thinking about how to move forward on this though. I am thinking about how I could create certain parameters in which the students will have to fill a finite plane of some shape and they will have to make some sort of prediction. I feel something is missing in my project that requires them to take it further than just designing their own. Although it is true that tessellations can be found both in the natural world as well as in more synthetic (man-made) products/ art/architecture. I am stuck in how to make this project more authentic to the students though. This entails an understanding in transformations, interior angles of a polygon and I differentiated by creating different roles: some students had to design a mutated figure that would tessellate with an equilateral triangle, square, regular hexagon, irregular triangle, and irregular quadrilateral. I am an 11th Grade math teacher and I have done a larger project with my students in which they have to design their own tessellation using Geometer's Sketchpad. I agree with John Golden, in that you could extend the idea to have student think about the "so what". I really like the idea of using pattern blocks to work with semi-regular tessellations.
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