Wednesday, October 29, 2008

Wednesday Math, Vol. 45: Regular tilings of the plane

A tiling of the plane is the covering of all of a flat two-dimensional surface with a repeating pattern. The fancy word is tessellation.

Think bathroom tiles. Think M.C. Escher. Think mosaics.

If you are also thinking about the renowned physicist Roger Penrose and his tilings, you get extra mathy Matty Boy bonus points.

Let's start simple, with tilings that involve regular polygons, straight line shapes where all the line segments are of equal length and all the angle measures are also equal to each other.

If we use only one regular polygon repeated over and over, there are only three possible tilings of the plane. The most famous is the square grid. Also well known is the hexagon tiling, which is the shape bees use in a honeycomb, and the third is a triangular tiling. All the angles at any corner must add up to 360 degrees, so with a triangle it's 6x60, with a square it's 4x90, and the hexagon gives us 3x120.

So far, so good.

There are also tilings where we mix and match two or more regular polygons to fill all the space. The most common of these uses octagons, where an octagon is surrounded north, south, east and west with other octagons, and the gaps created are squares. This is a well known linoleum tile pattern.

We can do something similar the with twelve sided regular polygon, known as a dodecagon. If it has dodecagons as neighbors at six of the sides, the gaps are regular triangles.

If we have dodecagons meet each other north, south, east and west, the gaps created can be filled with four triangles surrounding a square.

There are several ways to use triangles and hexagons together to tile the plane.

Of those, this one is my favorite.

Likewise, there are several ways to use triangles and squares together, and this one is my favorite of these.

The last mix and match tiling I present here is this pattern using hexagons, triangles and squares together. If you look at a hexagon and the squares and triangles that surround it, those shapes together define a dodecagon.

There are plenty more. If you want to try your hand at finding some, knock yourself out. It makes for fun doodle time.


pissed off patricia said...

I think I just dislodged something in my brain.

dguzman said...

I feel a little dizzy...

Undersquid said...

My favorite way to play with shapes such as these is called quilting. I love matching two-dimensional fabric shapes that will also keep me warm.

Spiffy entry!

Matty Boy said...

PoP, it's just pretty pictures. Go with it.

dg, some of the patterns are a little dizzying, especially the second to last. Still, I like them.

U-Squid, patterns on quilts can be very cool. Keep up the good work.

BobManDo said...

Good Stuff Mathy Boy... Just a minor matty correction :-)

You wrote: " with a triangle it's 6x60, with a square it's 4x90, and the hexagon gives us 3x120...."

Should be:
" with a triangle it's 3x120, with a square it's 4x90, and the hexagon gives us 6x60...."

Matty Boy said...

Hey, Bob. What I was talking about was the degrees at a corner. In the triangular grid the angles are 60 degrees at each corner, and it takes 6 triangles at each place they meet. You're right about the squares, but with the hexagons, there are three at each corner, and each has 120 degrees.

namastenancy said...

Pretty graphics but I'll be damned if I can understand the math. Still, you could take this art and promote yourself as one of the new geometric artists. You are that good!

BobManDo said...

Right you are MattyBoy... I was looking at the space inside and you were talking about the meeting point.