Tuesday, August 3, 2021

When vaccination rates are high, most positive cases will be among the vaccinated (and that's okay)

Two recent COVID outbreaks have made headlines because of the high proportion of "breakthrough" cases, where people test positive for COVID despite being fully vaccinated.  San Francisco hospitals in July reported 50 staff members tested positive for COVID, despite 75% of them being vaccinated.  In Provincetown, MA, a July outbreak of the delta variant led to several hundred new cases, the majority of whom were fully vaccinated.
How can the majority of new cases be among the vaccinated?  Some people wrongly (if understandably) worry this means the vaccines aren't working, particularly against the delta variant. 

The truth is the numbers show that the vaccines ARE very effective, even against the delta variant.

The problem here is the reporting.  News agencies are going for provocative headlines, and in the process are falling for what is known in statistics as the "low base rate fallacy" or base rate bias.  In this case, breakthrough infections have a very low base rate, and all of other numbers are meaningless without keeping that in mind.  

The low base rate of breakthrough infections should be the focus in reporting.  For instance, the headline does not mention that the 50 new cases in July among SF hospital staff is out of 7,500 staff members, a breakthrough base rate of less than 1%.  They don't mention that until later in the article.

How can positive rates be higher among the vaccinated?

What does it mean that, in the delta outbreak in Provincetown, there were a higher proportion of positive cases among vaccinated folks compared to unvaccinated?  Does it mean the vaccine is ineffective agains the delta?  NO.  Because of the low base rate, when vaccination rates are high, more positive cases will be among the vaccinated.  

What it really means is just that most people in MA are vaccinated! To understand this, a picture is worth 1000 words. I came across this diagram that clearly shows what's going on. 

source: https://i.redd.it/cpjwvqv2s0d71.png

The left half of the diagram shows what happens in a highly vaccinated population, based on real-world data from England.  There was a 2% chance of getting symptomatic COVID among the unvaccinated (red dots).  For the vaccinated, it was a much smaller percentage who got COVID (only 20% of 2%, aka 0.4%).  But since so many more people were vaccinated (over 10x as many), that led to a higher number of vaccinated people who tested positive (blue dots).  This explains the counterintuitive result that even though the vaccine is working, more people who test positive are vaccinated.  There's just so many more vaccinated people! (Notice there are still way fewer vaccinated who end up hospitalized.)

The right half of the diagram shows what happens when a lower percentage of the population is vaccinated.  First - and most importantly - there are more overall cases and hospitalizations.  Second, since there are more unvaccinated people, the higher proportion of people who test positive are unvaccinated.  This matches our usual intuitions.

In summary: When most people are vaccinated, it brings down the total # of positive cases drastically (yay!), but can end up with a higher proportion of positive cases among vaccinated folks. This can be confusing, because humans aren't great at doing statistics when there's a low base rate involved.  It really helps to visualize it. 

Remember that breakthrough cases are normal and to be expected, and extremely rare in reality (much less than 1% in MA and nationwide). The vaccines help in multiple ways: they reduce positive cases and also lead to a lower rate of hospitalization among those who test positive (only 1% of vaccinated positives from the recent MA delta outbreak were hospitalized). 

TLDR: Vaccines are working well, even when a higher percentage of positive cases are among the vaccinated!

Epilogue: Simpson's Paradox
The base rate fallacy leads to all sorts of funky, counterintuitive results:  
  • My favorite comes from baseball - In 1989 Andy Van Slyke had a higher batting average than David Justice (0.237 vs. 0.235).  In 1990 Van Slyke again had a higher average than David Justice (0.284 vs. 0.282).  But when you look at their combined 2-year battering average 1989-1990, David Justice is the one with the higher average (0.278, compared with Van Slyke's 0.261).  How can this be?  David Justice had far fewer at bats in 1989, so his combined average is skewed closer to his 1990 season.  
  • Simpson's paradox is what philosophers and statisticians call an association between variables at the population level that reverses when divided into subpopulations.  Understanding Simpson's paradox has important ramifications for epidemiology and for equity.
  • At the population level, vaccinated people are less likely to get COVID.  But when comparing vaccinated to unvaccinated, a higher proportion of new cases can be among the vaccinated.  This is Simpson's paradox.  Of course, it's not truly a paradox - it is just so counterintuitive you have to think it through carefully. Every. Time.

Saturday, January 26, 2013

Who are we? Don't ask our neurons, they're a bunch of idiots. And don't answer with your gut, because your gut isn't even human.

As the philosopher Daniel Dennett likes to remind us, each one of us is made up of trillions and trillions of living organisms (we call them 'cells'), and not a single one of these living things that make us up has any idea who we are, or even cares.

Take a moment to let that sink in.

Now, don't blame the cells for not caring; they are all living organisms and thus have their own metabolism, their own respiration, and their own reproduction to manage.  They don't have the time to worry about the bigger picture.  Nor do they have the mental capacity.  Even neurons--the living organisms that do our thinking for us--are quite dumb, when considered on an individual level.  Even though it is our neurons that allow us to think and learn, no individual neuron can think or learn.  Each one follows very simple rules of behavior: basically if a certain number of its neighbors are firing, they fire too.  That's it.  And yet somehow, out of all those mindless firings of neurons, a mind emerges--a mind that can even write a blog post about the neurons mindlessly making a mind.

Three stupid neurons.

More than human.
It gets even weirder.  A good proportion of the cells that make us up aren't even human!  Before I tell you what percentage, I invite you to take a guess: What percentage of the cells in our body are human cells?  90%? 70%? 50%?  So far, you're way off.  (Are your neurons even firing correctly?  Sheesh.)  Truth is, human cells are the in the minority amongst humans, by a long shot.

Human cells comprise only about 10% of the 100 trillion cells in a human.  10%.  The rest of the cells in our body are mostly bacteria, which generally take up residence in our gut.  Far from being invasive, our bacterial friends are an integral part of our digestive system, breaking down food and passing along the energy and vitamins to us after they're done with them.

Are you the same "you" that you were 10 years ago?
Biologically speaking, not exactly.  The living organisms that make us up generally do not live long enough to see us grow old.  Our body may grown and develop at a steady, slow pace, but that is due to a complex flux of cells dying and new cells being born to take their place (here's a nice NYT article on this).  The skin that covers your entire body--the thing that makes you look like "you" with your dimples and wrinkles and all--is not even the same skin you had two weeks ago.  The stable patterns in our skin, including our fingerprints and dimples and wrinkles and all, are sort of like traffic jams that persist even after all individual the cars that started the jam travel on down the road.

Down another level, they say that most of the atoms and molecules that make up the cells are replaced by other atoms (of the same variety) every year or so.  Since the new atoms are functionally equivalent to the old atoms, we would never know the difference.

So what are we to make of this?
So, if the neurons that make us think are all stupid, and the cells that make us up are dying all the time, how do we think?  How do we persist?

Part of what makes this all so mysterious is that we are so far removed from the microscopic level of the goings-on of our body that we have no sense of how the micro-level goings-on give rise to the macro-patterns of our daily life.  But there are some systems in nature where the levels are not so far removed, and so we can learn something about this mystery by studying them.  I will briefly introduce you to two of them: slime molds, and ant colonies.

Slime molds
The slime mold may be one of the grossest organisms on earth, but it is also one of the most fascinating.  We humans like to think of our life cycle in terms of our whole body, not our cells, but the life cycle of the slime mold doesn't work like that.

The slime mold spends much of its life as a single-celled organism, not unlike an amoeba, generally sliming around.  But there comes a point in the life cycle where tens of thousands of individual cells come together and act like one big organism.  First this aggregate organism looks like a slug, slithering around, eating, sliming, and doing whatever it is slugs like to do.  Then during it's reproductive stage the slime mold self-organizes into a mushroom-like stalk, releasing new spores into the surrounding environment and starting the whole process over again.

Although individual slime mold cells are just about as dumb as our individual neurons, the aggregate of slime mold cells can be pretty darned smart.  For example, they can find their way through a complicated maze to get to food:

Ant colonies
Although ants may be smarter than slime mold cells, they are still pretty dumb.  What I mean by that is that ants follow a very simple algorithm for guiding their behavior, a very simple decision-making procedure based entirely on pheromones.

If, in their walking around, an ant happens to cross paths with another ant, they give each other an antennae high-five and sniff out the pheromones.  There is a pheromone that signal one of a few basic meanings: "I found food" and one for "I'm working on the hive" and another for "there's a threat to the colony".  If they sniff the pheromone for "I found food", then they head in the direction that ant was coming from, repeating the process with other ants, until it ultimately finds the food.  This is not much different than our dumb neurons' algorithm of "if my neighbors are firing, I will too".

Now, despite being made up of dumb ants, an ant colony is pretty darned smart.  For instance, they can solve the complicated problem of allocating the optimal number of ants to various food sources at various distances with varying amounts of food.  (here's a nice TED talk on the emergent intelligence of ant colonies)

Also, ant colonies persist for decades, even though the individual ants generally die after several months or at most a few years.  (The queen is the exception to this rule, she's around for up to 21 years).

So, ant colonies and slime molds each give us insight on how a form of intelligence can emerge from the collective behavior of dumb individuals, much like mindless neurons can collectively make a mind. They also lend insight onto the persistence problem, how we can be around for decades when most of our cells are long dead by then.

The bigger picture.
Now that we've seen there's no real magic to how we can be composed out of a bunch of dumb organisms but still be relatively smart, there are more questions to consider.  Should we feel bad if we scratch our cheek and kill thousands if not millions of living organisms?  Is it possible that individual humans could be part of some larger system that is even smarter than any individual humans?  If so, should that system feel bad if it happens to "scratch it's cheek" and wipe out thousands if not millions of humans?  And what should we call that system?  I propose that we call this hypothetical system "Frank."  Frank, if you're listening, please don't scratch your cheek!!!

Extra goodies:

  • this article also reports on a finding that human cells even forage for food in a pattern similar to other scavengers, including sharks and um, penguins
  • another video of slime mold finding the shortest path to food through a maze 
  • here's a research article on the how ant colonies solve complicated problems with dumb ants

Saturday, April 14, 2012

Do you have the time? Some surprising facts about how consciousness works

Surprising fact: The neural signals from a drummer's foot take almost of a third of a second longer than do the signals from the hand when they are playing the same beat.

That's not all.  Whenever you stub your toe, there are pain signals that get to the brain almost a third of a second slower than the others, even though they contribute to the very same pain sensation.

How does the brain work this out to end up with one conscious experience of pain, or of a funky drum beat?  Does the brain wait for the last signal to come in before it puts it all together into a feeling?  Or can it revise and resubmit sensory information to the running journal of conscious experience?

An illusion called 'the phi illusion' helps to further illustrate the problem, and how the brain works it out. The result is really quite crazy.

The Phi Illusion

The phi illusion is the reason why we can watch movies and see them as motion rather than as a sequence of still pictures, which is all a movie really is.

For example, here is the first movie ever made: 1st movie ever.  In 1878 Eadweard Muybridge filmed it by lining a stretch of track 12 cameras that went off as a horse ran by.  When we experience the horse's continuous motion means that we are, in a sense, filling in the blanks between pictures.

The illusion is hard to appreciate, because we are so familiar with motion pictures.  But a way to notice the illusion more clearly is to focus on one or two features changing between frames.  Here is an animation of a dot that looks like it's moving back and forth while changing colors, at least for certain settings of the framerate.  But really, it's just two dots, one red and one green, flashing on and off with just the right delay between dots.

When you experience that dot moving back and forth, it seems you must been extrapolationg, somehow, where the dot was between frames of the animation.  And what color it was.  But just what color was it?  And where exactly did you "see" it in between frames?  Are those things somehow registered somewhere in your brain as sensory information?  Or are you just telling yourself a story about what happened and the details are left ambiguous and unexamined by your consciousness?  According to the philosopher and cognitive scientist Daniel Dennett, conscious experience is more like a story you tell yourself about what happened.  There is no "halfway dot" registered anywhere in your visual cortex.

Daniel Dennett has explained this illusion with his "multiple drafts" model of consciousness.  This theory says that we are always making up narratives of "what's going on".  There are usually more than one versions of the story floating around, and its never too late to revise...your consciousness experience is what emerges from an evolutionary-like battle between ideas of what happened.  In the phi illusion, stories that involve motion get selected for so that they end up in the later "drafts" of the story.

The dot appears to move back and forth because when it goes from say, being a red dot on the left to being a red dot on the right, because once that red dot appears we revise the story to include motion, and a color change.

So when a drummer plays a funky beat, he does not have to wait for all the neurons to reach his brain to from his foot as it hits the kick drum before he truly experiences the beat.  And the beat is not experienced as "smeared out" in time, either.  This is despite the fact that multiple kinds of neural signals contribute to the same sensation, some that arrive up to a third of a second later.  Once those straggling neurons deliver their message, the drummer's conscious experience of "the beat" is revised accordingly.

You may have experienced for yourself how the experience of the past can be "retrospectively presented" to the present version of consciousness, say, when you suddenly notice a church bell has been ringing and you are still able to count the chimes, even though you didn't notice it when it started ringing.  You just submitted a new draft that includes revised information about the past.  The versions of your story of consciousness that did not contain bells ringing got beat out by versions that included bells ringing, even though the battle took some time.

Okay, that's all I have time to write.  Or at least that's what I'm telling myself.

Here are some more links for the phi illusion, and related illusions:

Saturday, December 3, 2011

What's wrong with the Bill Gates' foray into educational research & reform

#1 Bill Gates is not an education researcher, let alone an educator, or a researcher.
Don't get me wrong.  I love that Bill Gates is identifying education as a key issue to focus on.  I love that he is taking an interest in improving education.  I am not questioning his motives.  Just his methods.

What I hate is that his approach is the same know-it-all attitude that many take towards education and education research.  It's the old "I turned out okay, and I know what worked for me, therefore I am an expert on education" approach.

Good ol' Uncle Bill knows all there is to know about education reform

Which would be bad enough if he were just Uncle Bill, telling-it-how-it-is from his comfy chair at a holiday party.  But instead, he is using his name to garner authority on a subject he knows little about.

#2 His ENTIRE research program is explicitly based upon students' scores on standardized tests
The following is taken from a recent report by Gates' MET project:
The MET project is based on three simple premises: 

  • First, whenever feasible, a teacher’s evaluation should include his or her students’ achievement gains.
  • Second, any additional components of the evaluation (e.g., classroom observations, student feedback) should be demonstrably related to student achievement gains 
  • Third, the measure should include feedback on specific aspects of a teacher’s practice to support teacher growth and development. 
NB: "Achievement gains" are edu-jargon for increases in standardized test scores.  There are so many problems with this, I don't know where to begin.  For one, there is the obvious: these standardized tests DO NOT MEASURE what students are learning.  But a more important issue to me is that: THESE TESTS DO HARM TO STUDENTS.  And I'm not talking just about "they get stressed when they take tests."  I'm also talking about the WEEKS of instructional time lost to testing throughout the school year, and the MONTHS of instructional time lost to test-preparation throughout the school year.  I'm talking about the science classes, the history classes, the art classes, which ARE NOT TAUGHT so that reading and math test scores can improve.

This is just a little slice of the Test-Mania inflicting the schools where I make my rounds.   I wonder what the kids make of this poster, as they pass it in the hallway every day.  And I wonder how it makes them feel.
I know what
I think when I pass it: 
58.3% is SOARing?? 
See, the biggest damage done is not from the 5-hour-long sessions of sitting at a desk to take these tests; it is in the life-long damage to the students done by the test-mania they are subjected to in school.  At the heart of the issue is the damage done to students' conceptions of what it means to learn something.

As a science educator, I am most concerned about how the tests distort students' views on what it means to learn and to do science.  The  students take away such distortions from this test-crazed school culture when it is working as it is supposed to.  That means under Gates' ideal conditions, students' conceptions of science (and who knows what else!) will be destroyed, or at least distorted beyond recognition.  I have seen this first hand in my time in the classroom as an instructor, and as a researcher.

The fundamental principle of educational reform should be the same as it has been in medicine: DO NO HARM.  Gates' program, despite its best intentions, is a way of institutionalizing harm to children.  This is what happens when you jump from your own biases to institutionalizing them as recommendations.  Which brings me to my next wag-of-the-finger:

#3 Gates is jumping right from making assumptions to making recommendations
It's one thing to do descriptive statistics.  It's another to make causal claims.  But causal claims must be substantiated BEFORE making recommendations.  And the research that's already been done must be addressed first.

Yet somehow, Gates decided that he can simply dismiss all the research that shows the positive correlation between small class sizes and student learning (one of the most robust findings in educational research).  His recommendation: INCREASE class sizes, so we can force teachers to do more for less, and with less resources to help them do it.

Again, if this were just Uncle Bill ranting at a party it would be one thing.  But this is a person who is determining the future of the educational system with billions of dollars of investment in harmful and misguided policies and their proliferation.

#4 He wants traffic cameras in the classroom...
...except instead of being used to issue speeding tickets, they will be used to punish teachers.  Big brother, anyone?  What a great way to encourage teachers to enter into the profession and stay there (which, by the way, is really the key issue, if you ask Uncle Luke).  And by "great" I mean "idiotic."

Gates knows when you've been boring, Gates knows when you've been late...
Gates knows when kids are snoring, so give them tests for testing's sake!

Okay, enough ranting.  Here is something good I think Uncle Bill is bringing to the table, although he is horribly misusing it: using videos of classrooms to improve instruction & learning.  Video of classroom practice can be extremely helpful for teachers, UNDER THE RIGHT CONDITIONS.  Larry Ferlazzo has written a wonderful article on how video can be used productively to improve teaching.  Mr. Ferlazzo *voluntarily* used video of his teaching in order to have an open discussion with an external evaluator *and his students* about his teaching *in that video* (not to summarily characterize his teaching).  That sort of practice has great promise, if you ask Uncle Luke.

Monday, October 10, 2011

What are the lessons to learn from Columbus?

Was Columbus a hero or a villain?  

We all know that Columbus' legacy is a controversial one.  Many history books have painted him as an intrepid explorer and overlooked the facts that would paint him in a negative light.  Some Columbus scholars have admitted that he encited a devastating genocide, but dedicated no more than a sentence to that fact (see, for example, Samuel Eliot Morison).  

So which is it--was Columbus a hero or a villain?  And how do we decide?  One thing that would be important to determine is what Columbus himself thought he was doing.  Perhaps he was trying to do good.  Perhaps he did not mean to exploit and kill all those natives...maybe it was his lower officers that took things into their own hands and got out of control.  

Fortunately, we have Columbus' log, as well as the log of several of his viceroys, and so we can find some answers to this question.  This, for example, is what Columbus wrote about his first encounter with the natives:

They ... brought us parrots and balls of cotton and spears and many other things, which they exchanged for the glass beads and hawks' bells. They willingly traded everything they owned... . They were well-built, with good bodies and handsome features.... They do not bear arms, and do not know them, for I showed them a sword, they took it by the edge and cut themselves out of ignorance. They have no iron. Their spears are made of cane... . They would make fine servants.... With fifty men we could subjugate them all and make them do whatever we want.

Well, he pretty much just comes out and says that he is intentionally subjugating them and making them do whatever they wanted.  And, also according to Columbus' own words, they wanted them for two things: gold and slavery.  So he took many as slaves and set everyone else to the grueling work of looking for gold where there wasn't any to write home about.   The working conditions for finding the gold were miserable, and the penalty for not meeting one's gold quota was chopping off the hands and bleeding them to death.  

The situation was written about extensively by one of Columbus' appointed governors, Bartolome de las Casas, a young priest who initially took part in the conquest and held slaves and such, but became disgusted by the cruelty of the Spaniards.  Here he describes the working conditions for the Arawaks faced under Columbus' rule:

Thus husbands and wives were together only once every eight or ten months and when they met they were so exhausted and depressed on both sides ... they ceased to procreate. As for the newly born, they died early because their mothers, overworked and famished, had no milk to nurse them, and for this reason, while I was in Cuba, 7000 children died in three months. Some mothers even drowned their babies from sheer desperation.... in this way, husbands died in the mines, wives died at work, and children died from lack of milk . .. and in a short time this land which was so great, so powerful and fertile ... was depopulated. ... My eyes have seen these acts so foreign to human nature, and now I tremble as I write. ...

When Las Casas arrived in the Bahamas in 1508 he documented the unbelievable scale of the devastation:

"there were 60,000 people living on this island, including the Indians; so that from 1494 to 1508, over three million people had perished from war, slavery, and the mines. Who in future generations will believe this? I myself writing it as a knowledgeable eyewitness can hardly believe it...."

And so the question of whether Columbus' behavior was villainous seems pretty settled, based on his own writings as well as accounts from his officers.  What is not yet settled, for me at least, is how Columbus justified his behavior to himself.  With his reports to the King and Queen of Spain he could talk of subjugation without needing to explain how that is a good thing.  But what was going through his head when he went to sleep each night?  Was there any recognition of the evil actions let alone remorse?  Did he think he was being a hero?  Did he, as a devout Christian, think he was saving the souls of these natives?  If so, did he somehow twist this reasoning to render his maniacal methods as somehow forgivable, even noble?  Is there a way we can ever find this out?

I don't know the answer to these questions, perhaps nobody does and nobody ever will; but I think it is an extremely important set of questions with which to struggle.  You see, Columbus is not the only instance of such a horrible turbulence between two cultures as they newly meet.  If anything, subjugation and genocide seem to be the norm of two intersecting cultures, especially in the Americas.  The same sort of despicable treatment of human beings, from slavery to torture and genocide, accompanied the missions of Cortes in Mexico, Pizarro in Peru, and English settlers in English Settlers in Virginia and Massachusetts with the Powhatans and Pequots.

The lessons we learn (or do not) from such conquests could make or break the survival of the Earth, if extraterrestrial beings ever pay us a visit.  Is there a way for two cultures to intersect and mutually reinforce each other, or is war and death the only inevitable outcome?  Or maybe we will one day venture out to the stars, and we could end up being Avatars, i.e., Columbus 2.0.  These are big IF's, of course, but I think it is worth thinking about how we might do things better next time, whichever end of the ordeal we end up on.

Columbus apparently went down a slippery slope, thinking it his spiritual duty to convert these natives and save their souls, which let him to thinking, how better to convert people than by the club, and what makes a better club than a sword?  By turning to the sword, Columbus ultimately succumbed to blind and arrogant self-righteousness which nothing--not the absence of gold nor the decemation of an entire people--could snap him out of.  Let's learn to avoid this in the future, what do you say, fellow humans?

Saturday, August 20, 2011

Electrons have their cake and eat it too...

A zen monk named Yogi Berra once said, "If you see a fork in the road, take it."  Quantum mechanics has shown us that elementary particles, such as electrons and photons, seem to take Yogi's advice.  That is, when an experimental physicist gives a quantum particle such as an electron or a photon a choice between two paths, it can take both at once.  This is evidenced by an interference pattern that results when you recombine the paths at the other end of the experimental apparatus.  The interference pattern shows that not only does the particle take both paths, but it actually "gets in its own way" as it does so! 

But it gets weirder.  The electron does not always take both paths.  Sometimes it takes one path or the other.  But this only happens when you try to find out which path it took!  In fact, the more you know about the path of the electron, the less likely it is to show an interference pattern.  In other words, the electron seems to respond how much we *know* about it!  If we set up our experiment to get any "which path" information, it will only take one path.  But if we set up our experiment so that we do not know which path it took, it will give an interference pattern showing that it took both paths.  So we can decide, based on how we set up our experiment, whether an electron took one path or both paths. 

But it gets EVEN weirder!  Not only can the experimental physicist decide whether it took one path or both paths by changing how they set up the experiment, but they can do so *retroactively*, i.e., after the electron has made its way through the path(s)!!  This is called the delayed-choice experiment.  Here is a description from John Wheeler, who came up with the idea:

In the 1970's, I got interested in another way to reveal the strangeness of the quantum world. I called it "delayed choice." You send a quantum of light (a photon) into an apparatus that offers the photon two paths. If you measure the photon that leaves the apparatus in one way, you can tell which path it took.
If you measure the departing photon in a different way (a complementary way), you can tell if it took both paths at once. You can't make both kinds of measurements on the same photon, but you can decide, after the photon has entered the apparatus, which kind of measurement you want to make.
Is the photon already wending its way through the apparatus along the first path? Too bad. You decide to look to see if it took both paths at once, and you find that it did. Or is it progressing along both paths at once? Too bad. You decide to find out if it took just one path, and it did.

The delayed-choice experiment may be weird, but it is not mere science fiction.  It was first carried out by physicists at the University of Maryland, where I go to school.  Actually one of the physicists who did the experiment works across the hall from the office where I spent most of my time here, Dr. Alley. 

Shih & Alley carried out the delayed-choice experiment in the late 1980's using photons, and it has been replicated several times since then.  They decided, retroactively, whether the photon took one path or both paths by shifting the experimental arrangement after the photon was well on its way along the path(s), several nanoseconds after it would have had to "choose" one path or the other.

But wait, it gets EVEN WEIRDER!!  In principle, as Wheeler has pointed out, this experiment could be carried out using astronomical sources.  A photon from a distant quasar, for instance, could have the option of taking two paths towards Earth due to an effect called gravitational lensing.  The photon could go straight to Earth or it could be pulled around another path by the strong gravitational force by a galaxy along the way.  Since we can still decide whether the photon took one or both paths by how we set up the experimental apparatus on Earth to measure "which path" information or to find an interference pattern, we can decide which path(s) the photon took BILLIONS OF YEARS AFTER it had to have taken them!

Good luck getting to sleep tonight.

Friday, July 8, 2011

A nifty little use of big binder clips...

It both keeps the toothpaste squeezable and also holds it upright.

I imagine someone more artsy than I could think of ways to improve the aesthetics...maybe some designs with white-out on the black part?

And yes, I am a total nerd.

- Posted using BlogPress from my iPhone