Tessellation describes the covering of a surface with geometric shapes, fitted together without any gaps or overlaps. These formations are not human-made designs; they appear frequently throughout the natural world in various contexts. The resulting patterns are regularities of form that recur across different environments and can be described with mathematical principles.
Understanding Tessellating Shapes
For a pattern to be a tessellation, the points where the corners of the shapes meet, called vertices, must align to fill the entire plane. This requirement means that the sum of the angles around each vertex point must equal 360 degrees.
Certain regular polygons can accomplish this on their own. For instance, squares have four 90-degree angles, so four squares meeting at a point create a perfect 360-degree angle. Similarly, six equilateral triangles, each with 60-degree angles, can join at a single vertex. The hexagon is another shape capable of forming a regular tessellation, as three hexagons with their 120-degree angles meet precisely to fill the space.
Animal Kingdom Tessellations
Tessellating patterns are observable in the structures built and inhabited by animals. The honeycomb, constructed by honey bees, is a well-known example where hexagonal wax cells are built side-by-side. This arrangement of repeating hexagons forms a strong storage structure for honey and pollen.
Protective outer layers on many animals also display these patterns. The shells of turtles and tortoises are composed of plates called scutes that grow in a tessellating arrangement, providing a durable shield. A similar pattern can be seen in the scales of many fish and reptiles. The compound eyes of insects, such as dragonflies, are another instance, where hundreds or thousands of individual lenses are packed together in a hexagonal grid to provide a wide field of view.
Plant Kingdom Tessellations
The plant world offers numerous examples of natural tessellation, often visible in the structure of fruits and flowers. The skin of a pineapple exhibits a distinct repeating pattern of diamond-like shapes that are individual fruitlets fused together. This patterning is a result of the pineapple’s growth process from a cluster of flowers.
Another striking example is found in the seed arrangement of a sunflower head. The seeds form interlocking spirals that create a tessellated pattern, allowing for a high density of seeds to be packed into the circular space. The florets of a Fritillary flower also show a characteristic checkered, or tessellated, pattern on their petals. Similarly, the kernels on a cob of corn are arranged in tightly packed rows that cover the surface completely.
Tessellations in Geological Formations
Non-living systems also produce tessellations through physical processes. Columnar basalt, found in locations like the Giant’s Causeway, consists of towering columns of rock, most of which are hexagonal. These columns formed as thick lava flows cooled and contracted, causing fractures to develop. The cracks propagated downwards, creating a network of polygonal columns.
Another common geological tessellation occurs in the form of desiccation cracks, or mud cracks. When a layer of mud or clay dries, it shrinks and cracks, forming a network of polygonal shapes. Initially, these cracks may form varied shapes, but over repeated cycles of wetting and drying, the junctions between cracks tend to settle into angles of about 120 degrees, favoring the formation of hexagons.
Why Nature Chooses Tessellations
The prevalence of tessellating patterns in nature is often a result of efficiency and physical constraints. In many biological systems, tessellations represent a solution for packing cells or other units together with minimal use of materials. The hexagonal cells of a honeycomb, for example, enclose a maximum amount of space for a given amount of wax, making it a highly efficient shape for storage.
This principle of resource optimization also applies to structural integrity. By fitting together without gaps, tessellated components create stable and strong surfaces, such as the protective shells of turtles or the scales of fish. Physical forces at play during formation processes, like the cooling of lava or the drying of mud, also naturally lead to these patterns. Cracking releases tension in the most energy-efficient way, which often results in hexagonal patterns.