The Schläfli Symbol Exploration

Objective: Complete the classification of regular tessellations in the three geometries.

The Schläfli symbol for a regular tessellation is written $\{n,k\}$, where $n$ is the number of sides on each face and $k$ is the number of faces coming together at a vertex.

Table of Regular Tessellations

The goal of this exploration is to fill in the table below. There is one square for each Schläfli symbol $\{n,k\}$. The columns are numbered at the top with $k$ and the rows are numbered at the left with $n$.

Regular Tessellations
$n\backslash k$ 2 3 4 5 6 7 8
2
3
4
5
6
7
8
1. What are the Schläfli symbols for the five regular spherical tessellations? Put an 'S' in the corresponding squares of the table, and write the name of the corresponding platonic solid in the square as well.
2. What are the Schläfli symbols for the three regular Euclidean tessellations? Put an 'E' in the corresponding squares of the table.
3. Put an 'H' in squares that correspond to regular hyperbolic tessellations. You may use the applets HyperbolicApplet and PoincareApplet to check.
4. There is a symmetry of the table itself. How does this symmetry relate to the tessellations?

Degenerate Regular Tessellations

Tesselations with $n=2$ or $k=2$ are called {define|degenerate tessellations}.

1. What sort of polygons would the {2,6} tessellation require? Is there a geometry which has those polygons?
2. Draw the {2,6} tessellation. Draw the {2,3} tessellation.
3. What are the corner angles of the hexagons in the {6,2} tessellation? Is there a geometry where such polygons exist?
4. Draw the {6,2} tessellation. Draw the {3,2} tessellation.
5. What does the {2,2} tessellation look like? Draw it.
6. Fill in the $n=2$ and $k=2$ portions of the table with the appropriate labels.

Formula for the Geometry

The ${n,k}$ regular tessellation is made of n-gons.

1. What are the corner angles of one of the n-gons in the ${n,k}$ tessellation?
2. What is the angle sum of one of the n-gons in the ${n,k}$ tessellation?
3. What is the angle sum of a Euclidean n-gon?
4. Compare your answers to questions 12 and 13, and find a formula involving $n$ and $k$ that determines when ${n,k}$ is spherical, Euclidean, or hyperbolic.
5. Handin: The Regular Tessellations table and a sheet with answers to the other questions.