This project combines some very interesting mathematical topics, equilateral triangles, fractals, and pyramids. Students enjoy learning about fractals, and there are many interesting applications of this fascinating topic.
When studying triangles and their properties, constructing just one face of the pyramid (an equilateral triangle and the Seirpinski Gasket within the triangle) will provide an interesting hands-on lesson in geometric construction. Students can do computations in finding the perimeters and areas of the triangles within the Gasket, and exploring the sequence formed by counting the number of triangles in each iteration of the fractal.
When studying 3-dimensional geometry, this pyramid will be helpful in discussing the methods of finding volume and surface area. The pyramid itself will be a valuable visual aid in explaining the definitions: pyramid, altitude and slant height of the pyramid, lateral surface area, total surface area and volume. You can find the definitions for each of these geometric terms at the following web page: just type in any word and you will find the definition: http://dictionary.reference.com/
For example, from that website here is the definition of Pyramid: "Geometry. a solid having a polygonal base, and triangular sides that meet in a point". Here is an example of a pyramid:
You can find out more about 3D drawing at the following web page: http://mathforum.org/workshops/sum98/participants/sanders/Geom3D.html
A pyramid always has triangles for sides, but the base of the pyramid can be any geometric figure: a triangle, a square, even a hexagon. If the pyramid has equilateral triangles for sides and the base, it is called a "tetrahedron".
If you find 3D geometric solids interesting, and would like to learn more about 3D drawing, visit the following web page: http://mathforum.org/~sanders/mathart/MACch4drwg.html
The Sierpinski Gasket is a "fractal" image. A fractal is a geometric design formed by repeating a geometric shape in a pattern. The definition of fractal is a fractal is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole" and you can read more about fractals, and see some beautiful fractal images at http://en.wikipedia.org/wiki/Fractal.
If you follow the steps below, you can create your own "Seirpinski Pyramid", a tetrahedron with fractal images for base and faces.
First, you will need to know how to construct a "Seirpinksi Gasket", by following the instructions below:
Step 2) Construct the midpoints of all 3 sides and join them.
Step 3) Construct the midpoints of the sides of all the 3 outer new triangles.
Step 4) Again, construct the midpoints of the sides of all the 3 outer new triangles.
You can continue this process indefinitely. This is a fractal, a geometric figure formed by continuing a process and repeating it over and over again. This particular fractal is called a Sierpinski Gasket, named after the mathematician who first created it. The Sierpinski Gasket has many interesting mathematical properties.
Construct four copies of the Sierpinski gasket, attached to each other , by following the instructions below.
Step 1: Begin with one Sierpinksi gasket:
Step 2) Mark side BC of the triangle as "Mirror". Select the entire figure (including it's interior) and reflect it as shown. (In each of the steps, the new reflected triangle is shown in a lighter shade, so you can tell one from the other.)
Step 3: Mark side BD of the triangle as "Mirror". Select the entire triangle BCD (including it's interior) and reflect it as shown.
Step 4: Mark side DE of the third triangle as "Mirror". Select the entire triangle BDE and reflect it as shown.
Score on segment BC, BD, and DE. Then fold inward (mountain fold) on all 3 lines to create a three-dimensional pyramid shape, as shown in the three-dimensional color drawing below:
You can see some very nice models of the Sierpinski pyramid at the following web page: http://www.nths.net/academics/math/Connections/patterns/sierpyr.htm