Math in Bloom

In mathematics, a tiling is a collection of subsets of the plane, covering a flat surface with non-overlapping shapes with no gaps between them. This is also called tessellation.

Tessellation specifically implies a repeating, predictable pattern of polygons, where tiling is a broader term for any arrangement of shapes covering a surface, which may or may not be repeating.

Tiling can be found abundantly in the world around us: stone walls, honeycomb, the patterns on a turtle’s shell. Even our bodies can be thought of tilings in which the ‘tiles’ are cells!

These repeating, gapless patterns of shapes are important in real life for optimizing structural stability and maximizing space efficiency while also being aesthetically pleasing. For example, the hexagonal geometry of honeycomb maximizes strength while minimizing material usage. By sharing walls, the cells distribute stress evenly to prevent weak points and support heavy loads. This pattern found in nature serves as a model for engineering materials in aerospace, construction, and packaging.

A large part of mathematics is the identification and exploration of patterns, and symmetry is a pattern that appears repeatedly in many places.

For two-dimensional geometric shapes, there are four fundamental types of symmetry: reflection symmetry, rotation symmetry, translation symmetry, and glide symmetry.

• Reflection symmetry: a shape can be mirrored on a line and look the same as it did before.
• Rotation symmetry: a shape can be rotated around a point and look the same as it did before.
• Translation symmetry: a shape can be shifted a distance in a direction and match up with its previous image.
• Glide symmetry: a shape can be shifted a distance in a direction and then mirrored on a line to match up with its previous image.

Symmetry is frequently used in garden design to create balance and uniformity. Symmetry when planting can make the space feel more formal. Think of a row of hedges lining a pathway. Asymmetry in garden design creates a more organic and relaxed feel. Imagine a naturalistic cottage garden featuring a single off-center tree.

What types of symmetry can you spot in the garden bed designs around you?

In mathematics, infinity is a concept that represents boundlessness and is unmeasurable because it has no endpoint.

Some examples of infinity in math include:

• The endless sequence of natural numbers (1, 2, 3, 4, 5…count as high as you possibly can, the numbers go on forever!)
• The decimal representation of Pi (3.14159…) never ends and never repeats.
• The Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13…) is an infinite series of numbers, where each number is the sum of the two preceding ones.

The water cycle is an example of a natural, infinite, closed-loop system. It is an “endless loop” or continuous process driven by the sun, where water is perpetually evaporated, condensed, and precipitated, rather than an infinite supply.

While water on Earth is recycled, it is a finite resource, with 97% being saltwater and only a fraction of fresh water available for human use.

Rain gardens restore the natural water cycle by capturing polluted stormwater runoff from impervious surfaces like roofs or sidewalks into a shallow depression. They filter contaminants through soil and plant roots, promoting groundwater recharge, reducing erosion, and preventing flooding. The native plants planted in this rain garden absorb water through their roots and release it back into the atmosphere as vapor.