A little horn tooting is in order.

Two things happened in June that bear mentioning.

This blog, now four and a quarter years old, reached the 500,000 hit plateau. Thanks to all my readers for sticking with me for whatever silly nonsense pops into my head.

The other blog, which has only existed a year and a half, has also reached the 500,000 hit mark. As you might expect, it's growing very fast. Since the end of June last year, about 200,000 hits have been registered at Lotsa 'Splainin' 2 Do, while over 470,000 visits have been marked at It's News 2 Them™.

As of midnight on June 30, this blog has had about 508,000 hits, while the other blog is just over 507,000. By Saturday, possibly even late Friday, this blog will be surpassed, an event I knew would happen eventually. If there is some small consolation, people who visit here tend to stay a slightly longer time and more people comment here, though I do not have a squadron of loyal commenters the way my blog hero/heroine Princess Sparkle Pony does. Peteykins' terrific slice of the Interwebs will soon have its two millionth visitor, and there are few places on the web more deserving of loyalty in my never humble opinion.

Anyone who reads my stuff knows how I love numbers, so I couldn't let these milestones pass unnoticed. I expect the next milestone will be the millionth visitor over at the gossip blog, which if things stay as they are going now could be about a year away.

UPDATE: 1 July 2011 at 2:00 p.m. Pacific Time.
It's News 2 Them™: 508,200
Lotsa 'Splainin' 2 Do: 508,199
Game over, man.

Could I have been remiss once again?

Yes, I could have and have been.

I've been forgetting to pander!

You might think, a blogger would no more forget to pander than a fish would forget to breathe! But it's not quite as second nature as it looks to the untrained eye.

For example, the last time I posted a picture of Indira Varma was Christmas, and it's the end of June.

I'm going to lose standing in the Creepy Internet Stalker Guild if I keep up this kind of lax performance.

And what about giant women, Matty Boy? Don't you realize that many of your readers are actually Your People who share your bizarre Agenda?

Yes, hypothetical question asker, I know My People read my blog. I am also aware that I haven't posted a giant woman picture since last February, which means two giantess posts in 2011 so far, compared to nine in all of 2010.

Gracious me, I don't know where the time goes. My heartfelt apologies to fans of pandering everywhere.

The new toys have arrived, Part 2.

Yay, even more new toys have arrived. These are regular polygons that can be used to fill an infinite plane with a repeating pattern.

These shapes are not magnetic like the Penrose tiles, but they come in four colors and five shapes, regular polygons with three sides (equilateral triangles), four sides (squares), six sides (regular hexagons), eight sides (regular octagons) and twelve sides (regular dodecagons).

A few years back, I drew pictures of some of the possible mix and match patterns that create regular tilings of the plane, and now I can re-create these with the new toys. This shot shows that the number of pieces used is finite.

But if I take a close-up, you can see that this could be part of a pattern that repeats indefinitely.

Thanks again to the folks at SeriousPuzzles.com, where I bought my new toys.

New toys! So much fun.

Brilliant marketing ploy with only a minimum amount of evil.

The Trader Joe's reusable bag.

99 cents. In 2011, that is officially ridiculously cheap, the way stuff costing nickels and dimes used to be when I was a sprout. Cheap is not evil.

Sturdy, cheap re-cycled plastic. Recycled is not evil.

You paid to carry around the company logo, so there's some evil.

If you go to TJs with a bag you can sign up for a lottery for a $25 gift certificate, but Orwell's Nineteen Eighty Four made too deep an impression on me. I'm not convinced anyone actually wins these lotteries. So, also some evil.

But when I go to Lucky, they give me a nickel off for any re-usable bag, even one with somebody else's logo on it. These things pay for themselves in about a month for me, and I keep them way longer than a month.

So TJ's has competitors pay customers to buy advertising for TJ's. Evil, but they have cut the consumer out of the "who gets screwed?" equation, so significantly not evil.

And then there's the brilliance of finding a distinctive bag design and sticking with it. I can spot one of these bags one hundred yards away, easy peasy. I expect that you can as well. This is marketing genius up there with The Swoosh and the Golden Arches.

So, the power of marketing used to save the consumer money and reduce the amount of wasted paper and plastic in the world. In the Matty Boy version of this equation, this is big on the plus side with only a tiny residue of evil.

And to toot my own horn just a li'l bit, when Matty Boy makes an equation, he gets it right at least nine times out of ten.

The new toys have arrived, part 1.

Yay, I have my new toys!


Have you ever wondered what 648 Penrose tiles look like in a big clump on a table?

Wonder no longer.


Here are all the yellow tiles in neater piles. Each color has 132 kites and 84 darts. The rules of Penrose tilings make the bigger pieces more useful.


And then I put each color and shape in its own container because I'm anal.

(As Diane Keaton said in Annie Hall, "Anal is the nice word for what you are.")

In any case, I want to give a shout out to SeriousPuzzles.com, the online store where I was able to get all these cool toys. If looking at the odd things I create gives you the itch to have a set of tiles or two or six, drop them a line.

Penrose pattern #1, 27/6/11: The Violet

Title: The Violet
Date: 27 June 2011
Type: Pattern
Number of tiles: 300
Color breakdown: 165 purple, 75 yellow, 60 blue
Shape breakdown: 195 kites, 105 darts
Kosher Penrose tiling rules: no
Supplier: SeriousPuzzles.com

Notes: Penrose patterns mean I want to make a finite shape that may or may not include gaps. This one has little white triangle gaps and a central pentagonal gap.

There will also be Penrose tilings, which means patterns without gaps that when repeated will fill an entire plane.

Rules of Kosher (or Halal, no prejudice): Gaps are not allowed in Kosher. Rhombi, where a dart and kite meet to form a parallelogram, are the other unacceptable pattern in Halal.

You might see faces in these patterns rather than a five fold symmetrical flower. That's called pareodolia, the name for the human tendency to see faces in even completely random patterns. It was a valuable evolutionary tool waaaaaay back in the day, and it's still part of our standard mental skill set.

This gives you an idea of the size and what it looked like in the room.

Changing the focus and light intensity in iPhoto gives you an idea of the geometry in my head when I designed this.

Get used to these. Or not.

Either way, there's going to be a lot more of them.

I'd be glad to know what you think of them, positive or negative.

Hello, Ladies!

Some of you might recall I caught a low grade infection last summer diagnosed as World Cup Fever. For many people, this disease can only be caught in even numbered years not divisible by four, but I also like to follow the Women's World Cup, the latest installment starting today in host country Germany.

It's almost not fair that the Germans are hosting, because they are the dominant team in women's football this century. They won the last two World Cups in 2003 and 2007. In 2007, they won the tournament by not giving up a single goal. (They didn't win all their games, as England played them to a 0-0 draw.)

Unlike the English and French, German national teams tend to look very European, but not this year. One of the German goal scorers was Celia Okoyino Da Mbabi, born in Bonn of a French mother and Cameroonian father. The other noticeably non-Aryan name on the squad is Fatmire Bajramaj, whose parents were refugees from Kosovo. Fatmire, known by the nickname Lira, is the new star being given the front and center position in the German press, though she didn't start the game today. All three of today's German subs, Bajramaj, Inka Grings and Alexandra Popp, are superstars that would have a starting job on any other squad playing in the tournament.

That's how good Germany is.



The Germans beat the Canadians in the opening match in Berlin, but the big news of the day is that the Canucks were not shut out. Trailing 2-0 in the second half and badly outplayed, Canada got a free kick from 25 yards and gave the honors to Christine Sinclair, Canada's greatest scoring star by a considerable margin. She hit a nasty twisting shot any man would be happy to mimic, just over the tall German wall and just under the crossbar. The goalie Nadine Angerer had no chance at all.

She is also the big news because she will not be playing in Canada's second match, a must win game against the French. Sinclair had her nose broken in the first half and only stayed in the game because she begged the coach Carolina Morace. The team doctor says it's too risky for her to play against the French on Thursday.

Some football fans don't like the women's game for the same reasons some basketball fans don't watch the WNBA. They aren't as big, fast or strong as the men, but that doesn't stop them from being tough, talented and fierce competitors. A slight advantage to the women's game is that flopping is not the pandemic it is in the male version.

The games from Germany will be airing in the morning here on the West Coast, so I won't be able to see a bunch of games on Mondays through Thursdays, but I'm going to do all I can to catch the games when I'm not working. I'm guessing that the experts predicting a German coronation know what they are talking about, but I expect a lot of drama along the way.

Way to go, New York!

I nicked this picture from the front page of The Huffington Post this morning, a celebration in front of the famous Stonewall saloon when the news came out that the New York legislature had legalized gay marriage. If you click on the picture for a larger version, you'll see several same sex couples celebrating, but let me note that the really cute blond girl with the big smile near the front of the crowd looks to be smiling at a dude!

Oh, yeah, that's right. Straight people can be happy that all adults are given the right to enjoy the legal protections of marriage offers.

Like the straight guy writing this blog, for example.

Congratulations to the New York legislature for doing the right thing.

Gene Colan, 1926-2011

Gene Colan, an artist born in the Bronx who became best known for his work for Marvel and DC comics, has died at the age of 84. The writer Clifford Meth has devoted a blog to Colan's work where original drawings are for sale. The proceeds went to Mr. Colan's care in his final days and works sold now will be the start of a scholarship in his name.

Colan worked on a lot of titles, as did all of the artists back then. I was amazed at how much work these guys did every month and missing few deadlines. In later years, Colan admitted to abusing amphetamines to keep up with the schedule.

He worked on some titles I didn't regularly read, like Iron Man and Captain America, but he was also the artist on Daredevil, Howard the Duck and others.

When he started, Stan Lee told him what he told all the new artists. Draw like Jack Kirby or Steve Ditko or John Romita or some other established artist. Colan refused, saying wasn't physically capable of doing that. He got to work in his own style and it worked very well for him.



Some of my favorite work of his was in Dr. Strange. There was another artist coming up in the late 1960s named Barry Windsor-Smith, who drew very intricately but much too slowly. There were several titles that Smith was supposed to draw, but because of missed deadlines, other artists would be forced to fill in. As I became obsessed with Windsor-Smith's work, I considered it a letdown when someone else had to fill in for him on an issue, but over the years I came to appreciate Colan's work on its own merits and stopped wondering what a story would have looked like if Barry Windsor-Smith had just been fast enough. Colan's work with the inker Tom Palmer was especially remarkable.

Best wishes to the friends and family of Gene Colan, from a formerly fickle fan.

As the young people say, Jon Stewart FTW.

I like Jon Stewart. If I still had cable, his show is one of the few things I would make time to watch. As it is, several websites I visit have clips of The Daily Show and I watch them several times a week.

Chris Wallace had him on as a guest on Fox News Sunday. As you might imagine, Mr. Wallace did not like the idea that he worked for a "relentless agenda-driven 24 hour news opinion propaganda delivery system". Stewart stood by these words.

Wallace did something a lot of newspeople do, criticized Stewart for not being a serious newsman. Stewart said the comparison was silly, that he is a comedian just as Will Rogers was, and that people take him seriously because they are disappointed in the news they get.

Wallace said Fox News viewers weren't in the least disappointed. Stewart said that poll after poll shows that they are the most consistently misinformed.

In other words, if they were bright enough to be disappointed, they'd be disappointed.

Wallace made the point that other news outlets are biased in the liberal direction. Stewart countered that the actual bias of the news media in general is not ideological, but weaknesses for conflict, sensationalism and laziness.

The Huffington Post had this story as the top of the page early this Sunday. They had a huge picture of Stewart, slightly grainy and looking somewhat upset, with a 48-point headline "YOU'RE INSANE!"

Let's see. Conflict, check. Sensationalism, check. Laziness, check.

I gave Mr. Wallace's quotes first and Mr. Stewart's quotes second because Stewart correctly countered everything Wallace brought up. Jon Stewart is one of my heroes. Chris Wallace is a corporate tool and he got owned on his own show.

Here endeth the lesson.

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 sketches, 16.6.11: Five fold symmetry with three kites.

Here are three kites put together. If there were five kites centered around the angle at the bottom, it would create a regular decagon, a ten sided polygon. The angle for the gap at the bottom is 144°, which is an angle that can be filled many ways. The idea is to take five of these three kite shapes, twist them by the same angle each time to make the patterns below. The first is using the regular kite shape and a two color pattern, the second is using the larger kite I call the Mama Kite, again using just two colors, and the last is using the Papa Kite and three colors.

Completed: 15 June 2011
15 tiles, all kites
5 yellow, 10 purple

Completed: 16 June 2011
45 tiles
30 kites and 15 darts
20 yellow and 25 blue

Completed: 16 June 2011
120 tiles
75 kites and 45 darts
40 yellow 40 purple and 40 blue

The long wait is over.

The Dallas Mavericks beat the Miami Heat in convincing fashion to become this year's NBA champions. While it is not one of the greatest upsets in sports history, the Heat definitely looked better on paper. Their three stars, LeBron James, Dwayne Wade and Chris Bosh, are all in their seventh seasons playing, which would put them in their prime. The stars of the Mavericks were all at least ten year veterans and their starting point guard and playmaker Jason Kidd was in his 16th season and 38 years old, making him one of the oldest players in the league currently.

By winning, this gives Kidd his first championship in his long career, almost all of it as a starter, ten times on the NBA All-Star team. By an odd coincidence, if the Heat had won it would have been the first championship ring for Juwan Howard, also in his 16th year and also 38, but Howard is now a bench warmer and not considered a major factor.

Another famous All-Star in his sport who waited a very long time for his first championship was John Elway. He had been in the league 15 years before his Denver Broncos won the Super Bowl, and just to show it wasn't a fluke, they repeated the next year in his 16th season, after which he retired.

Unlike the other players on this list, Elway played for the same team his entire career.

But the longest wait for a championship I've heard of, and a tip of the hat to my friend Art Velasquez for remembering it, is hockey's Ray Borque. Widely recognized as one of the greatest defensemen ever to play the game, Borque played twenty one seasons for the Boston Bruins, was traded to the Colorado Avalanche late in the 2000 season, then helped the Avalanche win the cup in 2001, the culmination of his 22nd and final season.

Congratulations to Jason Kidd, who is a Northern California product by the way (St. Joseph's High School in Alameda, college at Cal), for joining this remarkable list of persistent and talented athletes.

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

Shame. Not dead, not sleeping.

I was driving on Friday last, listening to the radio version of The News Hour on PBS. Shields and Brooks were giving us their measured, reasonable and usually wrong headed views on the news of the day. Shields, when asked about the Anthony Wiener unpleasantness, said "Shame is officially dead."

Au contraire, Mr. Shields. It is still with us. It just has to applied in the proper forum in the correct way and shame still possesses a crushing force.

Obviously, Mr. Shields would not have said what he did without some data on his side. In past generations, what happened to Bill Clinton or David Vitter or Alec Baldwin or Kanye West or Kobe Bryant or Anthony Wiener would have driven them from the stage, possibly forever. But we live in the post-Nixon era, and if even Tricky Dick can come back from the grave, so can nearly anyone with enough time and intestinal fortitude.


Submitted for your disapproval: Donald Trump. He made some nonsensical noises about running for president, and since we are still more than six months away from an actual delegate in the nomination process to be assigned to a candidate, the press was unable to ignore him.

Then came his decision to hang his hat on the birther issue, to be followed by the long form birth certificate being released.

This did not shame him. He took credit for a thing he did not do, a completely predictable move from such a pompous swine.

No, he was shamed out of the race on C-SPAN (and more importantly, You Tube) by comments made by Barack Obama and Seth Myers at the White House Correspondents' Dinner. In an odd twist, Trump had been to a modern Friar's Club roast only a few weeks before, where the jokes were much more vicious. But at the Friar's Club, it's supposed to be an honor to be mocked, and the target gets the last word. Instead, the Leader of the Free World used irony to crush him while a room of "important people" laughed out loud, completely unimpressed by the fact that the target of the mockery was in the room. Minutes later, Seth Myers tucked in to what was left of the warm and not yet rotting corpse.

And Myers, not Trump, got the last word and the last laugh.

Arnold Schwarzenegger never really wanted a long term career in politics, but he did plan to return to the movies when his stint as the savior of the Californian Republican Party was over.

Those plans are currently on hiatus.

Kind of like Manimal is on hiatus.

People are coming out of the woodwork to tell stories of his swinishness now, and the press is eating these stories up.

It's funny what just one secret child fathered out of wedlock can do in this day and age.

And in this day and age, the "secret" part is far worse than the "fathered out of wedlock" part.

Tiger Woods will not be at the U.S. Open this week. He has to be seriously injured to make such a move. But if you have followed his career, you'll know the last major championship he won, The 2008 U.S. Open, was won on what was effectively a broken left leg, and he then missed the last two major championships that year. He came back in 2009 after surgery to win many regular tournaments, but no majors.

And then, around Thanksgiving 2009, the world found out Tiger did have a hobby outside of golf.

Shame has turned Tiger Woods into Robert Gamez.

Here is sweet Reese Witherspoon. She has been married, her husband Ryan Phillippe cheated on her, she left him. His career has stalled, hers continues to move forward. On my silly gossip blog, she has had 18 cover stories about her since the beginning of 2010, none negative. I looked online to see if she had ever been involved in a feud, and in 2008 she went out of her way to deny any bad blood between her and co-star in Four Christmases Vince Vaughn.

If you have ever seen Jon Favreau's show Dinner For Five, you know Vince Vaughn is an unstoppable flaming asshole in real life. But Our Reese does not want to be seen as a difficult person to get along with.

And then earlier this month, she went on the MTV Movie Awards and said this.

"I get it, girls, that it’s cool to be a bad girl but it is possible to make it in Hollywood without doing a reality show. When I came up in this business, if you made a sex tape, you were embarrassed and you hid it under your bed.”

I am of the opinion that sweet little Reese Witherspoon has just done to Kim Kardashian what Barack Obama and Seth Myers did to Donald Trump. She will finally feel shame. Maybe I'm wrong and she will weather it, but I really think the right words were said on the right venue at the right time and Kim is going to have to be satisfied with being rich and beautiful instead of rich, famous and beautiful.

It's already happened to Paris Hilton. She tried a new reality show and it bombed, she's through.

All things come to an end, pleasures and plagues alike. If you find yourself wondering in 2012 whatever happened to Kim Kardashian, look back at Reese Witherspoon's comments and know that shame is not dead and it is not sleeping. It just has to be applied properly.

Here endeth the lesson.

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.

Justice in Oakland.

Chauncey Bailey was a reporter for the Oakland Post, a local independent newspaper. He was killed in August of 2007. Nearly four years later, an Oakland jury has Yusuf Bey IV, a sociopath who ran Your Black Muslim Bakery, of ordering his death when Bailey was looking into improprieties in the business.

This case took a hell of a long time to get in front of a jury, but thank Odin, Vishnu and the li'l baby Jeebus that it got in front of an Oakland jury. The last big public murder in my hometown was Omar Grant being killed by an incompetent BART cop. The trial went to Los Angeles, where an incompetent judge in a deeply corrupt town told all cops everywhere there was no need for them to be competent, even if their bungling might cost a citizen his life, especially if that citizen had a less than optimal skin color.

I hope the judge here throws the book at Bey IV and I hope the book is a very heavy one.

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.

Hell's top tourist attraction.

If you haven't seen pictures of the Chilean volcano erupting, here's a link to several at Talking Points Memo. Some of the photos, especially the night time shots, are more beautiful and awe-inspiring than the best special effects I've ever seen in any Big Stuff Blowing Up movie, but I don't think anyone wants to risk getting up close and personal to get a better view.

Nature at its most remarkably nasty. The most devout skeptics might scoff at the idea of angry gods, but I get the feeling that seeing this in person, angry gods would be a really convincing explanation.

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?"

Yah'weh, you stinker!

He was quite the scamp back in the day.