Penrose sketches, 19.6.11: Five fold symmetry in nature and with tiles.

Five fold symmetry appears in many shapes in nature. The most famous is probably the starfish, but many flowers like the one pictured above have a pattern that starts at a central point and radiates out just about equally in five directions, each one a 72 degree turn away from its nearest neighbor.

Here is a simple Penrose tile pattern with five fold symmetry that resembles a five fold symmetry flower. I used fifteen yellow kites and topped of that central pattern with ten blue darts just to highlight the yellow more, since the pattern is currently fixed to my off-white refrigerator. Right now, I am experimenting with this fifteen kite pattern to see what can be made with it. Recall that we can make larger shapes similar to the kite using Penrose tiles.

This is my favorite design using fifteen kites. Depending on my mood, I see a gear in a machine or a modern logo. More whimsically, I see a hitchhiking cartoon bird whose head, feet, tail feathers and hitchhiking fist with thumb are all the same size.

Here is a sketch for a larger work using the fifteen kite shape, this time with fifteen Daddy Kites, each made of five kites and three darts. My first idea was to make the outside of the pattern all purple, but with my 216 tiles, it was not possible.

I call this a sketch because some time this week, I should be getting another 432 tiles from SeriousPuzzles.com, as well as some other toys I will be playing with. Then I will be able to make the final version of my original concept using Grandaddy Kites, which should be made of thirteen kites and eight darts, but instead are composed of twelve kites, seven darts and a gap the shape of a large dart, which is not possible to make, as proven here last week.

I'll be giving SeriousPuzzles.com some link love on all my Penrose tiling posts, and they will be using some pictures of my shapes and patterns on their website.

Mutual backscratching aside, I've been very happy with the customer service from them, whether things were immediately in stock or on back order. My thanks to Chris Dillon and all the rest of the staff.

Penrose sketch, 14.6.11: Five Cats

The extra sets of Penrose tiles are still on back order, so I am content for now with small sketches that fit on my refrigerator. I like working with the five fold symmetry, and it would be nice if there were five colors of tiles instead of three, but I'm still having fun with toys that exist.

Completed: 14 June 2011
95 tiles
45 kites and 50 darts
36 yellow, 36 purple, 23 blue

The math of Penrose tiles, part 3: Two proofs of impossible similarity.

I'm about to prove a couple of negatives about Penrose tilings. Recall Donald Rumsfeld proudly and stupidly saying you couldn't prove a negative when it became obvious to everyone the weapons of mass destruction ruse was a complete phony. I had to wonder exactly how many classes he slept through when he got his degree at Princeton.

Of course you can prove a negative. The only place where real proof exists is in math and we prove that things are impossible all the time.

Let me give a couple examples.


It is impossible to build a larger shape similar to a dart using kites and darts.

The dart is the Penrose tile with the dent, and angle of 216°. It is also the only Penrose tile that has the sharp 36° angle. Those angles are adjacent to each other, which means if you need a 36° angle when you are building something, you have to use a dart and you have to plan for the fact the 216° will be right next to it at the distance of short.

If we want to build a bigger dart, it will have to have two 36° angles and a 216° angle, but the distance between these will have to be at least the length of long.

We can't do this with these pieces, or if we achieve this, we will not have a long enough straight line to make the outside of the dart.

This proof takes no math skills really. If you had some Penrose tiles to play with, you would see pretty quickly the problems involved trying to make a shape similar to the dart.



It is impossible to build a shape similar to a kite bigger than Papa Kite.

Yesterday, I showed this picture of a regular kite, a slightly larger kite made of a dart and two kites (a shape I call Mama Kite) and a third larger shape made out of five kites and three darts I call Papa Kite.

Notice this. Each of the straight lines that make up a side of all three of these kites has at most one side of the short length. Because of the angles available, one short is all you can have if you are building a straight line that is empty on one side and completely filled in on the other. The problem is that to make a straight 180° angle from a 72° angle, we need 108°, which in Penrose tiles can only be done by combining a 72° and a 36° angle. Just as we saw in the earlier problem, the 36° angle is a little clumsy when trying to continue a straight line because it is so closely tied to the dent, the 216° angle, known formally in geometry as a reflex angle.

Here is my best attempt at making Granddaddy Kite, the next size up of similarity. The Fibonacci sequence tells me how many pieces I need, 13 kites and 8 darts. I used 12 kites and 7 darts and the shape of the empty space that caused the problem has a 36° angle that we can't negotiate with the shapes available.

Notice that the unfillable space is exactly a Big Dart, the shape we can't make with the two standard Penrose tiles. If a third Penrose tile existed that was the shape of the Big Dart, with side lengths long and long+short, the number of things we could do with the new system would increase dramatically, though it wouldn't help with making a dart bigger than Big Dart. That would still be impossible.

Instead of Big Dart, another "third" Penrose tile that could help in this situation would be a triangle with sides short, short and long, which would have angles 36°, 36° and 108°. With this addition, Big Dart would be these two triangles put side by side along one of the short sides, and suddenly bigger darts and bigger kites would be much, much easier.

In math, we call this "prove or disprove or salvage". When you prove something can't be done, you try to find the simplest changes you could make to the problem where you could do what was asked. The most famous early example of this was Archimedes proving that trisecting any given angle was impossible with a compass and straightedge, but it could be done if you were allowed to put one mark on the straightedge.

This is one of the reasons mathematicians put Archimedes head and shoulders over other ancients like Euclid or Pythagoras. Nobody else was "thinking outside the box" like our Sicilian pal.

Not that I'm telling Sir Roger what to do with his tiles. He is a Big Damn Deal in physics and I'm a blogger.

Not that I'm comparing my salvage to Archimedes' method for trisecting angles. That is a work of stunning beauty.

I'm just sayin'.

And, oh yeah, Donald Rumsfeld is still a pinhead who planned two wars he didn't know how to finish and he can bite me.

I'm just a blogger, but I'm a shitload smarter than he ever was.

If you ever read this, Don, quod erat demonstrandum, you ugly, murderous little pencil pusher.

The math of Penrose tiles, part 2: The Golden Ratio phi and its relation to the Penrose tiles.

Yesterday, I discussed the two shapes of the Penrose tiles, the kite and the dart. The dart in this picture is the one in light blue. If alliteration helps you remember, the dart is the one with the dent. The angles on the kite are 72° three times and one obtuse angle of 144°. The dart has one angle of 72°, two sharp angles of 36° and a reflex angle of 216°, which is the one that causes the dent.

There is no hard and fast rule as to how big the two tiles should be, but because they follow the geometric rules of kites (four sides, only two lengths, sides of equal length are adjacent), the ratio between the long and short sides is set in stone. It is phi, also known as the Golden Ratio. The exact value is (1+sqrt(5))/2 and the approximate value is 1.61803398875... on your calculator. Using 1.618 as an approximation is not too bad.

Here are the capital and lowercase versions of phi. Blogger software is .html based and doesn't have a lot of symbols from the Greek alphabet, so I will type out phi every time I mention the number. It's pronounced "fee" not "fie" if we want to be close to the Greek, but some people want it to rhyme with pi. Technically, pi should be "pee" when we say it, but then it would be confused with the letter p in our alphabet.

Phi has many interesting properties, and most of the ways it shows up in the real world involve ratios, some big number divided by a smaller number is equal to the Golden Ratio. Another way phi can be generated mathematically is as the solution to this algebraic expression.

phi² = phi + 1

Phi is not the only number that satisfies the condition that the square of a number is the same as adding 1 to the number, but the other solution is negative, so it can't be the description of a length or an area or some other real physical property.

When we have an equation like the one above, we can use it to find the value of higher powers of phi as well.

phi³ = phi² times phi = (phi + 1) times phi = phi² + phi = 2*phi + 1

Using similar methods to change higher powers of phi into combinations of phi and 1 we get the following pattern.

phi to the fourth power = 3*phi + 2
phi to the fifth power = 5*phi + 3
phi to the sixth power = 8*phi + 5

Some people may recognize the numbers 1, 2, 3, 5, 8... as the start of the Fibonacci sequence.



Here's how phi and the Fibonaccis are tied to the Penrose tiles. Not only is the ratio of the long side to the short equal to the Golden Ratio, but likewise the area of the kite divided by the area of the dart is phi. What this means is that if I want to make a bigger kite that is similar to the original, it can be done, but only by multiplying the side lengths by phi and the area by phi².

For these next statements, remember that long/short = phi and (area of kite)/(area of dart) = phi.

baby kite
Side lengths: long, short (or phi and 1)
Area: 1 kite (phi)

mama kite
Side lengths: long + short, long (or phi² and phi)
Area: 2 kites and 1 dart (phi³)

papa kite
Side lengths: 2 * long + short, long + short (or phi³ and phi²)
Area: 5 kites and 3 darts (phi to the fifth power)

Here's the thing. We can't make the next size up of kite, and there is no way of making a bigger dart with Penrose tiles.

Understandable proofs (knock wood) of these statements tomorrow.

The math of Penrose tiles, part 1: Definitions and angle measures.

Sir Roger Penrose, the world class physicist, is also a recreational mathematician. He came up with several combinations of tiles that could be used to fill the plane with non-repeating patterns before developing the kite and dart system, the two shapes of refrigerator magnets I am using in the posts with the label "Penrose tilings". The words kite and dart are actually standard geometric terminology. A kite is any four sided polygon (quadrilateral) that has two sides of one length and two sides of a different length, the same length sides meet at a corner. A dart is a kite that is concave, or we might say has a dent in it. The dent means an interior angle that is more than 180°. The math term is reflex angle.

The first special thing about the Penrose kite and dart is if we call the side lengths long and short, the long on the kite and dart are the same, as is the short. This means they have several ways of fitting together nicely.

Making such a kite and dart pair is easy if we start with any parallelogram where all the sides have the same length. The standard term for this is a rhombus, but it is also sometimes called a lozenge. (Some books use lozenge to mean only a rhombus whose angles are 45° and 135°.) A rhombus is to a parallelogram as a square is to a rectangle. In fact, rectangles are special parallelograms where all the angles are 90° and a square is a rhombus.

In any case, we can take any old rhombus and cut it in a variety of ways to make a kite and dart pair that will have the same length of short side and the same length of long side.


So there are infinitely many ways to make kite and dart pairings that can be combined into rhombi, and any old rhombus can be use as a tile that when repeated infinitely will fill the entire plane, a method called tesselation in math.

Here is the decision that made Penrose tiles more interesting than your run of the mill kite and dart that make some random rhombus. Sir Roger chose the angles carefully and the one angle both the kite and dart share is 72°. Since 72 times 5 is 360, five of these corners can be put together to fit perfectly, making a ten sided polygon, which is called a decagon. The convex decagon in yellow made of darts is called the same thing both by mathematicians and by actual people, a five pointed star.

Penrose could have chosen another angle that divides evenly into 360 so the kites and darts could be combined to make regular polygons or stars with some number of points, but 72° has some nice properties. The angles of the kite are 72°, 72°, 72° and 144°. The angles of the dart are 72°, 216° for the reflex angle and 36° at both the pointy ends. This means that in some situations, we can replace a 72° angle with two 36° angles put together, and similarly two 72° angles can be replaced in some situations with the 144°.

The math of the angles of the Penrose tiles is really more arithmetic, nothing harder than 36+36=72 and 72+72=144. Choosing these particular angles means the side lengths long and short have a relationship known as the Golden Ratio, or phi, and the math for that steps up from grade school level to high school level. Tomorrow, we will look at phi, the Fibonacci numbers and the several ways these interesting math concepts are linked to the Penrose tiles.

The Princely Purple Cat

In a kingdom across the sea
Live a Princely Purple Cat
Who wanted to be king.
And what's so wrong with that?
























But then his nose turned blue,
And the cat was all at sea.
"For if I am not constant
Then what will become of me?"

Still playing with the new toys.

I'm still in that lovely honeymoon period with my new toys, the Penrose tiles I bought online at seriouspuzzles.com. With tiles, the "usual" idea is to fill up the plane with patterns, possibly repeating and possibly not repeating. If I put a kite and a dart together like above, I make a rhombus. It's easy to tile the plane with any rhombus in a repeating pattern, and even the non-repeating patterns using rhombi are usually simple variations on a theme.

One way to break away from the rhombus is to make this shape with a dart and two kites, which is really a bigger kite. I started making patterns with this as my main basic shape.



In this first tiling, I started with the nearly round shape in the middle (actually a decagon, a ten sided polygon), and started building out from it using the new big kite shape and the "bow tie", which is a two big kites that share a small kite.



Here I was trying to see what tiles would have to be used if I surrounded one decagon with five other decagons that have a bow tie buffer between them. As you might be able to see, I had to use some rhombi as buffers between the new decagons.

Messing around with even this size of puzzle makes me wonder if I need to buy one more container of tiles.

If anyone knows the name of a local Penrose Tile addiction support group, please send it to me discreetly.


I also did something much simpler on an unusual tack. Instead of trying to tile the plane, I started looking at simple shapes that could be made with relatively few tiles that would use negative space. Here five darts are put together to make a negative space regular pentagon with a pointy star created in purple.

Trying to think of star shapes that have a hole in the center, I thought this kind of looked like a ninja throwing star, known as a shuriken.

I call this the Thin Penrose Shuriken.


This is to distinguish it from a similar idea using the kites instead of the darts, which is the Thick Penrose Shuriken.


With kites, there is a second Thick Penrose Shuriken. To me, this looks more like a rotating gear, so I'm also calling it a Regular Penrose Cam.


This is to distinguish it from a mix and match use of kites, the Irregular Penrose Cam.

I Googled "penrose tiles negative space", but I didn't find anything looking like these last four patterns anywhere on the net. I may have stumbled on a new idea.

New toys for Matty Boy! YAY!

Way back in January 2010, when I was still doing my (almost) weekly Wednesday Math posts, I wrote about Penrose tiles, the two shapes designed by physicist Roger Penrose that fill the plane in repeating patterns or non-repeating patterns, depending on how clever you want to get.

I bemoaned that there was no way to get a nice set of reasonably priced Penrose tiles to play with and mulled over the idea of having some made by a plastic fabrication shop.


Well, this is one of those times my natural laziness and broke assedness (which springs naturally from laziness, thank you very much) paid off big time. After putting this on the back burner for over a year, I searched last week and found a company that sold the thing I was looking for at a very reasonable price. SeriousPuzzles.com sells 108 Penrose tiles for $20, and better than that, they come with magnetized backs, so you can put them on your refrigerator or, for us teachers, tack them up on the white board in your class.



My friend Mark was nice enough to hold a pair of these in his hand to give you an idea of scale. They are nice soft bendy plastic, so I would say they are safe for any child who has already learned "Just because I can hold something in my hand, it does not follow that I have to put it in my mouth."


The two shapes are called the dart (top) and kite (bottom). The sets I got came in blue, yellow and purple. They appear to be easy to clean and in my experience, rubberized magnets maintain their stickiness nearly forever. (They are magnets, so tell the kids to keep them away from the computer.)


The angle between the two long sides is 72°, so if I put five pieces together, long side to long side, I can start a tiling of the plane. Five kites make a regular ten sided shape called a decagon. Five kites make a five pointed star, but not exactly the pentagram that we see on the stars of the American flag.



If you put a kite and a dart together on the short sides, they make a rhombus. Five of these rhombi correctly placed can make a regular pentagon with a pentagram in the negative space. The pentagram is the shape of the five pointed stars on the American flag.

Next: Matty Boy has way too much fun with Penrose tiles.

Growing a pattern with Penrose tiles.

We start with a yellow five pointed star.


I add a blue kite into each one of the gaps.


Now, a purple kite and a yellow dart in the new gaps, so that tiles sharing an edge do not share a color. (That was my basic marching order after the very beginning. It's a guideline not a rule.)


Blue kites for the yellow darts.


Yellow darts cap the purple kites.


Purple kites in the gaps.


"Got to have more of that sweet, sweet dart!"


That last caption was for my buddy Abu Scooter. If he didn't read this far, there wasn't much point to that gag.


So now it's growing more or less algorithmically, which is a fancy way of saying "These are the orders architects give to construction workers."


The purple and blue shapes together start to look like M.C. Esher fishies, don't they?


Notice how five straight lines lead out from the middle of the original five pointed star. At this point I thought that was a little too predictable, so I started changing things up.


Once I did that, all chances of tessellating the plane went out the window and I just had fun.

Actually, the whole process was fun for me. I hope my students enjoy it as well this next term.