Geometry in Nature

Background

The Geometry in Nature unit we designed is fairly straightforward and the level of difficulty is not very advanced.  However, there may be some terms and information that you are not familiar with.  Thus, we have included some background information about certain aspects of the unit.
 

Fractals¹

Fractals exhibit a property of being self-similar.  That is, fractals are composed of copies of itself that repeat on forever.  For example, when looking at the Sierpinski Triangle you will notice that the outline of the figure is an equilateral triangle.  If you look inside the triangle you will also see that the figure is composed of smaller and smaller equilateral triangles.  This is what it means for a fractal to be self-similar.  All fractals exhibit this property.

Fractals are also formed by a process of iteration.  Iteration is repeating the same process on a figure over and over again.  For example, on a Cantor Dust fractal it starts out with a line segment.  The first iteration is to remove the middle third of the line segment.  You are now left with two line segments that are the same length.  The second iteration is to repeat the process you just performed, removing the middle third of each of the two resulting line segments.  This is fractal iteration.  The more iterations a fractal exhibits the more complex the fractal.

Crystal Symmetry²

The defining property of a crystal is its inherent symmetry, by which we mean that under certain operations the crystal remains unchanged.  For example, rotating the crystal 180 degrees about a certain axis may result in an atomic configuration which is identical to the original configuration. The crystal is then said to have a two-fold rotational symmetry about this axis.  In addition to rotational symmetries like this, a crystal may have symmetries in the form of mirror planes and translational symmetries, and also the so-called compound symmetries which are a combination of translation and rotation/mirror symmetries.  A full classification of a crystal is achieved when all of these inherent symmetries of the crystal are identified.

Geometer’s Sketchpad

Geometer’s Sketchpad is a geometrical software which comes with an extensive manual, as well as a “Help” menu installed in the program.  Additionally, there are online resources and technical help at http://www.dynamicgeometry.com/technical_support/index.php.

Other

Additional helpful internet sources are included in the resources page.  These sources are also cited in the individual lesson plans.

¹Fractal Information provided by the PBI Fall 2003 Fractal Unit by Erick, Ash, & Derek.

²Crystal symmetry information is taken verbatim from Wikipedia.  The website can be accessed athttp://en.wikipedia.org/wiki/Crystal_structure.

Good Luck and if you have questions please contact us.

Larkin Campbell
Reem Kattura