Semiregular Tilings


A Topic Supporting Diverse Math Practices by Nina White and Hanna Bennett


Q


uestions about polygonal tilings of the plane can utilize a classical area of math- ematics to highlight and connect middle and high school mathematics content


standards, mathematical practices, and the nuanced nature of mathematical justification. Inspired by an MTC of Austin session on Escher-like tilings led by Altha Rodin, we ran a session on semiregular tilings of the plane at the Wayne County (MI) MTC last fall. Here we describe what mathematical features naturally arise at different points in the inquiry.


Semiregular Tilings A tiling of the plane covers the (infinite) plane, with- out gaps or overlaps, using congruent copies of one or more shapes. We can consider special categories of tilings and ask what possibilities exist under those con- straints. For example, we might ask that only one shape be used, that only quadrilaterals be used, or that only regular polygons be used. A semiregular tiling is a tiling of the plane with the following constraints:


• Two or more regular polygons are used. • Polygons meet “edge-to-edge.” Tat is, no vertex falls in the middle of an edge of another polygon— it always meets at other vertices. An immediate consequence of this is that all of our edges must be the same length.


• Te pattern of polygons around every vertex is the same. (Note that the vagueness of the wording “the same” in this definition is intentional, as we discuss below.)


Te overarching question of this session is: Can you find all possible semiregular tilings of the


plane? Manipulatives are a useful tool for exploration as


well as communication (see link to printable .pdfs at www.mathteacherscircle.org/newsletter); we suggest printing them on different colors of paper so that


different shapes are easily distinguishable when used in a tiling together. Te overarching question can be scaffolded by posing it as two consecutive questions:


Question 1: What possibilities exist around a single vertex? Approaching this preliminary question creates a few intellectual needs:


1. the need to recall (or re-find) interior angle sum formulas. Using the fact that the interior angle sum of a triangle is 180o


, there are several different jus-


tifications of the interior angle formula for convex n-gons that participants or facilitators might know. Tere is also a simple justification starting from the (equivalent) fact that the sum of the turn angles in any convex n-gon is 360o


. Depending on the time


available, this could be an opportunity to share some of these justifications.


2. the need to articulate what makes two vertex patterns the “same’’ or “different.” Tis creates the opportunity to discuss rotational and translational symmetry, a middle school content area.


3. the need to systematically approach the ques- tion case by case. Tis third need is sometimes called organization, and is a centrally important mathematical problem-solving strategy (see Zucker article at www.mathteacherscircle.org/ newsletter). If you try to align this particular skill with the eight Common Core State Standards for Mathematical Practice, you can find supportable overlap with at least six of the eight standards (nos. 1, 3, 5, 6, 7, and 8). If this practice of organization is made explicit in the course of the session, this alignment could make for an interesting discussion with teacher participants familiar with the CCSS Standards for Mathematical Practice.


To scaffold this question further, the facilitator can ask participants to first consider the case of triangles


American Institute of Mathematics · Summer/Autumn 2016 · MTCircular 11 ⊇


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