How did Escher make his tesselations?

Question

Question for the: Topic Challenge #1 – Famous Designers

I remember as a kid my favourite book in the whole house wasn't a book of fairy tales or colourful illustrations. It was The complete works of M.C. Escher. I could stare at his works for hours on end. Trying to wrap my head around how those little people could go up and down the stairs forever.

But what intrigued me even more were his Tessellation pieces, where he would divide a plane into interlocking figures, such as here:

Later, in his Metamorphosis pieces, he would take this a step further and morph one tesselation into another, like so:

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Nowadays, it's pretty trivial to get a computer to calculate an optimal tesselation, but Escher didn't have any of that. And the metamorphosis works add a whole new level of complexity even a computer would find hard to do. So my question is quite simply: How did Escher make his art? If I wanted to do something similar today, how would I go about it?

Answer 1

With a pattern.

In this case a 3 axis grid (triangular).

Once you know what to draw on each piece, you need to repeat this. You can have and use sub-patterns or smaller ones to be more exact.

These patterns are pretty easy to draw, and they are used for example in architecture in different cultures. We are used more to a square pattern, but this triangular pattern can produce hexagonal and rhomboidal patterns as well.

And you can play with it to start building ripples, but still, you repeat the internal objects on this now deformed patterns.

Look how many patterns you have with this triangular grid.

Grab a paper and a ruler, draw some pages and find some more patterns!

This is a typical example of introductory classes at the University. We called it "Little Squares 101" Or "Sticks and Balls II" (That was the second course) and yes, we drew this by hand.

You can also see these patterns in 3D often used in Architecture.

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Regarding the comment:

I really see no difficulty to draw this lizard by hand. Look at the second image, it clearly marks the middle of the triangle and where the legs should intersect them. I would probably have a reference drawing but draw those by hand. Especially if the next lizard will turn into a duck... Metamorphosis...

Additionally, comparing two lizards they are not exactly the same.

When you are using a pattern, you let the pattern guide you.

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Edited some years later. Let us explore how Escher used these grids to develop part of his work.

Here is a screen capture of the website: https://mcescher.com/gallery/symmetry/

where you can see how he used a two-axis grid rotated 45° next to a work using a three-axis grid. The grid even shows in pencil.

The grid is only a starting point. You have some other resources like mirroring, rotating, and scaling. But art is about taking the resources you have as a guideline, not as a limitation.

Here the "deformation" of the grid is extreme, the grid itself has its own new shape:

Source: https://mcescher.com/gallery/mathematical/

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Remember that you can create subpatterns (smaller ones) or superpatterns (bigger ones)

Answer 2

Actually this lizard is created on the basis of a hexagon, not a triangle! Here is the hexagon:

Here is a movie how to deform the hexagon to get the lizards:
https://youtu.be/T6L6bE_bTMo?t=10

You might be tempted to think a triangle would suffice to make lizards because we may divide a hexagon into six triangles. How do we know a triangle would not suffice? Because then each triangle has to have a different pattern painted on it and we need six such triangles. Let's check for example the triangle made with vertexes where all three lizards heads meet:

Of course, we can tessellate this triangle (or any triangle cut out from original drawing) but it will not produce lizards. The poor beasts made from this triangle would not have limbs on the right side. We need all six triangles to make the Escher's tessellation complete. The hexagon is the smallest geometric figure which makes the lizard tessellation possible with a single pattern.

Here might be interesting reference for more on tessellation:
https://www.livescience.com/50027-tessellation-tiling.html

You might be interested in this app:
https://math.stackexchange.com/a/3289908/624901