Spring is here

Beautiful, unseasonably warm weather last week here in the Hudson Valley (70s yesterday).  Vernal equinox occurred over the weekend - the Sun is now passing over the equator and moving into the Northern Hemisphere as we now are tilted more toward the Sun with each passing day.

Built a fire last night (fire ring in my backyard) and sat outside listening to the calls of the spring peepers (Psuedacris crucifer) and wood frogs (Rana sylvatica).  Today will be chilly and rainy - normal March weather - but more beautiful days are coming.

Bats on the Wane

Thursday, March 18, I went to see a talk called Bats on the Wane by NYS Department of Environmental Conservation biologist Carl Herzog at SUNY New Paltz.  It was sponsored by the Shawangunk Ridge Biodiversity Partnership.  It was one of the most depressing things I've seen in a long time.

Bottom line - the species of bats that overwinter in caves and mines in the Northeast are on the verge of extinction from the fungus Geomyces destructans which causes White Nose Syndrome (WNS) disturbing the bat's hibernation and causing them to essentially starve to death.  In some bat hibernacula, the mortality rate is virtually 100%!

Geomyces destructans image from Tom Volk's Fungus of the Month

WNS was first seen in Howes Cave up in Schoharie County in the winter of 2005/2006.  Within a year it was seen in other caves in the Albany/Schoharie area and within two years throughout the Hudson Valley and into Vermont and the Adirondacks.  Then it started spreading down the Appalachians and is currently moving west toward caves in KY, MO, and IN.  There's really no stopping it.

Some statistics I jotted down during the talk.  WNS affects bats which hibernate in mines and caves over the winter - there are about 500,000 of them in NYS based on previous bat surveys by the DEC.  A surpising 75% of the bats were found in only 5 hibernacula in the State - this is really the problem since a disease will then spread rapidly.  At those sites, 85% were little brown bats (Myotis lucifugus) and 11% Indiana bats (Myotis sodalis).  The Indiana bats are an endangered species which is why the DEC had been studying them prior to WNS.

Two important bat hibernacula are near Williams Lake - an old hotel in the Town of Rosendale where I live.  These are abandoned limestone cement mines that the bats have taken over in the past 100 years.  Surveys in these mines have shown that bat populations dropped from 10,336 bats to 268 bats (97% mortality) in one and from 97,084 bats to 10,000 bats in another (90% mortality).  This is over two years!

So what is WNS?  It's basically a white fungus that grows on the bat (on areas of exposed skin like the face leading to the name "white nose").  This fungus somehow disturbs the bat's torpor during hibernation causing it to awaken multiple times over the winter and burning off it's fat reserves such that it starves to death before it can come out in spring and find insects.  That's why one of the signs of WNS in the area are bats seen flying around in February and March - they're starving to death and looking for food (last winter I found a dead bat in my yard - probably from the Williams Lake mines).

Bats with WNS (Al Hicks - NYSDEC)

Of the six species of bat which overwinter in NYS, the numbers are grim.  Here are the mortality rates:

  Little brown bat (Myotis lucifugus) - 90%+
  Northern long-eared bat (Myotis septentrionalis) - 90%+
  Tricolored or Eastern pipistrelle bat (Perimyotis subflavus) - 90%+
  Small-footed bat (Myotis leibii) - 78%
  Indiana bat (Myotis sodalis) - 57%
  Big brown bat (Eptesicus fuscus) - Relatively unaffected

Two notes - it's ironic, but for some reason the federally endangered Indiana bat is least affected (but a 50%+ mortality for an engangered species is not good news) and the big brown is relatively unaffected apparently because it doesn't hibernate in large cave colonies like the others do.  There are a few species of forest bats that live in NY over the summer but don't overwinter and they're fine.

So, what's going to happen.  We can't treat this fungus, caves apparently stay infected even after all bats die (the fungus becomes established), and 5 of the 6 species above are very likely to become extinct in NYS.  Bats can eat over 50% of their weight each night in insects (mostly moths and beetles, not mosquitos as many people think).  I used to see lots of bats on summer nights swooping around the lights.  Now I don't.  The entire night ecosystem will change.

Dead bats in a NY cave (US Fish & Wildlife)

Like I said, a depressing talk.

Happy Birthday

March 14, 1879 - April 18, 1955

Einstein's work is still relevant today, 100 years later.

Happy PI day

Today, March 14, is pi day (3/14, 3.14 - get it?).  I'll try to post this at 1:59 (3.14159).

The origin of pi comes from the geometry of circles.  The diameter (D) of any circle, no matter what its size, multiplied by pi (p), equals the circumference (C) of that circle.  In other words:

 

p = C/D

If you want to determine pi, all you need to do is measure the circumference of a circle and divide that by the diameter.  That generally doesn't give you a very accurate value, however.

Let's supposed you measure something circular in your house, a  bowl for example, with one of those flexible tapes people use to measure their waistline.  As carefully as I can, I measured one in my kitchen and it had a circumference as 1070 mm and a diameter of 341 mm.  Doing the division C/D gives me 3.137 which I'll round to 3.14.  Not bad.

A small bit of difference in the measurement, however, can give quite a variation in pi.  It's easy to be a millimeter off when measuring with a flexible tape measure which would give me pi values ranging from 3.13 to 3.15.

Archimedes figured out how to calculate pi without any direct measurements over 2,000 years ago.  It was an ingenious method.  Archimedes knew that it was easy to determine the perimeter of a polygon.  He then reasoned that if you took a unit circle (radius = 1 unit), you can inscribe a polygon within the circle and one just outside the circle.  The circumference of the circle will be between the perimeter of the two polygons.  If you increase the number of sides of the polygon larger and larger, you'll more closely approximate the circumference of the circle (which will be pi/2 if the radius = 1).

With a 96-sided polygon, Archimedes was able to show that pi was between 3.140845 and 3.1428858.  Not much better than my simple measurement but ingenious since it allows for improvement (remember that Archimedes didn't have a calculator - calculating the perimeter of a 96-gon is quite a lot of work by hand!).

Today, there are many formulas for computing pi that have nothing to do with circles (pi pops up in all kinds of surprising places).  One neat way is a formula discovered by Liebniz, a German mathematician of the 1600s.

(PI / 4) = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 ...

There are dozens of other formulas for calculating pi, some converge slowly (like the one above), others much more quickly.

Fabrice Bellard, a French computer scientist, calculated 2.7 trillion digits of pi just this last January on a home computer running Linux.   If you're interested, here's 100,000 digits of pi.  Pi is infinite and non-repeating.

The tradional way to celebrate pi day is by having some pie.  Enjoy.

ISS Transits the Sun

Cool picture taken by some folks through an amateur telescope down in Argentina a few days ago.  It shows the International Space Station (ISS) transiting in front of the Sun (click for a larger image).

Most people don't think about this, but the ISS currently has 5 astronauts aboard.  Read about where they're doing here.  Where is the ISS now?  Here's a tracker.  Want to watch the ISS pass overhead?  Click here for predictions for your area.

Picture of the Week - Millstones

I grew up in a little town called High Falls which is about half way between Kingston and New Paltz in the Mid-Hudson Valley of NY.  The town is named for a waterfall, which is now the site of a Central Hudson hydroelectric plant, but has a long history of industrial use.  Down by the falls is a decorative millstone laying on the ground.

The millstone is made of a local rock called the Shawangunk Conglomerate - it's a hard, white rock that forms a ridge extending from near High Falls southwestward between the Rondout and Wallkill Valleys past Port Jervis and into the Delaware Watergap area of PA and NJ.  Nearby, in the village of High Falls proper (across from the church), is another millstone with a slightly different pattern.

Perhaps they were used to grind different types of grain, I don't know.  What amazes me about these millstones is that they were carved by hand!  The Shawangunk Conglomerate is composed entirely of quartz - quartz pebbles and sand cemented by quartz (silica) cement.  It's harder than granite.

Wandering the Shawangunk Ridge (in this case on Mohonk Preserve property), it's not uncommon to stumble upon old millstone quarries like the one below.

Millstones were likely mined here by the local homesteader because of suitable fractures in the rock.  Slabs of rock would be taken out and then shaped into a rough disk.  Why was the one here left unfinished?  Who knows.  Maybe the chip in the lower-right ruined it.  Imagine working for months on this and then having it crack the wrong way!

Here's a nicely finished disk (you can't see it, but there's a hairline crack in it making this one useless as well).  This one's at the Mohonk Preserve Visitor's Center on Route 44/55.

One of these days, I plan to write something up more formally about the old millstone industry here.  There's a lot I don't know and I think it would be interesting to research a bit.

Interesting Math Column - VI

Yet another weekly column in the NY Times series by Cornell mathematician Steven Strogatz. This week it's Finding Your Roots and about complex numbers. This one gets a bit more challenging with its mention of Newton's method and fractals.

The images shown in this article are very similar to the better known Mandelbrot Set which is also plotted in the complex plane by continued iterations (long ago, in college, I wrote a Turbo Pascal program to generate the Mandelbrot set).  I love zooming into fractals and exploring these images of infinite complexity generated by such simple equations.  Fractals describe many diverse forms of geometric objects including branched trees, coastlines, cloud shapes, and fault surfaces to name only a few objects that are self-similar at different scales.

One interesting thing not mentioned in the article is Euler's Formula.  Leonhard Euler (1707-1783), pronounced "Oil-er", was one of the most prolific mathematicians who ever lived.  He discovered an amazing formula:

e^ix = cos(x) * i sin(x)
 
The constant e is the base of the natural logarithms and is equal to 2.71828183 (like p, it goes on forever).  The formula cos(x) + i sin(x) denotes a set of complex numbers (the number depends, of course, on what x is equal to).  What's so amazing about this formula?  Well, if you set x = p, it becomes:

e^ip = cos(p) + i sin(p)

e^ip = (-1) + i (0)

e^ip = -1

It's a formula that now relates two of the most fundamental constants in modern mathematics, e and p (both infinite non-repeating decimals), with i, the square root of -1.  The brilliant Nobel-prize-winning physicist Richard Feynman called this "one of the most remarkable, almost astounding, formulas in all of mathematics."

By the way, this formula also pops up in the book discussed in the previous post, The Housekeeper and the Professor.

The Housekeeper and the Professor

A few weeks ago I wrote about a math column where a book called The Housekeeper and the Professor (Picador, 2009) was recommended.  Just finished reading it and I have to say that it's a very enjoyable little book.  It's the only novel I've ever read that features number theory as a plot element.

The story is set in in Japan in the 1990s and features a brilliant math professor who suffered a traumatic head injury in 1975 and, as a result, can't remember anything for more than a short time since then (his memory is like a looping tape that erases every 80 minutes).  A single-mother housekeeper and her latchkey kid form an unlikely bond with the professor despite their very different backgrounds.  The beauty of number theory (and a love of Japanese baseball) is a central theme throughout the book.

One interesting part of this book is that the characters are never given proper names - the Professor, the housekeeper (who narrates the story), and Root (her son, nicknamed because his flat-topped haircut reminded the professor of a square root sign).  You never really notice.  The book is an easy read - the language is lean and flows well - but leaves you thinking about it for a long time afterward.  The author, Yoko Ogawa, is an award-winning Japanese writer but a lot of credit must also go to Stephen Snyder who beautifully translated this book into English.

Don't let a fear of math or disinterest in baseball keep you from this book.  While I'm interested in math, I know nothing about baseball (let alone Japanese baseball), but the real story in this book is the relationships between everyone in this unlikely ad hoc family.  Highly recommended.

Mountain building and climate change

I was talking to some students recently about a large mass extinction event which occurred at the end of a period of time that geologists call the Ordovician Period (around 444 million years ago).  The extinction event correlates fairly well to evidence we have of an ice age occurring on a southern supercontinent called Gondwana around that time.

(What most people think of as the "Ice Age" occurred between 2 million years ago and 10 thousand years ago and is only one of several ice ages the Earth has had in its long history).

Continental glaciation lowers sea levels globally (water evaporates from oceans, falls as snow, and doesn't flow back into the oceans).  During the Ordovician, much of North America was actually under water (epicontinental seas).  The cooling Earth had colder oceans (stressing marine life) and dropping sea levels exposed shallow sea floors killing off the animals living there.

North America during the Late Ordovician Period (450 Ma)

Dr. Ron Blakey (http://jan.ucc.nau.edu/~rcb7/nam.html)

North America during the Early Silurian Period (430 Ma)
Dr. Ron Blakey (http://jan.ucc.nau.edu/~rcb7/nam.html)

One of the interesting questions one can ask about glaciation is "Why then?"  What was significant about this period of time such that global temperatures dropped?  Well there is some good evidence that global carbon dioxide (CO₂) levels dropped significantly before this glaciation event (this was a time when carbon dioxide levels in the atmosphere may have been 15 times what they are today but that's a different story).

Anyway, this drop in carbon dioxide levels seems to correlate with an event occuring right here in New York at the time - the Taconic Orogeny (from the Greek "oros" meaning mountain and "genesis" for birth or origin).  During the latter part of the Ordovician Period, a chain of volcanic islands collided with the edge of the proto-North American continent due to plate tectonic convergence.  This collision thrust up Rocky Mountain sized peaks just east of where I'm sitting right now in the mid-Hudson Valley.  These were the beginning of the Appalachian Mountains.

Before the collision, there were volcanoes with massive eruptions spewing huge amounts of carbon dioxide into the Earth's atmosphere (several of these events spewed ash layers we can trace over much of eastern North America!).  After the collision, those volcanoes shut down.  What we're left with are high barren mountains (there was still essentially no significant amounts of life on land at this time - maybe a few small plants and arthropods near the shorelines).

Feldspars are minerals common in igneous and metamorphic rocks found in mountains belts such as the ancient Taconics.  Three common types of feldspar are potassium feldspar (orthoclase or microcline - KAlSi₃O₈), sodium plagioclase feldspar (albite - NaAlSi₃O₈), and calcium plagioclase feldspar (anorthite - CaAl₂Si₂O₈).

Here's a picture of a typical igneous granite, for example:

http://www.beg.utexas.edu/mainweb/publications/graphics/granite.htm

This pink mineral grains are potassium feldspars and the white mineral grains are the sodium plagioclase feldspars (there are other minerals in this rock as well such as glassy quartz and black amphiboles).  When feldspars are exposed to water (H₂O) and carbon dioxide (CO₂), they chemically weather into a type of clay called kaolinite (Al₂Si₂O₅(OH)₄) as well as releasing various cations (e.g. Ca⁺², K⁺, or Na⁺), silicic acid (H₄SiO₄), and the bicarbonate anion (HCO₃^-).  Here are the equations.

CaAl₂Si₂O₈ + 2CO₂ + 3H₂O = Al₂Si₂O₅(OH)₄ + Ca⁺² + 2HCO₂^-

2KAlSi₃O₈ + 2CO₂ + 11H₂O = Al₂Si₂O₅(OH)₄ + 4H₄SiO₄ + 2K⁺ + 2HCO₃^-

2NaAlSi₃O₈ + 2CO₂ + 11H₂O = Al₂Si₂O₅(OH)₄ + 4H₄SiO₄ + 2Na⁺ + 2HCO₃^-

So we have high mountains and the mountains are exposed to the elements.  Minerals in the rocks chemically weather into clays and free ions - a process which uses up carbon dioxide from the atmosphere.  So much carbon dioxide gets used up in the weathering away of these mountains that global temperatures start to fall.  This triggers the Late Ordovician ice age which causes a mass extinction of marine life.

A neat story.  The lithosphere (solid Earth), hydrosphere (water), atmosphere, and biosphere (life) are all interrelated in the Earth system.

Interesting Math Column - V

Another column in the NY Times series by Cornell mathematician Steven Strogatz. This week it's The Joy of X and about algebra.  One thing that popped out for me is his mention of thinking about how units will cancel out when discussing a conversion of feet into yards problem.

Converting units involves dimensional analysis (it's easier than it sounds).  One of the complaints among science faculty at the community college where I teach is that students coming out of high school have no idea how to do dimensional analysis and needlessly struggle through what should be very simple algebraic problems.  A common refrain heard around here is "What exactly are they teaching these kids in high school science classes?"

Even in my relatively math-free introductory geology classes, students often struggle to do the simplest problems.  For example, working with maps, we might have to convert feet to meters.  Maybe we have an elevation of 750 ft and we need to know how many meters that equals.  Their lab manual may list a conversion factor 2.54 cm = 1 in.  What now?  Well, it's assumed that people know there are 12 inches in 1 foot and that there are 100 centimeters in 1 meter.  Many students will then take 750 ft and proceed to semi-randomly multiply and divide numbers until they get some sort of answer. 

Sometimes, they even write down, as an answer, a numerically ridiculous answer like 0.246062992 m (they did all division and reported all the decimals on their calculator screen) or 2,286,000 m (they did all multiplication).  I've seen it over and over again.  That's what the calculator said so it must be true.

I always stress that these problems become trivially easy if you simply multiply through by 1 and pay attention to units you're crossing off.  What do I mean by multiplying by 1?  Well, if there are 2.54 centimeters in 1 inch (2.54 cm = 1 in), then we can set up the fraction (2.54 cm / 1 in) or (1 in / 2.54 cm) and both fractions equal 1.  Why do they equal 1?  Because the top and bottom terms are equal to each other.  We can set up similar fractions for 12 in = 1 ft - (12 in / 1 ft) or (1 ft / 12 in) - and 100 cm = 1 m - (100 cm / 1 m) or (1 m / 100 cm).  Now we just set up a series of fractions so we can cross off units going from feet to meters.

750 ft x (12 in / 1 ft) x (2.54 cm / 1 in) x (1 m / 100 cm) = Answer in meters

Why does this work?  Because units cross off as you go left to right.  First feet (ft):

750 ft x (12 in / 1 ft) x (2.54 cm / 1 in) x (1 m / 100 cm) = Answer in meters

Then inches (in):

750 x (12 in / 1) x (2.54 cm / 1 in) x (1 m / 100 cm) = Answer in meters

Then centimeters (cm):

750 x (12 / 1) x (2.54 cm / 1) x (1 m / 100 cm) = Answer in meters

And, finally, the only units left are meters (m).

750 x (12 / 1) x (2.54 / 1) x (1 m / 100) = Answer in meters

Now just multiply all the top terms together, all the bottom terms together, and divide top by bottom:

(750 x 12 x 2.54 x 1 m) / (1 x 1 X 100) = 22,860 m / 100 = 228.5 m

So 750 feet is equal to 228.6 meters.

It's a little more laborious to write out this way, but you'll never again wonder "Do I multiply by 100 or divide by 100?"  It's obvious if you set the units up so that cross out top and bottom.  Here's a website that has more simple examples.

Picture of the Week - North-South Lake

Below is a panorama image (click for full-size) of North-South Lake near Catskill, NY.  The picture was taken from a place called Sunset Rock (also called Bear's Den) off the blue escarpment trail (a highly recommended hike along the edge of the Catskill escarpment).

Between 1824 and 1963, there was a famous hotel called the Catskill Mountain House on the cliffs to the left of these lakes (there were actually several hotels in the area - all long gone now). 

View from the Mountain House by W.H. Barlett (1836)

During the latter half of the 19th century, the Catskill Mountain House was visited by the rich and famous who traveled up the Hudson on steamship and to the hotel on rail or a horse and carriage.  If you want to learn more about the Mountain House, there's a great book The Catskill Mountain House by Roland Van Zandt.  This area was also made famous by the Hudson River School group of romantic landscape painters.

A View of the Two Lakes and Mountain House,

Catskill Mountains, Morning by Thomas Cole (1844)

Catskill Mountain House by Jasper Cropsey (1856)

It's neat to stand on such a beautiful spot with my digital camera and feel a connection with these artists who stood there 150 years earlier with an easel and paints and try to immortalize the same scene.