Question

Find examples of tessellations in nature, in magazines, or on the Internet. a. Tell whether the tessellations are regular, semi-regular, or neither. b. Explain which transformations can be used to create the tessellations.

Short answer

Honeycombs form regular tessellations using translation and rotation.

Step-by-step solution

Understand Tessellations

Tessellations cover a plane without any gaps or overlaps using one or more geometric shapes. Regular tessellations use congruent regular polygons, semi-regular tessellations combine two or more types of regular polygons, and tessellations that do not fit these categories are neither regular nor semi-regular.

Identify Natural Examples

Look for examples of tessellations in nature. Some examples include honeycomb patterns created by bees, turtle shells, and pinecone scales, which are formed by repeating shapes covering an area without gaps.

Check Regularity of Tessellations

Examine each example for regularity. The hexagonal cells in a honeycomb are a regular tessellation because they are all congruent regular polygons. Turtle shells may not always form regular tessellations as they can have varying irregular shapes.

Choose Transformations

Identify transformations used in creating tessellations. Common transformations include translations (sliding shapes), rotations (turning shapes around a point), and reflections (flipping shapes over a line). For honeycombs, translation and rotation transformations are evident as each hexagon repeats and rotates to fill the space.

Key concepts

Regular Tessellations

Tessellations are fascinating patterns formed by fitting together shapes without any gaps or overlaps. A regular tessellation uses congruent regular polygons to cover a plane. This means that each shape has equal sides and angles, and only one type of polygon is used.

There are only three regular tessellations:

• The equilateral triangle tessellation.
• The square tessellation.
• The hexagonal tessellation.

These patterns are identical and can continue infinitely in all directions on the flat surface. For example, the hexagon patterns seen in honeycombs are examples of regular tessellations. Each hexagon fits snugly with its neighbors, repeating consistently throughout the structure.

Understanding these regular patterns helps us appreciate the uniformity and consistency required in tessellating a plane with just one congruent shape.

Semi-Regular Tessellations

Semi-regular tessellations are intriguing as they incorporate two or more types of regular polygons. The key character of these tessellations is that the pattern at each vertex is identical, even though more than one shape is used.

There are precisely eight semi-regular tessellations, where examples include:

• A tessellation featuring alternating squares and triangles.
• A tessellation using octagons and squares.

These tessellations create a varied yet systematic pattern. Each vertex where different polygons meet maintains a repeating sequence of shapes. This variation allows for interesting designs while still maintaining a consistent vertex structure.

Semi-regular tessellations are often found in art and architecture, making them a wonderful intersection of mathematics and design.

Geometric Transformations

Geometric transformations play a crucial role in creating tessellations. These transformations include translation, rotation, and reflection. Each transformation manipulates shapes to fit together perfectly, filling a surface without any overlaps or gaps.

• **Translation** involves sliding a shape along a plane. For example, moving a hexagon in a honeycomb uniformly in one direction.


• **Rotation** turns a shape around a point. This allows shapes like tiles to rotate into positions that seamlessly fit with each other.


• **Reflection** flips a shape over a line. This can create symmetrical tessellation patterns, essential for certain artistic designs.

These transformations enable the endless continuity and beauty of tessellations seen in both natural formations like turtle shells and human-made structures.