Archimedean Solids By: Nicole Strauss, Carrissa Texley, and Marlene Stockton

What are Archimedean Solids? Archimedean solids belong to the semiregular polyhedra family. There are 13 Archimedean solids. They are like platonic solids but differ in the sense that they have two or more regular polygon faces. 7 of the 13 Archimedean solids are derived from platonic solids.

History Archimedean solids are named after the Greek mathematician, Archimedes who discovered and discussed them in a book that was lost. The solids were “rediscovered” during the Renaissance period by Johannes Kepler.

Truncation According to Dictionary.com truncation means “to shorten by cutting off a part; cut short” By truncating certain platonic solids you create 7 Archimedean solids (which is what we will be doing today). Different Archimedean solids can be made but truncating the same platonic solid at different depths.

Euler’s Formula Do you think Euler’s Formula holds true for Archimedean solids?

Unearthing Archimedes' Lost Solids 5th Grade (but can be adapted for middle school as well). The topic of our activity is finding Archimedean solids. The purpose is to prove that seven of the Archimedean solids come from truncating Platonic solids and show the relationship between these two classes of polyhedra.

The standard we are addressing in this activity is a NCTM Geometry standard for 3-5 grades, which states that students should be able to; “Analyze characteristics and properties of two- and three- dimensional geometric shapes and develop mathematical arguments about geometric relationships and should— Investigate, describe, and reason about the results of subdividing, combining, and transforming shapes.”

We think that this standard applies to our activity because we are truncating, which is a way of transforming, a three-dimensional shape in order to create a new three- dimensional shape. It is also relevant to our activity because students will be analyzing the shapes they start with versus the ones they end with, therefore leading them to look closely at the various characteristics and properties that make up both figures.

In a normal classroom this activity would probably take minutes, in our class we foresee it taking about 15 minutes; since our class is familiar with the material and are skilled in building solids from nets at this point. The class should be split into six fairly equal groups. Each group will need 3 nets for the Platonic solids (should give them 3 different Archimedean solids when they are finished), 3 nets for the finished Archimedean solids, scissors, and tape.

If this activity was implemented in a classroom we would suggest that teachers do the following: Break the lesson into 3 parts – Discuss Platonic solids – Build the Platonic solids – Truncate Platonic solids into Archimedean solids This way the students are not overwhelmed with too much information, and can focus on each section at one time. If you go younger than 5th grade students might need assistance in cutting out the nets and putting them together due to dexterity issues. A teacher might want to guide their students a little more on where they should truncate in order to get the correct Archimedean solids.

To conclude the activity each group can present a different Archimedean solid to the rest of the class, and explain from which Platonic solid they got it. They can also go into detail about what regular polygons make up the faces, and the number of edges and vertices their Archimedean solid contains.