Transformation Tessellation Tuesday

Last week I demonstrated how you can transform a single-shape tessellation to make additional single-shape tessellations. This week I will demonstrate transforming a multiple-shape tessellation.

First, I begin with a 4-patch block as shown below.

This is what nine blocks looks like.

There are four distinct Shapes in this tessellation. I have outlined them below.

Now, in order to transform these shapes, you will have to "borrow" a piece of one Shape to add to another. Below, I have shaded a piece of the smaller Shape and shaded where that same piece touches the larger Shape. The lower tessellations show how the Shapes have been transformed by adding the shaded pieces to the larger Shape, and deleting the shaded pieces from the smaller Shape.

Below, I have shaded a piece from a different Shape and also shaded where that same piece touches the larger Shape. The lower tessellations show how the Shapes have been transformed by adding the shaded pieces to the larger Shape, and deleting the shaded pieces from the medium Shape.

You are neither limited to the number of pieces you trade between Shapes, nor the number Shapes you transform. Below, I have shaded two different pieces of two of the Shapes. The remaining tessellations show these pieces added to two different Shapes. Yet it still remains a multi-shape tessellation, though quite different from the Shapes I outlined at the beginning of this post. Amazing!!


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Transformation Tessellation Tuesday (with a nod and a wink to LoveBug Studios*)

Today, I will discuss tessellations and how to transform them . . .

What is a tessellation? A tessellation is "one or more congruent geometric shapes that cover an infinite plane without gaps or overlaps." Huh? How about "one or more shapes that fit together to make a repeating pattern"? The Dutch graphic artist, M.C. Esher (my hero), is well-known for his tessellations.

I will begin with my design tile arranged as a 9-patch block . . .


All nine tiles maintain the same orientation, that is, they are all facing the same direction. This is an example of the operation of symmetry called Translation. Any given tile can be shifted up, down, left, right, or diagonally onto its neighbor without changing the overall pattern.

Speaking of pattern, do you see the Shape outlined below? (Notice that I refer to this as a capital "S" Shape.)


This qualifies as a tessellation because it is a single Shape that fits with itself to make a repeating pattern. Here are four 9-patch blocks, with complete Shapes outlined.


Let's take another look at the basic Shape . . . it is made up of six contiguous shapes, that is, six smaller shapes that touch each other. In fact, these are the six shapes that make up the original design tile!! One shape each from six different design tiles make up this larger Shape. (These individual shapes are lowercase "s" shapes.)


Here's where the "transformation" comes in . . .

Below, I have shaded in the shape at the bottom of our Shape . . . I have also shaded in the same shape at two other points where they touch the Shape.


By using one of the alternative shapes, I can make two different Shapes that are also tessellations.





Try it yourself with one (or more) of the other shapes in the Shape . . .


Next week: how to transform multiple-Shape tessellations!!

*You can check out LoveBug Studios here . . .

As always, thank you for reading my blog . . . please feel free to leave a comment or follow this blog!!