The Magic of Symmetry: From Repeating Patterns to Tiles That Never Repeat

Welcome. This site contains weekly lecture notes for the Cascade 2026 class on symmetry. The notes for each week will be available shortly after that week’s class and will generally include avenues of further exploration. Questions can be addressed via email to either of your instructors:

Lyman Hurd Glenn Hurd

Schedule Overview

Week 1: What are symmetries? We introduce pictures from art, architecture and geometry of symmetric students and we ask the students to try classifying those with the same symmetry without formally defining what that means. We will then discuss the kinds of observations that lead us to associate one pattern with another, e.g., the type of rotational symmetries, or the existence of mirror symmetries. If there is time at the end, we will show the students other objects that can be symmetrical such as Temari balls for spherical symmetry and 3-dimensional crystals. We will also show some simple fractals. The goal of this lesson is to push the idea that a symmetry is a transformation that leaves a pattern unchanged.
Week 2: Transformations and Symmetry of Finite Patterns Students explore the symmetries of bounded figures. We will characterize what it means for a pattern to be rotationally symmetric and then expand to mirror symmetries. Students will be asked to classify the symmetry group of example patterns taken both from geometry and decorative arts, and also will construct patterns of their own with a fixed symmetry which will be a good way to introduce the notion of a fundamental domain as the formal term for the observation that the higher the degree of symmetry, the fewer degrees of freedom we have when creating a pattern. Another takeaway of this lesson is that all possible symmetry groups are part of two infinite families (cyclic and dihedral).
Week 3: Frieze Patterns Students create repeating border patterns that have one dimension of translational symmetry. The main new type of symmetry in these patterns is the “glide reflection”. Without proving the assertion, we will introduce the seven types of frieze symmetry. The mathematician John Conway has described the seven possible patterns as dance moves: “hop”, “step”, “sidle”,… which almost irresistibly calls for attempts to replicate physically. Students will help organize types into a decision tree. We will attempt to classify example patterns into their respective classes and the students will be challenged to create friezes with a given type (one way of doing this is to constrain the pattern to consist of repetitions of specific capital letters).
Week 4: Rotational Symmetry We give a heuristic argument showing that lattice rotational symmetry can only be 1, 2, 3, 4, or 6. Students will classify existing pictures noting that some tilings can be rotated around more than one center point. Again we will try to construct some example symmetries.
Week 5: Kaleidoscope Symmetry Delving deeper into wallpaper patterns, we introduce points with kaleidoscope symmetry and incorporate actual kaleidoscopes into the session for hands-on exploration. Without proving it, I wanted to introduce the “Magic Theorem” showing how one can derive from rotation points and mirror points combined with translations and glide reflections, all possible combinations.
Week 6: Aperiodic Tiles Having learned about wallpaper groups and periodic tilings, the students will be introduced to the notion of aperiodic tilings. Using rectangles, we can illustrate nonperiodic tilings and then make the distinction between tiles that can tile periodically or nonperiodically and those that can only tile nonperiodically (aperiodic). We will discuss a brief history including the reduction of the number of tiles required leading finally to unveiling the hat and spectre aperidoic monotile. For many of these the students will be provided with physical representatives so they can attempt to tile themselves.