An Afterthought on Tessellations

Check out this awesome tiling I saw today - this place had its floor tiles cut like Escher's lizard tessellations! 

The four types of tessellations - translations, reflections, rotations, and glide reflections

Food for Thought

Here's a thought - take a slice of cheese. With n number of cuts, what is the maximum number of pieces  you could cut your cheese into? Observe...

Notice how as we add each cut, starting at 0 cuts with 1 slice, we get 1, 2, 4, 7, 11... 

A lovely pattern can be seen in the maximum number of pieces you get where you take the previous number and add 1, then 2, then 3, then 4... perhaps more familiarly as the triangular numbers + 1 or n(n+1)/2 + 1

Continuing with that thought, what would happen if you extended this problem to say, a block of cheese? (going from 2D to 3D). Given a certain number of planar cuts, what is the maximum number of pieces you can create? With 2D cheese, when adding a cut, to maximize the number of pieces you get, your line has to cut through each of the existing cuts/lines. The same is true in the case of 3D cheese, in that each new plane must cut all of the existing planes. The following pattern emerges: 

Notice how with 2D cheese, each time you were adding 1, then 2, 3, etc. but then in 3D cheese, the numbers your adding each time are the numbers of pieces per cut from the 2D cheese problem. With the previous formula, you can derive the following pattern in 3D: (n^3+5n+6)/6

Additional notes: 

• This is actually called the lazy caterer's sequence in 2D and cake numbers in 3D 
• It's pretty cool how for n dimensions, the first n slices will be consecutive powers of 2, and the next one will be 2^(n+1)-1
• For any given number of slices, can you cut the 2D or 3D cheese into equally sized pieces? If so, how, and if not, why not? (I actually have no idea) 

Twisted Tessellations IV - Glide Reflection

Original artwork by M. C. Escher - image source.  To finish off the series of tessellation posts (see translation, reflection, and rotation for more), here's in my opinion the coolest one - glide reflections. It is, like it sounds, a combination of flipping and sliding; notice how here you could take one of the white knights, flip it, and slide it to get to a grey knight. 

Step 1 - Take your square, draw your usual squiggly bit on one side and cut out. 

Step 2 - Flip the piece over and attach to the opposite side. 

Step 3 - Choose one of the two untouched sides, draw another squiggle and cut out. 

Step 4 - Flip it over and attach to the opposite side like you did with the first piece in Step 2. Now your lovely tessellation is ready to trace! 

This one is possibly the hardest of the four to trace because it involves a lot of flipping and fitting squiggly bits together, but if you chose to use paper that has different colors on each side, you can think when you traced the first shape (see the picture for Step 4), the top edge was the red edge with the triangle poking out, so when you flip the piece over, you'd align the piece so the red edge with the space for the triangle would be on the bottom. 

It takes a bit of practice, but the results are quite pretty! 

Further exploration - 

• I happened to use squares as the base shape, but you can definitely use others too - triangles are nice, so are rhombuses. 
• That said, can you start with any shape? 
• These tessellations were all done with only one shape, but can you do something with two, three, four, etc. shapes? 

Mathspiration

Basically, I found this tonight and it deserves way more views than it has. It's pretty awesome.

Twisted Tessellations III - Rotation

Original artwork by M. C. Escher (can you tell I have an Escher obsession?) Image source. A third type of tessellation (in addition to translations and reflections) is rotation, where instead of sliding or flipping the pattern, you spin it about a pivot point. 

Step 1 - Starting again with a square, mark off the midpoint on one of the edges, and draw and cut out any pattern or squiggle you'd like, starting at either corner and going to the midpoint. 

Step 2 - Give the square a quarter turn and, using the piece you just cut out as a stencil, trace and cut out the same piece from the new side, making sure you stay consistent with the first edge (i.e. if you started cutting from the left corner, don't switch to starting from the right corner). Repeat with the remaining two sides.

Step 3 - Now you should have four identical squiggly pieces cut out. On each edge of the square, there is one half that wasn't cut - position the four pieces on the uncut half as shown. If you've done it correctly, you should be able to hold the midpoint and pivot the piece about that point until it fits back into where it was cut from. 

Step 4 - Tape the pieces accordingly and now your shape is ready to trace!

Can you see how this tessellation is both a rotation and a translation?