Symmetry and Butterflies

Hope you are enjoying the summer. Recently I moved to Edinburgh and I have started visiting random things. One of the first things I wanted to see was the Edinburgh Butterfly & Insect World. Butterflies are some of my favorite creatures ever. Beside their incredible beauty, I find them extremely interesting (especially the process caterpillar – pupa – adult).

Butterflies have a special place in mathematics, especially when it comes to early school years. I think most of you remember seeing at least one image of a butterfly when studying symmetry. Looking at a butterfly’s wings, it is easy to understand the concept of symmetry, more specifically reflection. With this in mind, the next definition is again very easy to understand: “ The line of symmetry is the imaginary line that divides a figure into 2 congruent parts, each of which is the mirror image of the other. You could fold the image and have both halves match exactly.” Looking at any butterfly, this definition becomes basic knowledge and a trivial statement.

But there is something we need to be very careful about when talking about symmetry. Butterflies are just an example of symmetry in biology, where this notion is mostly used explicitly to describe body shapes. Butterflies are, theoretically speaking, bilateral animals (like humans) and they are more or less symmetric with respect to the sagittal plane which divides the body into left and right halves (this sounds incredibly similar to the mathematical definition for line of symmetry). But, symmetry is way more than this.

Mathematically speaking, we say an object is symmetric if it is invariant (doesn’t change) to any various transformation (including reflection, rotation or scaling: enlarging or reducing). More specific, it means that however you want to transform the object it will always be similar to its original version. If we want to be very rigorous (mathematically), we can say “A mathematical object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation preserves some property of the object.” For mathematics, maintaining the same property after some sort of operation is key to understanding many advanced concepts.

Looking at the beautiful patterns of a butterfly’s wings is the first step at understanding concepts from calculus (even vs odd functions), linear algebra (think about matrices a little), abstract algebra (symmetric group), statistics and way more. Moreover, symmetry is not a concept that appears just in mathematics or biology; physics talks about invariance (lack of change); chemistry uses it a lot; architecture and art breath symmetry; the list could go on and go. Feel free to comment bellow with any example of where symmetry is used.

Hope you enjoy this post. More nature related posts are coming soon. Let me know what you’ve been up to in the past month. Have a great day. You can find me on Facebook,  Tumblr,  Google+,  Twitter   and  Instagram. I will try to post there as often as possible.

Lots of love and don’t forget that maths is everywhere! Enjoy!