This is a part of the series of posts introducing an active research field called Tessellation – The Mathematics of Tiling, discussing some basic, intuitive tiling patterns. Subscribe now (link) to get notified about more posts in coming days!

• #1: Some Basic Ideas
• #2: Let’s Start Tiling!
• #3: Irregular Polygons
• #4: Monohedral Convex Pentagonal Tiling

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Before we begin, let’s look at a new word we will be using more often in coming days:

A primitive unit is a section of the tiling that generates the whole tiling using only translations, and is as small as possible. For example, a square or regular hexagonal tiling can be achieved with a 1-primitive unit.

Classification

As discussed earlier, there are 15 types of convex pentagonal tilings. They are classified based on 1. Interior angles (a, b, c, d, e) and 2. The side length (A, B, C, D, E).

Just to be sure we are all talking about the same thing, we will be taking the angles and sides as in the image below:

Type 1 Convex Pentagonal Tiling

Type 1 is one of the five types found by German mathematician Karl Reinhardt in 1918.

Condition 1: B + C = 180°; A + D + E = 360°

Sample tile:

Primitive units: 2-tile

Tiling:

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Condition 2: a = c, d = e
A + B = 180°, C + D + E = 360°

Sample tile:

Primitive units: 2-tile

Tiling:

Puzzles

1. Verify if the second sample tile satisfies Condition 1.
2. Also, check if first sample tile satisfies Condition 2.
3. Find one or two tiles satisfying each condition – try to create them in Geogerab 2D.
4. Which of the following tiles is of type 1? (How will you measure the sides and angles?)

Extra Reading

• Pentagonal Tiling, Type 1 (Wikipedia)
• Karl Reinhardt (Wikipedia)
• Primitive Cell (Wikipedia)
• Geogebra.org
• Pentagonal Tiling (Poster) (Dartmouth College)
• Pentagon Type 1 Tiling (Wolfram | Alpha)
• Interactive Demonstration of Pentagonal Tiling (Wolfram)
• Online Protractor (To measure angles)