Art, Design, and Biology

Geometry in Nature

Shapes in nature are literally everywhere, and given that this part of the unit is mostly a review of concepts covered in earlier grades, students will be asked to find their own examples of shapes in the real world, both in nature and the man-made world. Polygons and polyhedra are the first subject addressed in the PA state standards, but is addressed on a higher level that just identifying and classifying shapes so this section is very much being used as a hook to pull students in and grab their attention with things they’ve already learned and haven’t looked at since middle school. “In nature we can witness the beauties of form fitting function according to the principles of mathematical physics and see… that they are ‘good.¹⁴’”

Symmetry in nature is very common, and is more apparent the more you look for it. Bilateral symmetry can be found in insects and leaves, just to name a few. Butterflies may be the first example students think of, and for good reason. Butterflies are big, colorful, and attract the attention of students and elementary teachers alike.  Radial symmetry can be found in most echinoderms¹⁵ (think starfish and octopuses) as well as the interior of many seeded fruits (like a kiwi or an apple.) These are limited examples, but they are sufficient for students to study in class to gain an understanding of symmetry as something that’s an important part of their lives and not just a concept in class. The image below, photographed by the author, shows symmetry of leaves growing evenly on both sides of a stem.

Plant with symmetrical leaves. Photographed July 2025 in New Haven, CT, by author.

(fig. 4)

Patterns in nature are just about everywhere you look; animal skin is often patterned (zebra, cheetah, leopard,) as are tree branches, seashells, and even our skin! Take a close look at the back of your hand, and you can see all the itty bitty shapes that come together to form human skin. Because the definition of pattern is so broad, we can look for patterns and find them just about anywhere.The more we look, the more we find when it comes to patterns, and seeing patterns in front of us allows our brains to take a break, as Oscar Wilde wrote: “The repetitions of patterns give us rest.¹⁶”  Sequences are also prevalent in nature, especially in the center of flowers, and can be added into this unit as an extension for students who want or need a little bit more than a gentle entry into a year of high school geometry.

Tessellations are precise patterns that are often symmetrical, which makes tessellations a great ending point to bring all of this together. We find tessellations in nature in honeycomb, snake skin, turtle shells, pineapples, and more. These also have practical application into architecture, construction, tiling, and other creative endeavors while reinforcing measurement skills, problem solving, reasoning, and attention to detail. The photograph below, taken by the author, shows the tessellating detail on the skin of a pineapple.

Pineapple Skin. Photographed July 2025 in New Haven, CT, by author.

(fig. 5)