Geometry in Nature

By Megan Hancock

What would our world look like if our students thought about Geometry every time they looked out the window? If we teach our students how Geometry connects to the world around them, this could become a reality.

The Rory Meyers Children's Adventure Garden at the DallasArboretum and Botanical Garden is a great way to show children of all ages how Geometry and nature are connected. While at the children’s garden, I was able to view mathematics in nature through children’s’ eyes. The Kaleidoscope Garden in the children’s garden has different stations that talk about mathematical concepts such as fractals, tessellations, rotations, reflections, and the Golden Ratio.  Children are able to read about these math concepts and see different plants that showcase these concepts.

In a high school geometry classroom, students learn about tessellations, rotations, rotations, and fractals.  Students often struggle with these concepts because they are not able to picture them or relate them to concepts they are already familiar with.  By taking students to the children’s garden, they are able to see these concepts come to life.

While visiting the Dallas Arboretum, I chose to focus on fractals and tessellations.  Lornell and Westberg (2014) described fractals as having the characteristic of self-similarity, which means the part of the whole closely resembles the whole.  Objects can be broken into small pieces that resemble the whole, and this process can be repeated multiple times. Trees, flower centers, and different plants are great representations of fractals in Geometry.  Fractals are often a difficult concept for students to understand, so by showing students examples of fractals in nature, they are able to better understand this concept. 

Their article, Fractals in High School: Exploring a New Geometry, goes into detail about the Koch curve and the Koch snowflake.  These concepts allow students to learn about the growth pattern of the perimeter of the snowflake under iterations.  In a higher level mathematics class, students can begin to think about what happens to the perimeter of the snowflake as it approaches infinity.  They support this understanding with examples of how to teach fractals in a high school classroom using these concepts. 

Retrieved from http://mathworld.wolfram.com/

To take students deeper into the idea of fractals and the Koch snowflake, Linda Bolte described an activity she did with her students that found the area and perimeter of a Koch snowflake.  In the article, Sharing Teaching Ideas: A Snowflake Project:Calculating, Analyzing, and Optimizing with the Koch Snowflake, Bolte provided readers with the activity, the assignment, assessment methods, and sample solutions.  This is a lesson that could be used in its entirety, or specific parts could be used depending on the focus of the assignment. 

 I also focused on tessellations while at the children’s garden.  The tessellation station in the garden described a tessellation as a “repeating pattern made of flat shapes that fit together with no spaces between them.”  At the garden, students are able to test their knowledge of tessellations and create their own.  Texas Instruments (TI) has created an interactive lesson for the TI-Nspire that allows students to use regular polygons to determine which will tessellate.  They use this information to determine why some objects will tessellate and why others will not.  Part 2 of the activity guides the students through dihedral tessellations, a tessellation using congruent copies of two different shapes.  They use this information to determine if their rules for tessellations still hold when two shapes are used. 

By taking students to the Rory Meyers Children’s Adventure Garden, they are able to gain insight into how concepts from Geometry are present in nature.  Making these connections ensures students will never look at nature the same way again.  They will see fractals when they look at trees and leaves and tessellations when they look at honeycomb, plants, and animal fur.  Students would begin to think about mathematics not only in their math classroom, but in the rest of their world. 

Bolte, L. (2002).  Sharing teaching ideas: A snowflake project: Calculating, analyzing and optimizing with the Koch snowflake.  The Mathematics Teacher, 95(6), 414 – 419. 

Lornell, R. & Westerberg, J.  (1999). Fractals in high school: Exploring a new geometry.  The Mathematics Teacher, 92(3), 260 – 265.