Picture yourself in Northern Africa, on a tour of one of the mosques. The first thing you notice is the interesting tile pattern on the floor. Next, walking around the property you admire all the intricately designed carvings on the various columns and panels. You can't help but wonder, who created these patterns and why they chose to make them the way they did. The vibrant colors, simple yet intricate shapes, and soothing repetitious pattern both astound and calm you. It's an interesting feeling you've never quite experienced before. What were these marvelous artistic patterns? How did they come to be?

We call these tessellations. They have slightly different definitions whether you're talking about a mathematical or artistic standpoint. Robinson (2019) defines these as "shapes, patterns or figures that can be repeated to create a picture without any gaps of overlaps" when referring to the artistic side. When talking about the mathematical component, she goes on to say that mathematically, " tessellations are recognized as coverings of a plane or surface without any gapes in the ' tiling '." When we combine these two ideas, we recognize that we have a two-dimensional surface that has some sort of pattern or repeated shape which leaves no gaps. These repeats can be either in shape, color, or both.

No one is quite sure where tessellations first originated. What we do know is that they been traced as far back as the Sumerian civilization in 4000 B.C. These people would use clay tile patterns to create decorative features in their homes. Others claim to have found tiling pieces along the Nile River that date back between 12,000 and 18,000 years. (Robinson, 2019). There are also tessellations that we can see in Greek and Muslim architecture, especially in their temples. This practice of using tessellations in sacred places later spread into Moor and Christian artwork. There have also been various tessellations found in Roman, Persian, Egyptian, Arab, Japanese, and Chinese art and architecture. Each civilization modifying the pattern slightly to help fulfill their culture and tradition. Some created these new tessellations while others simply observed their natural occurrences to help explain some phenomenon's they were experiencing, later leading to many geometrical breakthroughs ("From Mathematics", 2016).

No one knows for sure who put the first tessellation together for sure. However, more of the work is contributed to Mauritus Cornelius (M.C.) Escher in the artistic realm and Sir Roger Penrose in the mathematical context. M. C. Escher was a Dutch graphic artist who lived from 1898 to 1972. He originally wanted to study architecture but later transitioned into graphic design. Sir Roger Penrose was a mathematics professor at Oxford.

As previously discussed, tessellations are "any repeating pattern of symmetrical and interlocking shapes" (Khaira, 2009). Some places you might be familiar with these in your everyday life would be a decorative backsplash in a kitchen or a fun tile pattern in a bathroom. Outside the home, common examples appear in honeycomb, flower petals, reptile skin, giraffe spots, or decorative fencing (Fathauer, 2015). Some occur naturally, such as the honeycomb or animal skins. Others, we end up creating ourselves. However, there isn't just one type of tessellations. There are varying types that we enjoy using for various purposes.

When people think of tessellations, they commonly think about what we refer to as regular tessellations. This is the simple regular polygon set in a repeating pattern, often classified by the number of sides the polygon being used has. Some examples of this are trihedral and tetrahedral, for triangle or quadrilateral shapes ("From Mathematics", 2016). This is what we typically think of when we talk about bathroom tiling or honeycomb. When we start moving into semi-regular tessellations, we start to see some more intricate patterns immerge. These tessellations have more than one kind of regular polygon (Coolman, 2015). This would include patterns like you would see on a typical soccer ball or a pattern interior designers enjoy using where they have a regular octagons and use squares to fill in the spaces between some of the octagons. These are just the simple patterns that people can easily wrap their heads around.

We then start to move up in complexity of both the patterns and interactions of the shapes we use. The first type in these more complex tessellations is called a monohedral tessellations. Monohedral breaks down to mean "one shape". This means that it is similar to our regular tessellation. The only difference between those two is that in our regular tessellation, the shape always faces the same direction and looks identical. When talking about our monohedral tessellation, the shape can be rotated or reflected (Coolman, 2015). This causes the same repeating shape to occur throughout the piece, however it would look dramatically different from our regular tessellation piece.

There is also another interesting point regarding the monohedral tessellations. This is in the concept of duals. When we talk about duals in a tessellation, they are actually simple to create. The dual comes from taking the given tessellation, drawing a dot in the center of each shape, connecting all of those dots to each of the neighboring shapes center dots, then erasing the original given tessellation (Coolman, 2015). In many cases, especially some specific cases, these duals are reversible. Let's take this in a context that will be relatively simple to visual. Picture a square grid, much like a checkerboard. In the center of each square, you draw a dot. You then taken each dot and connect it to the dots in the adjacent 4 squares. If you're drawing this out, you can see that you'll have a new square grid. There will be a chart from Coolman to the right where you'll be able to see just how these duals interact.

We can also bring Alhambra-style and Escher-style tessellations into the conversation at this point. Alhmabra-style tessellations are very similar to what we described as semi-regular tessellations. These Alhambra-style tessellations have the same repeating pattern; however they have multiple regular polygons that are used to create these patterns. These patterns were commonly used in Spain around the 14th century. Some famous works can be found in various cathedrals built around that time period ("The Beginnings"). Escher-style tessellations get a bit more complicated. Instead of easily recognizable polygons, Escher enjoyed using animals, plants, or even people as the basis of his tessellations. He would take something simple, like a regular hexagonal pattern, and then change it to something like a gecko pattern. A visual of this will also be provided (Coolman, 2015).

So now you may be asking yourself, "well how do we know if I can make a tessellation with a certain shape? Is there a way to know?" If done properly, the answer will most likely be yes. Khaira (2009) walks a through a fantastic explanation of the how and why our tessellations, particularly regular tessellations, work. When talking about a regular tessellation, the vertex of the shape must be able to meet and complete 360 degrees without any gaps or overlaps. What do we mean by this? Take, for example an equilateral triangle. We recall from our geometry experience, that all angles are 60 degrees. Let's say we picked one of the vertices and we wanted to create adjacent, congruent, equilateral triangles all centered at the one vertex. This is possible because basic algebra tells us that 360/60=6. Therefore, we would be able to fit 6 triangles around that one point without overlap or gaps.

There are a couple other formulas that are important to note, which Taalman and Hunsicker (2002) give to us. The first being a=180-360/n. This formula gives us the interior angle measure of any regular polygon where a is the angle measure and n is the number of sides. It is also crucial to recognize the rule the formula k(n)=360/a=2n/(n-2) plays in determining if a regular polygon will tessellate. Again, a is our angle measure, n is the number of sides of the regular polygon, and k(n) is the number of polygons needed to create the tessellation about a given vertex. Knowing those bits of information, we can then see how shapes like triangles, rectangles, and hexagons would easily tessellate. This doesn't work so well for shapes like a pentagon. If we were to work that out, we see that our a=108, which would mean that k(n)=3.333. That shows overlap, meaning we can't use pentagons to perform this type of tessellation.

Rotating about a vertex isn't the only type of transformation that is out there. We also need to consider other types of transformations and symmetries that exist. Hopefully most people are familiar with words such as rotations, reflections, translations, and glide reflections. Khaira goes on to explain these various techniques and their applications to creating tessellations.

Translations are used when we simply slide the entire image in a specific direction. These often get referred to as pre-image and image. How far we create the image from the pre-image is given to us by some sort of vector. This vector tells us both the angle and magnitude at which we create this new image from our pre-image.

We talked a little bit about rotational symmetry when we talked about rotating about one of the vertices of a regular polygon. Something crucial to note is that the rotation doesn't always have to occur about a vertex. It can also be around a designated point that is not a vertex of the shape. A subsection of this rotational symmetry occurs when we create a reflection. This reflection takes a specific axis that is some distance away from the given shape. It then has each point on the shape have equal distance from this axis line, on either side. The easiest way to help visualize this is imagining a point on a coordinate grid that is located at (2,1), with the y-axis as our reflection axis. The point would then be reflected and end up at (-2,1). This becomes true for every point on our shape we are trying to tesselate.

The final translation that was mentioned was a glide reflection. The name is pretty self-explanatory. The reflection part comes from the reflection as just described. The glide is simply a translation in some direction. It doesn't matter which occurs first. The glide reflection will result in the same image from a given pre-image.

This is all great for regular polygons that we can easily tesselate. But what about non-regular polygons? What about regular polygons that cannot be tessellated given our above definitions? This depends on the type of polygon you are interested in tessellating. Luckily for us, Kharia also gives us some insights into the methodology of tessellating non-regular polygons as well.

When we have triangles or quadrilaterals that are not equiangular, we need to figure out how we can fit these shapes in such a way that the angles of each shape at a particular vertex will be equal to 360 degrees. The other caveat is that all the sides need to be lined up to create a smooth tessellation. The best way to ensure this is possible is to create summation of angles that gets you 180 degrees, and then reflect that group to produce the full 360 degrees at the vertex.

The purposes of creating a tessellation will defer depending on the end goal. For a middle school or high school classroom, the purpose might be that the students understand the basis of translations and regular polygons. Interior designers want to maximize appeal of a home for someone to purchase it. Artists create them simply for the visual enjoyment of others. For this purpose, we would be thinking more in the direction of a school class. We want the students to start by creating a regular polygon. They can create a simple tessellation by tracing this shape end on end until they completely fill out their paper. That would be a very easy visualization and they would be able to comprehend that type of tessellation.

However, students usually aren't impressed by simple tessellations that look like something they'd see on a backsplash in a kitchen or as the pattern on a bathroom floor. They want something more exciting. This is when the students would get introduced to a more Escher style tessellation. A wonderful YouTube video by Mr. Otter Art Studio (2018) helps to walk us through the basics of creating an Escher style tessellation from a square piece of paper.

In this process, the individual takes the square and along the bottom side cuts out some sort of pattern, being careful to not cut into any other side. The old outside edge of the bottom piece is then translated and taped to the top of the square, making sure the straight edges line up perfectly. We can repeat this for our left side, cutting a pattern into that and taping it to the right side.

Once this is completed, the student would then trace out the outline and perform translations to continue with their tessellation. In some special cases, the student may even be able to perform a rotation with their tessellation, but that is something that would have to be looked at on an individual case-by-case basis.

Tessellations are used in many different realms. There are natural tessellations that just occur in different situations around us. We have tessellations that man creates for a visual appeal in housing and other architecture. There are also tessellations used for spiritual or religious purposes and decorations.

On a casual stroll through a park, we might notice blooming flowers, the way their petals all seem to align perfectly to create a beautiful image. Pinecones littering the walkway, their perfect symmetrical spiral awe-inspiring. If this stroll happens to be through a zoo, noticing the repetitive patterns on the feathers of different birds, or the perfectly partitioned scales on the backs of various reptiles.

Touring through various houses in search of your own place, there's many places in which tessellations can take place. Some as simple as regular tessellations, giving a little personality to a backsplash, but not overpowering the natural beauty of the kitchen. Construction workers had already taken the power of simple tessellations into account when creating a perfect driveway and basketball court in the backyard. The interior designer had gone above and beyond, creating a stunning semi-regular tessellation pattern using tiny tiles on the bathroom floor.

Various Islamic architecture and art contain these tessellation patterns. For example, there is a monument with various inscriptions from the Quran that is found in Iran. On this monument is an inscription that translated means "Did you not think we created you in jest?" Alongside this question in an inscription of a pentagon. As we discussed earlier, a pentagon does not follow a regular tessellation pattern. We can use it as part of a semi-regular tessellation pattern, but it alone cannot stand in a regular tessellation. Bier (2002) alludes to the idea that given this phrase with the accompanying pentagon, there is a deeper meaning as to why the pentagon was the chosen shape to accompany this quote.

In short, tessellations are all around us. Whether they are simple repetitive regular tessellations, complex Escher-style, or a unique dual of a semi-regular tessellation, we find them all around us. All we have to do is look. The more we understand them, the more we can appreciate their beauty.

Try creating your own tessellation by clicking here .

Bier, C. (2002). Geometric Patterns and the Interpretation of Meaning: Two Monuments in Iran. In Bridges: Mathematical connections in art, music, and science (pp. 67-78). Bridges Conference.

Coolman, R. (2015, March 04). Tessellation: The Geometry of Tiles, Honeycombs and M.C. Escher. Retrieved September 29, 2020, from https://www.livescience.com/50027-tessellation-tiling.html

Fathauer, R. W. (2015). Real-World Tessellations. Proceedings of Bridges, 107-112. Retrieved September 29, 2020, from https://archive.bridgesmathart.org/2015/bridges2015-107.pdf.

Khaira, J. (2009). What are Tilings and Tessellations and how are they used in Architecture? Young Scientists Journal, (7), 35-46. Retrieved September 29, 2020, from https://www.ysjournal.com/wp-content/uploads/Issue07/What-are-Tilings-and-Tessellations-and-how-are-they-used-in-Architecture.pdf.

Robinson, P. D. (2019, March 02). Who Invented Tessellations? Retrieved September 29, 2020, from https://sciencing.com/who-invented-tessellations-12759386.html

Mr. Otter Art Studio. (2018, February 1). How to make a Tessellation – step by step tutorial {Video}. YouTube. https://www.youtube.com/watch?v=5FIOF8xsPas

Taalman, L., & Hunsicker, E. (2002). Simplicity is Not Simple. Mathematical Association of America, 5-9. Retrieved September 29, 2020, from https://www.maa.org/sites/default/files/pdf/upload_library/22/Evans/september_2002_5.pdf.

Tessellation Patterns. (n.d.). Retrieved September 29, 2020, from http://www.spacemakeplace.com/tessellation-patterns/

Tessellation Patterns - From Mathematics to Art. July 10, 2016. Retrieved September 29, 2020, from https://www.widewalls.ch/magazine/tessellation-mathematics-method-art

The Beginnings. (n.d.). Retrieved September 29, 2020, from http://www.tessellations.org/tess-beginnings.shtml