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The Beauty of the Season (and Mathematics)

The Beauty of the Season (and Mathematics)

Ever wonder why some people are considered beautiful? Do you know why some painters can create incredible works of art that are considered masterpieces? Why do some buildings and room designs look so appealing? What causes us to pick out what we think is the “perfect” Christmas Tree? The answer is actually quite mathematical! It is two key mathematical concepts, Symmetry, and the Golden Ratio!


Let’s break this down a little. Many factors throughout history have contributed to what we believe adds to the beauty of a person. Height, physical fitness, and physical features have all gone in and out of favor for what defines a person as beautiful. What has never changed over the years is the more symmetrical a person’s face and body is, the more beautiful that person is perceived to be.


The brain reacts to facial symmetry specifically as pleasing and relates that symmetry to beauty. This is true for both males and females. Study after study has shown that a person with the most symmetrical facial features will be identified as the “most beautiful.”

face symmetry example

Mona Lisa

Mona Lisa painting symmetry

Paintings like the “Mona Lisa” are considered masterpiece works of art due to the use of both symmetry and the Golden Ratio by Leonardo DaVinci. In the painting the facial features are extremely symmetrical. They are painted perfectly. Most experts say they are painted better than the person who the painting is modeled after.


The proportion of her face and body are also in the perfect proportion. It is the proportion that is called the Golden Ratio. The Golden Ratio is a mathematical ratio. It is commonly found in nature. When used in a design or painting like this, it fosters organic and natural-looking compositions that are aesthetically pleasing to the eye.

Golden Ratio

The Golden Ratio shows up in so many places and in so many ways. There are many fun and interesting lesson plans that teachers in Math, Science classes like Biology and Anatomy, and Art classes like Drawing, Painting, and Sculpture can use. There are many teacher and classroom ready lesson plans about the Golden Ratio Texas Instruments puts out a couple of examples of these lesson plans. Golden Ratio Human Body and Golden Ratio Faces.

golden ratio example
golden ratio example

Christmas Trees

This brings us to the Holiday Season. Why do we pick the Christmas trees we like the best? You guessed it. It is because of their symmetry. A great activity for both art and math students is about learning symmetry by painting Christmas Trees. You can see this activity at:


There are actual instructions for the proper pruning of Christmas trees for Christmas tree farms so that they produce the “perfect” Christmas tree. Who would have thought that to be the best at growing Christmas trees you would have needed to have a keen awareness of mathematical symmetry?


And what about the ornaments we put on the Christmas trees?

A sphere or ball has infinite lines of symmetry. A Star has multiple lines of symmetry. Look at these ornaments. Can you see all the lines of symmetry?

wooden Christmas ornament shapes
wooden Christmas ornament shapes


Do you remember as a kid cutting out snowflakes to hang from the ceiling in the wintertime? Do you remember how you created the different snowflake designs? Usually, you started with a square piece of paper. Then you folded it in half and folded it in half again and again a few times. Then you cut geometrical shapes and designs. When you unfolded the paper, you had cut out an amazing snowflake. Each of those folds you made in the creation of the snowflake before you started cutting, created symmetry. There are many creative math, art and science lessons that can be done but cutting out those paper snowflakes. From cutting out simple paper snowflakes to very ornate and complex snowflakes online

paper snowflake instructions

Seeing the Beauty

As Math Teachers we need to make sure we show the students the beauty of the mathematics we teach. People laugh at me when I look around and see mathematics is so many places and talk about the beauty of mathematics. They comment on the beauty of math?!? Really?!? I tell them yes math is beautiful. At this point I get a minor tirade of how bad they are at math, how much they do not like math, and how boring and awful their math teacher was in middle and high school.


I then talk to them about all the math they see around them every day and just how beautiful it is! I will talk about the things I have mentioned here so far. If we are in a city, I ask them if they notice all the ornate frieze designs that are found on so many buildings in the city. Have you ever noticed the frieze designs on buildings? Do you even know what frieze designs on buildings are? You really should learn about these beautiful designs that are found all around us in so many places.


In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Such patterns occur frequently in architecture and decorative art. They are used frequently in architecture, clothing, borders around objects, and yes often during the holiday seasons. If you take the time to learn about frieze designs, you will start to notice them all around you and appreciate the beauty that they add to their surroundings.


I am always amazed when I am in the cities with all the ornate craftsmanship and incredible designs you will see. They are all different and actually can have cultural significance. There are fun and interesting lesson plans for frieze designs.  Here is an example of an interesting lesson plan for frieze designs. If you look at it, you too will be intrigued. Below is an example of some holiday frieze designs. Do any of them look familiar to you?

Christmas pattern
Christmas pattern


Overall, I hope that math teachers not only teach the structure of the mathematics but also the beauty that it holds. Some of the more common concepts we teach in math like symmetry and ratios can hold some of the most profound beauty. If we take the brief moments to look around, we will see just how often mathematical designs and ideas are used to create much more beautiful and pleasing surroundings for us all.


I hope that you will take the time this holiday season to notice and comment on all the mathematical beauty we have around us on a daily basis. My wish for you all is a very safe and happy holiday season. Please enjoy a little rest and relaxation and precious time with family and friends. Most of all notice all the incredible mathematical beauty that is everywhere you look.



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