Infinite series are weird — redux!

A bit over a year ago, I wrote a blog post about the mathematics of infinite series, and how weird such series can be, considering in particular the behavior of “conditionally convergent series”.  A recent post at Built on Facts covered similar oddities and gave a nice and different perspective on them.  In the comments of that post, though, an even more bizarre result from the theory of infinite series was introduced, namely the argument that

\displaystyle 1+2+3+4+\ldots = -1/12.

This result, if true, is enough to shake one’s faith in mathematics, and is completely non-intuitive for no less than three very big reasons:

  1. The sum of an infinite series of increasing positive integers should not converge to a negative value,
  2. The sum of an infinite series of increasing positive integers should not converge to a fractional value,
  3. The sum of an infinite series of increasing positive integers should not converge at all to a finite value!

So is the equation above correct?  Not exactly; it is based on a valid bit of mathematics centered on the Riemann zeta function, but that mathematics is being somewhat misinterpreted to get the paradoxical equation.  An explanation of what went wrong is interesting in itself, however, and allows me to describe a rather difficult concept in the theory of complex analysis known as analytic continuation.

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Posted in ... the Hell?, Mathematics | 14 Comments

ResearchBlogging editor’s selections: scale in oceanography, majestic Megatherium, strange atlatls and virtual slaps

  • Varieties of Oceanographic Experience. Sam at Oceanographer’s Choice looks at a classic paper that considers how the scale of a measurement influences — or distorts — the results.
  • The majestic Megatherium. When people think of paleontology these days, they think of dinosaurs; however, this wasn’t always the case.  Brian at Laelaps looks at the original “great beast” of the public’s imagination, Megatherium, and how perceptions of it have changed throughout the years since its discovery.
  • Atlatls to Bows: A Very Strange Atlatl from California. Continuing a fascinating series on the history of atlatls in the New World, teofilo at Gambler’s House looks at an unusual specimen that was acquired in California and the implications of the find.  (Don’t know what an atlatl is?  Read the introductory post by teofilo here.)
  • A virtual slap in the face (isn’t there an iPhone app for that?) Finally, NeuroKüz discusses recent research that highlights how real a virtual body in a first-person simulation can feel — through the experience of a virtual slap!

Check back next week for more miscellaneous selections!

Posted in General science, Science news | Leave a comment

Sabrina at rest!

I finally managed to get a picture of our cat Sabrina’s cute little habit!  Sometimes, when she sleeps, she ends up sticking her tongue out:

That is all for now; continue with your regularly scheduled day…

Posted in Animals | 2 Comments

Jeff VanderMeer’s Finch

I’ve had the good fortune to read many good works of weird fiction since starting this blog — in fact, one of the major motivations for starting the blog was to “force” myself to get back into reading strange and creepy stories such as those that had captured my imagination as a youth.  Every once in a while, though, I come across a work so wonderful and fascinating that it will permanently haunt the depths of my psyche.  Case in point: I was absolutely blown away by Jeff VanderMeer’s recent novel, Finch (2009):

The novel defies easy characterization: it is part detective novel, part science fiction novel, part war novel, part fantasy novel — and part horror novel.  Even with that mixing of genres, VanderMeer manages to tell a very serious, intricate, and mesmerizing tale.

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Posted in Horror, Science fiction, Weird fiction | Leave a comment

The Giant’s… ahem… Leviathan’s Shoulders #23 is up!

The special marine science edition of The Giant’s Shoulders, dubbed The Leviathan’s Shoulders, is up at Deep Sea News!  Many thanks to Kevin Zelnio for putting together this lovely oceanic edition!

What do you think of the idea of  “special editions” for the blog carnival?  Feel free to leave a comment!  If future hosts have suggestions for special editions, send me an email with your idea.

The deadline for the next edition is June 15th, and it will be held at Jost a mon.  Entries can be submitted through blogcarnival.com or directly to the host blog, as usual.

Posted in General science, Science news | 2 Comments

ResearchBlogging editor’s selections: Antikythera, sports fan anthropology, bad oil theories, and false memories

  • Planets and Anomalies in the Antikythera Mechanism. The Antikythera Mechanism is one of the fascinating relics of the ancient world — a 2000 year old clockwork “computer” that could be used to track the Sun, Moon, and calendar.  In a detailed post, Alun at AlunSalt describes recent research that highlights again how ingenious the device is.
  • For the Love of the Game: A Look at Fans and Disappointment. It’s easy to get wrapped up in the successes and failures of one’s favorite sports team, but we typically don’t think about it as an anthropological exercise.  Krystal at Anthropology in Practice relates some research on disappointment versus discouragement in baseball fans: why do we stick with a losing team?
  • Rethinking petroleum a little too hard… Most people realize that one day we’ll run out of oil, but there are some diehard skeptics who believe that new oil is naturally produced all the time in the Earth’s interior, and is effectively limitless.  Greg Fish at weird things throws some cold water on that hypothesis and describes the origins of this stubborn abiogenic theory.
  • Confidently Wrong: Correcting False Memories. Remember the time that you saw that thing happen — even though it never did?  Daniel at Ingenious Monkey talks about the phenomenon of false memories, and how we stubbornly cling to those memories.

Check back next Monday for more “miscellaneous” selections!



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Shocking: Michael Faraday does biology! (1839)

(This is my entry to the first “special edition” of The Giant’s Shoulders, dubbed “The Leviathan’s Shoulders”, with an emphasis on oceans and ocean life.  The post is actually about a river creature, but, hey, it’s still aquatic!)

I’ve spent a lot of time talking about Michael Faraday (1791-1867) and his scientific accomplishments on this blog.  His thorough investigations into the nature of electricity and magnetism paved the way for all of modern electromagnetics as well as optics, and he is rightly viewed as one of the greatest experimentalists of all time.  Among his monumental works are the observation that changing magnetic fields induce electric fields (electromagnetic induction) and the observation that light polarization can be affected by an applied magnetic field (Faraday rotation).

Though it is natural to think of Faraday as a researcher of electricity alone, in his era the study of electricity connected to almost every aspect of the natural sciences.  In the late 1700s Luigi Galvani had shown that an amputated frog’s leg could be made to move by electrical stimulation, demonstrating a connection between biological function and electricity.   By 1800 it was known that chemical reactions can be induced by electricity, in a process known as electrolysis; Faraday himself published fundamental results on electrolysis in 1834.  Electricity could be connected to thermodynamics through the observation that an electrical current heats the wire it passes through (Joule heating); this process was rather mysterious because neither the origins of heat (atomic motion) nor electricity (electrons) were established in Faraday’s time.

Electricity could be generated through atmospheric, chemical, and mechanical means, and it was by no means obvious that these different sources were manifestations of the same fundamental electrical phenomenon.  (In fact, Faraday himself did a significant amount of research to demonstrate that all forms of electricity are in fact the same. )

A researcher of electricity could therefore be expected to make forays into quite diverse areas of study.  In 1839, Faraday published the scientific results of one of his forays, “Notice of the character and direction of the electric force of the Gymnotus,” in the Philosophical Transactions of the Royal Society (pp. 1-12).

What is the “Gymnotus”?  The taxonomy of the species seems to have been changed over the years, but at this time seems to be referring to what used to be known as Gymnotus electricus, or the electric eel (image source):

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Posted in History of science, Physics | 24 Comments

5 days until The Leviathan’s Shoulders!

Here’s one last reminder:  There’s 5 days left to submit entries for the 23rd installment of The Giant’s Shoulders, a special edition dubbed “The Leviathan’s Shoulders”!  It will be hosted by Kevin Zelnio over at Deep Sea News, and will be a special “oceans edition” of the carnival.  All ordinary entries will still be accepted, but bloggers are encouraged to submit posts on the history of science that deal specifically with oceans and ocean life!   Entries can be submitted through blogcarnival.com or directly to the host blog, as usual.

Let’s make this an extra-special carnival!

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A trip to Savannah, Georgia

Last weekend, the wife and I celebrated our first-year wedding anniversary by taking a 3-day trip to Savannah, Georgia.  Savannah is one of those rare American cities that has maintained a significant portion of its historic neighborhood, and this neighborhood is highlighted by a large number of green open squares that were first planned when the city was founded in 1733.

Savannah is a rather hard city to photograph for a novice like myself: buildings are quite close together and there are a large number of very old trees, making it difficult to get nice scenic shots.  These same properties make it a very lovely area to walk and explore, however, and I thought I’d share some of the pictures I did manage to take, including shots of the wonderfully atmospheric Bonaventure Cemetery.

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Posted in Cemetery, Travel | 3 Comments

ResearchBlogging editor’s selections: bilingual animals, horse-hunting hyenas, the nature of dark matter and toiletology

  • Bilingualism as a preadaptation for language. Can animals other than humans be “bilingual”?  This fascinating question is addressed in research described by Sean at The Adventures of Auck.
  • The horse-hunting hyenas of Srbsko Chlum-Komin Cave. We can learn a lot by studying what an animal eats — and what parts it eats!  Brian at Laelaps explains how the bone piles of ancient hyena dens tells us more than just what animals were hunted.
  • Dark matter confronts observations. In a detailed post, The Astronomist at his eponymous blog discusses theory and observation relating to the specific nature of dark matter.
  • Toiletology. What can the signs describing a toilet tell us about the inclusiveness of a society?  Ingrid at Language on the Move explains.

See y’all next Monday for more “miscellaneous” highlights!

Posted in General science, Science news | Leave a comment