Archive for the ‘Polygons’ Category

Soccer Math

June 19, 2014

Buckminsterfullerene.svgThe World Cup is happening! It’s inspiring to watch excellent soccer players…inspiring us to write about some excellent math. We’ll venture out on the field with a few soccer and math tidbits.

  • The soccer ball that I think of as typical – in other words, the one that I remember from Days of Yore – is an Archimedian solid, made from 20 regular hexagons and 12 regular pentagons.  Specifically, it’s a truncated icosahedron because it can be built from lopping the corners off of a regular icosahedron.  It’s also a buckminsterfullerene, although that’s only the formal name:  friends can call it a buckyball.  The buckyball was Red Hot News in 1985, because it was a new way of putting Carbon atoms together.  Scientists Harold W. Kroto, Robert F. Curl, and Richard E. Smalley* named it after architect  R. Buckminster Fuller*, whose geodesic domes had inspired them to try and create such a carbon cluster.   But this soccerball-shaped soccerball doesn’t limit itself to Ancient Greeks and Modern Scientists, oh no.  It also dabbles in the arts, as shown in this photo below from Labor Park in Dalien, China.Giant Soccer Ball Sculpture
  • Soccer balls aren’t the only thing math-related in soccer: there’s also the number of people on a team.  Each team has 23 players, which means that on any team there is a 50% chance that two people on that team share a birthday.  With 32 teams playing in the world cup, you’d expect about half of them to have birthday-sharing teammates, and in fact, as the BBC pointed out earlier this week,  exactly 16 of the 32 teams do.    For example, tomorrow (June 20) six people have birthdays, including two (Asmir Begovic and Sead Kolasinac) on the team from Bosnia and Herzegovina.  Now oddly enough, even though you’d expect half the teams to have teammates sharing a birthday, the fact that it’s exactly half is actually rather strange:  with a 50% chance of two teammates getting to share cake, the probability that exactly 16 of the 32 teams satisfy that is only 14% – it’s just that at that point it’s equally likely to be more or fewer days.  Ironically, it’s rather unexpected to actually hit the expected value.
  • One final math fact about the World Cup: one of the referees for yesterday’s match between Chile and Spain is actually a former high school math teacher!    Not all that former, either:  Mark Geiger taught in New Jersey alongside his brother, winning the Presidential Award for Excellence in Math and Science Teaching, but eighteen months ago he left teaching in order to referee full-time, hoping for a shot at the World Cup.  Not a bad gig, and he always has those math skills to fall back on if he finds he misses teaching.

*Whenever I type “[Occupation] [Person’s Name]” I get the urge to add “renowned” and then go read Da Vinci Code again.

The photo of the sculpture is by Uwe Aranas, Creative Commons License.  And if you didn’t follow the link to the BBC article, “The Birthday Paradox ath the World Cup” by James Fletcher, it’s worth a read – it has a lot more detail about the birthday paradox and sports.

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Sierpinski hiding in the Sistine Chapel

December 2, 2010

It was the second day in Rome, an intense day of walking and walking and WALKING, made all the harder by the youngest member of our family twisting his foot near the Colosseum.  And in a bout of bad timing, this was also the day we had tickets to the Vatican Museum (tickets that cost significantly less than 10 Billion Euros, I’m happy to say), so sore foot or not we forged ahead.

The museums were absolutely amazing, with cool things like actual Babylonian script (no idea what it means because it wasn’t clearly numbers, but still):

Plus, because it was a Friday night and the Museums aren’t always open then (last we heard it was a summer thing, extended through October), there weren’t many people in the main part of the museum.  It was dark, and we could look out from nearly empty rooms into nearly empty courtyards:

But the museums are long.  Really long.  I can’t find the dimensions, but according to my city map they look about 1/3 of a mile, and you basically walk around near the entrance then then down one whole side the entire 1/3ish mile length on the second floor, and then you go return on the bottom floor.  By the time we reached the end of the second floor we were already carrying our younger son, and we still had to walk back to get to the exit [and then walk to the Metro, and then the hotel.  And it was almost 10pm.]  But still, at this halfway point is the Sistine Chapel, and that is not to be missed, no matter how tired.

So we went in the Sistine Chapel, which was the one area that was completely crowded, plus it was really loud in there because the guards kept saying SHHHHHHHHH into the microphones and then a recorded voice came on overhead to tell everyone that this was a place of worship and to be quiet, and this was repeated loudly in 8 different languages.  So after about 15 seconds of admiring the ceiling we decided to call it a day and begin the trek back.  But then, right near the exit, TwoPi suddenly whispered, “It’s a Fractal!”  And so I looked at the floor:

See all those Sierpinski Triangles???? They go all the way to Stage 3!

The entire walk back we stopped at every souvenir stand (they’re all over the museum) and had this conversation:

“Do you have any picture of the floor of the Sistine Chapel?”
“You mean the ceiling?”
“No, the floor.”
“No, sorry.”

But then the next day we went to the Mouth of Truth (a giant face where you stick your hand in the mouth, and it gets bit off if you’re a liar), which is part of the church Santa Maria in Cosmedin.  The exit from the Mouth of Truth area goes through the church itself, and lo, there were MORE Sierpinski triangles on the floor!

Here:

and here

and smaller ones here:

and curved ones here

There were some other neat shapes, too, like these

and these, which looked just like a quilt

and these

The pieces were all laid out in sections, like…well, I really did think of quilts every time I looked at the floor:

There were even swirly parts that formed a giant infinity.

After taking 800 pictures we finally left, but a few hours later we were at the Basilica of San Clemente, which is a medieval Church built on top of a 4th century church built on top of a Temple of Mithras, and at the most modern level the floor has the same kind of design.  We sat and rested our tired feet admired it, but didn’t take any pictures because a Mass was about to start and we didn’t want to intrude.

So what was going on?  It turns out that this style of floor is called cosmatesque, and Our Friend Wikipedia describes it as:

a style of geometric decorative inlay stonework typical of Medieval Italy, and especially of Rome and its surroundings. It was used most extensively for the decoration of church floors, but was also used to decorate church walls, pulpits, and bishop’s thrones. The name derives from the Cosmati, the leading family workshop of marble craftsmen in Rome who created such geometrical decorations.

So it’s not terribly surprising that we saw three similar floors within 24 hours, even though we’d never seen anything like it before.  Sierpinski is hiding out all over the place.

How common are pentagon buildings/rooms?

August 3, 2010

I read a news article that a 4th century Roman villa was recently [where recently might mean 4+ years ago] discovered near Aberystwyth in Wales.  According to this July 26, 2010 article from the BBC news, “It was roofed with local slates, which were cut for a pentagonal roof.”

Pentagonal roof?  That sounded really cool, though I wasn’t quite sure what it meant and assumed it referred to the shape of the building itself.  The article included an outline of the area, but I couldn’t really tell if it was a pentagon or not.

In searching some more, however, it turns out that the pentagonal refers to the individual pieces of slate.  In the photo on this Heritage of Wales site they explain:

Two pentagonal Roman roof slates from the Abermagwr villa, the one on the right nearly complete. Made from local stone, with square nail holes, these slates constitute what may be the earliest slated roof in mid Wales.

Dang — no pentagonal rooms after all [as a footprint on the Heritage site confirmed], though I bet the roof looked cool anyway.

(Though this roof is from France, published by Arlette1 under GNU-FDL.)

Still, the pentagonal roof question got me wondering how common 5-sided buildings are.  There’s THE Pentagon, of course:

But there are some older examples.  There’s a stone age temple in Sweden according to this article from the Archaeological Institute of America (which actually claims that it is “a near perfect pentagon”).

I also ran across a pentagon room from Flatland by Edwin A. Abbott:

but I feel a little guilty posting it because I haven’t managed to finish the book.  And in it’s current exhibit our Science Museum has a photo of a building that isn’t the Pentagon, but there aren’t any captions so I don’t know what it is.

I had a little more luck with pentagonal rooms.  The  1903 book Roof Framing Made Easy:  A Practical and Easily Comprehended Construction, Adapted to Modern Construction, for Laying Out and Framing Roofs by Owen B. Maginnis has a whole chapter on pentagonal roofs, beginning on page 82, although Owen confesses that “this roof is of a form rarely met with in building construction”.  Incidentally, despite Owen’s claim to make it as simple as possible, I’m thinking simple in 1903 is a tad different than simple in 2010.  [Incidentally, Owen also has a chapter on hexagonal roofs and no fewer than FOUR chapters on octagonal ones, but the heptagonal roof is skipped completely.  Poor heptagon.]

Still, as Owen pointed out, they aren’t exactly common and it makes me wonder if there really just aren’t all that many, or if I just haven’t looked.

Godzilla makes a hexaflexagon

May 13, 2009

When Godzilla isn’t trampling buildings, flipping pancakes, or making cookies, he likes to engage in the fiber arts.

G hexflex 1

So he decided to crochet a hexaflexagon.  This is a hexagon that seems flat, but can be twisted to show hidden sides.

Here’s the hexaflexagon that Godzilla made using a pattern from Woolly Thoughts (link updated 1/1/10). It initially looks like this:

G hexflex 2

but he can twist the inside…

G hexflex 3

and there is purple in the middle instead of blue sparkles!

G hexflex 4

Then he can twist it again…

G hexflex 5

And it’s orange in the middle!

And those aren’t the only colors.  If you look at the other side, there is this

and this

and, finally, this!

(Crocheting it can go pretty quickly, depending on how many meetings or TV shows are on your schedule; you can also make paper versions using patterns from here or here or, of course, here.)

Snowflakes

March 11, 2009

I think winter is over.  Seriously.   I know that we’ve had a many a snowstorm this late in the season, but this winter has been rather long, and I’m ready for Spring.

Just in case I’m wrong and I have to face yet another storm, here are some pictures of snowflakes that make snow look really appealing.  These aren’t the snowflakes I see, which look like Hey-I’m-going-to-be-late-to-work, but the stereotypical ones that have that pretty six-sided symmetry.  Here’s a great example by dpnsan, taken this January.

snowflake by dpnsan

And here’s one that CaptPiper took last January:

snowflake by CaptPiper

water_molecule_3dSnowflakes have six-sided symmetry because water molecules are made of  one Hydrogen and two Oxygen molecules, in a configuration that looks a little bit like Strong Sad.  According to this science site, the molecules get all cozy:

The oxygen atom has a particularly strong attraction to the electron clouds of the two hydrogen atoms and pulls them closer. This leaves the two hydrogen ends more positively charged, and the center of the “V” more negatively charged. When other water molecules “brush up” against this growing snowflake, strong forces between the negatively charged and positively charged parts of different particles cause them to join together in a very specific three-dimensional pattern with a six-sided symmetry. Each water molecule that joins the snowflake reflects this pattern until eventually we can see its macroscopic six-sided shape.

And that’s how we get results like this (taken with an electron microscope, from the US Department of Agriculture).

snow_crystals_2b

The neat thing about snowflakes is that they don’t have to be so spikey looking.  Sometimes they form regular hexagons:

snowflake by Wilson A. Bentley

The photo was taken by Wilson “Snowflake” Bentley.  He was the first person ever to photograph a snowflake, back in 1885, and he went on to take bunch of pictures that were published by all sorts of places, including Scientific American and National Geographic.  I’ll end with a collection of some of his photos.

snowflakes by Wilson Bentley

In case you’re wondering how people manage to take pictures of things that melt almost as soon as you look at them, there’s a description here.  For some really stunning examples of the finished products, see the photos by David Drexler, Mark Cassino, and F. W. Widall.

Update 3/12: I woke up this morning and the ground was covered in snow.  I might have been a tad premature in saying that winter was over.

Frankenstein, Great Expectations, and Polygon

October 29, 2008

What’s the connection? Mary Shelley (born Mary Wollstonecraft Godwin) wrote Frankenstein; or, The Modern Promethius when she was a teenager, in 1818. The original Dr. Frankenstein’s monster didn’t look like the guy to the left: in the 3rd edition of the book (published in 1831) he looked like this:

So what does this have to do with Polygon? Well, Mary Shelley was born in The Polygon! The Polygon was here:

Some sites indicated that The Polygon was the name of the actual house, but after surfing the net when I really should have been grading doing some research I’m pretty sure that The Polygon was that immediate neighborhood, not one single building.

For example, in a book (Memoirs of the Author of a Vindication of the Rights of Woman) that her dad (William Godwin) wrote about her mom (Mary Wollstonecraft, who died 11 days after Mary was born), The Poygon is mentioned twice:

It is perhaps scarcely necessary to mention, that, influenced by the ideas I had long entertained upon the subject of cohabitation, I engaged an apartment, about twenty doors from our house in the Polygon, Somers Town, which I designed for the purpose of my study and literary occupations. Trifles however will be interesting to some readers, when they relate to the last period of the life of such a person as Mary. I will add therefore, that we were both of us of opinion, that it was possible for two persons to be too uniformly in each other’s society. Influenced by that opinion, it was my practice to repair to the apartment I have mentioned as soon as I rose, and frequently not to make my appearance in the Polygon, till the hour of dinner.

In digging around some more, I discovered someone else who lived in The Polygon: Charles Dickens! He wasn’t born there, but moved to 17 The Polygon, Somers Town in 1827 (more than a decade after Mary had left) at the tender age of seven when his family was evicted from their previous abode. [He only lived there about a year before moving.]

Finally, The Keeper of All that is Good and True says, “In 1784, the first housing was built at the “Polygon”, now the site of a council block of flats called “Oakshot Court”.” So I’m convinced that The Polygon is that neighborhood, maybe the plaza (which would likely be in the shape of a polygon). And the word Polygon is mathy, and Frankenstein is a pretty Halloweeny book, and Charles Dickens has some scary stuff it it (not monster-scary, but those debtors’ prisons don’t sound like much fun), so it all seems to fit the season.

Fun with the Inclusive OR

October 12, 2008

I’m a little disappointed in header “SQUARE OR DIAMOND?” I think they could have added “OR RECTANGLE” for additional options.


more fail, owned and pwned pics and videos
[Note: Not always safe for work, although The Fail Blog now offers a G-rated option].

Godzilla’s Dinner Party

August 29, 2008

It’s a little known fact that Godzilla likes to throw dinner parties. Some of these gatherings are formal dinners, with no fewer than four forks, but others are more intimate. Upon occasion Godzilla throws a dinner with only two guests, but then at the last minute a friend drops by and naturally Godzilla invites him to stay for enchiladas with mole sauce (or whatever the evening’s menu). This means that Godzilla has to add a fourth place setting to his triangular table. What to do?

Fortunately, Godzilla’s table is hinged and can turn from an equilateral triangle into a square at a moment’s notice. Allow him to demonstrate. In the mock up below the separate hinged pieces are colored for easy demarcation.

Godzilla prepares by looking at the table.

He slowly starts to separate the pieces:

You can see where the three hinges are (on the outside of the triangle) connecting the four pieces. Godzilla continues to spread them out. There’s a pretty star outline in the center.

He swings those bottom three pieces around…

and up towards to the top

Now it’s starting to look like a square:

And voilà! He’s gotten a square table! The party is saved!

Here’s the final formation in color.

Isn’t that cool?

There’s a slight problem with this design, though, in that there have to be at least four table legs, all close together. Fortunately Greg N. Frederickson designed a triangular table with a large enough piece in the center so that a single pedestal would do, and the six tiny swinging pieces could all be hidden with a linen table cloth. It’s the lead article (“Designing a Table Both Swinging and Stable”) of this month’s College Mathematics Journal. He has some nifty spiff animations here and even more information at the bottom of this page.

For more on these dissections, check out Ivars Peterson’s January 27, 2003 Math Trek column about how chemists were doing exactly the same sort of table spinning as Godzilla (without mentioning him by name, of course) using little plates that would self assemble into different kinds of shapes (although to do the triangle-square hinged switcharoo they had to be connected with thread.)

Time for the dinner party. Pass the enchiladas.

Nonagon: The Video

August 2, 2008

They Might Be Giants has been doing videos and podcasts for kids. One of their recent releases (from earlier this year) is a video all about one of our favorite polygons: the nonagon! Several other polygons make guest appearances as well.

You can see the video below. The first minute or so is introduction, followed by the nonagon song, one about the letter O, and then a good-bye (5:32 total).

More glass polygons

July 3, 2008

Houston, we have nonagons! The house that we just stayed at on the beach, the one with the neat-looking but nonfunctional Tide Clock, redeemed itself mathematically by having glasses in the shape of nonagons:

Does that look like I faked it? Here’s the unedited picture, in case you don’t believe me:

And just so the other polygonal glasses don’t get jealous, there was also a hexagon glass:

These join the decagon mug in the polygonal kitchenware Hall of Fame.

Hidden Heptagon

June 28, 2008

We recently found a heptagon hidden on the underside of a Wegman’s rectangle cake, in a design all around the edges. (I’m no longer sure why we thought to look at the bottom of the cake, especially since I’m pretty sure we hadn’t cut into it at this point.)

The heptagons are subtle, because they’re not regular and they look like they’re masquerading as either hexagons or octagons. But they’re there:

In retrospect, it might not be too surprising. Wegmans is, after all, the store that thought to put heptagons and nonagons on the design of their cookie cakes (from back in the day when I thought heptagons were unusual, instead of all over the place).

Hooray!

Yet Another Heptagon

June 21, 2008

I recently found a heptagon in a surprise place: Wikipedia. Not on the heptagon page: I was actually looking up copyright info for the blue triangle I used in this post, and I noticed that the copyright statement, explaining that the figure was a simple geometric figure and so couldn’t be copyrighted, actually had a picture of a heptagon on the right to illustrate (click for a more legible version):

Go Wikipedia!

It does occur to me that the copyright notice might actually be copyrighted. Recursion anyone?

Hoorah for the Hexadecagon!

May 29, 2008

There was a Hexadecagon in the New York Times Wednesday, in an article about glassblowing. First the article talked about people blowing glass at the Corning Museum GlassLab:

Except the glassblowers weren’t from 1955. Then the article explored the creation of glass designs, including knit glass (!!!) and New York City inspired pretzels. And, as promised in the title, there was also a hexadecagon. It looked a little like this

Except in wasn’t in Las Vegas. And it had 16 sides.

With that inspiration, I looked up hexadecagon to see what I could learn about it. I found that a hexadecagon is also known as a hexakaidecagon, and can be constructed with a straightedge and compass. And I found that origami madness made a neat origami design with four hexadecagons:


Photo by origami madness. Some rights reserved.

Here it is unfolded:


Photo by origami madness. Some rights reserved.

And oschene made some Fujimoto cubes with hexadecagonal irises (irides?)


Photo by oschene. Some rights reserved.


Photo by oschene. Some rights reserved.

And finally, the Imperial Seal of Japan (Crest of Chrysanthemum) isn’t quite a polygon, but if it were then it would be a hexadecagon.


Licensed under Creative Commons Attribution ShareAlike

Polygons on Mars

May 27, 2008

The Phoenix Mars Lander arrived safely on Mars Sunday night! This is a particularly big deal because the previous Lander didn’t: the Mars Polar Lander was due to arrive on Mars in December 1999 but, for still-unknown reasons, communications stopped suddenly about 6 minutes before it was due to enter the Martian atmosphere and its exact whereabout remain unknown. (Incidentally, the Polar Lander was part of the Mars Surveyor mission. The other part was the Mars Climate Orbiter, which ran into a little $125 million problem of its own when the teams didn’t translate between imperial and metric units.)

But back to the Phoenix Lander: the Phoenix had no trouble, landed perfectly, and is already sending back pictures. Our newspaper this morning showed the following photo of polygons on Mars:

The newspaper made a big deal about the fact that there were polygons and when I looked at the picture my response was along the lines of, “Umm. Okay.” But then I found NASA’s image page and the caption for this picture explains that this polygonal pattern is “similar in appearance to icy ground in the arctic regions of Earth”. So then I went to My Favorite Source and found this photo from Canada’s Northwest Territories (taken by Emma Pike), where the polygons are more noticeable:

The polygons are formed by water getting in cracks, freezing, and then expanding. When it gets cold enough (-17°C, or close to 0°F) the ice contracts rather than expands, and that leaves even bigger cracks (called ice wedges) for more water to get in, etc. So finding polygons on Mars could be a big deal indeed.

There was also another neat NASA picture of the Phoenix Lander landing:

You can see the parachute and everything! According to the the FAQ page at the University of Arizona:

Phoenix is very grateful to the Mars Reconnaissance Orbiter (MRO) team for that otherworldly picture. It was very, very good math. MRO was moving about 3.4 km/sec (30,000 mph). Phoenix, at the time of parachute deployment, was moving between 700-130 mph.

Hooray for very, very good math and for the Phoenix Lander!

Hexagons in the News: Nanotubes

May 22, 2008

Nanotubes are back in the news! Nanotubes are sheets of carbon atoms, one atom thick, that roll up to form strong cables for tennis rackets and baseball bats. Very strong cables, in turns out, as in a possible material for a space elevator.

But even if the space elevator doesn’t work out, it turns out that carbon nanotubes can be used to make near-ideal black objects: things which absorb light completely, not refracting any of it. ‘Darkest Ever’ Material Created on BBC News in January explains that this is useful for creating solar cells and solar panels.

So what does does this have to do with mathematics? Quite a bit, if you look on wikipedia or if you actually build anything physical with it. But even at a simpler level, one neat property is that the carbon atoms bind in hexagons. You can see that in the picture below of a hexa-tert-butyl-hexa-peri-hexabenzocoronene. (How’s THAT for a word?)

The hexagons in the picture above are regular hexagons, meaning that all the sides and all the angles are equal. You can completely tile a plane with regular hexagons, without leaving any gaps. And this means that the hexagons in the carbon nanotubes must not be exactly perfectly regular, or else there would be no way for them to roll up. Either the angles or the sides must be a little off.

This irregularity doesn’t actually harm the carbon nanotubes. Unfortunately, they have bigger problems right now: BBC news reported Tuesday that when some fibers got into the lungs of mice, they caused inflammation and legions, like asbestos. Poor nanotubes. That doesn’t sound the death knell, however; it’s only one study, and so more research, more money, etc. etc. have to be dedicated to the topic. But it might be a while before we see that solar-powered space elevator.

Pictures used under GNU Free Documentation License.