TWSB: Hey baby, let’s go off on a tangent together
This is a super-cool dictionary.
A few short interesting entries:
“It is common in mathematics now to use the Halmos symbol to indicate the end of a proof. It is named for mathematician Paul Halmos who seems to have first applied it to mathematics. Halmos has stated that he got the symbol from popular magazines where it was used to indicate the end of an article. It is also frequently called the tombstone.”
“The symbol “÷” which is used to indicate the operation of division is called an obelus… By a misunderstanding of a credit to John Pell about other material in the book, many English writers started using the symbol and calling it “Pell’s notations”. It is one of the most territorial of all math symbols, appearing in regular use in both the US and Great Britain and yet nearly non-existant in the rest of the world.”
“The latin word for the greek term ortho, was rectus which was also used to mean both straight and erect. We see the imprint of rectus in many math words and common language with both the straight and erect meanings. The rectum is so called because it is the “straight intestine”, while a rectangle is a parallelogram with an “erect angle”. In fact, the word rectangle was sometimes used for a “right angle” into the 19th century.
The latus rectum in a parabola is the side (latus) through the focus that is straight (rectus – parallel to the directrix). Latus rectum is the Latin translation of the Greek orthea pleura for erect side which was the term Apollonius used in his books on the conic sections. As languages blended in the middle ages, rectos became “recht” and eventually became our word for “right” and for the right angle.”
“A mathematician named Klein
Thought the Mobius strip was divine
He said “If you glue
The edges of two
You can make a strange bottle like mine”.”
“Q: Why did the chicken cross the Möbius strip?
A: To get to the same side.”
“The shortest path between two points on the surface of the Earth is along a great circle arc, but this path is often not possible for ships. One reason is that a great circle arc takes constant changes of compass heading. Because it is not much longer in the middle latitudes, ships often sail a path of constant compass heading, called a loxodrome.”
Also, statistician Karl Pearson is a badass.
TWSB: I dare you to cross the line
WOAH, SCIENCE!
(Sorry, I’m hyper.)
Today I finished formatting a business textbook (barf) and actually started working on a fun textbook for once.
Astronomy! In the second chapter I read about something I’d never heard about before: forbidden lines.
What’s a forbidden line? According to Encyclopedia Britannica, it’s an emission line in the spectra of certain nebulae that is not observed for those same gases on earth. Why? Because apparently, on earth those gases cannot be rarefied sufficiently.
Forbidden lines result from electrons in the upper energy levels of gases transitioning to a lower energy level. This transitioning requires the atoms to be undisturbed (i.e., not bumping into other atoms) and takes a long time. The resulting photon emissions are very weak. In labs on earth, these transitions are even rarer (“highly improbable”) because the excited atoms have a much greater chance of hitting other atoms and disrupting the level transitions of the electrons.
In interstellar space, however, the atoms are able to be undisturbed long enough for the electrons to make these transitions. In fact, according to the Encyclopedia of Science, up to 90% of the visible brightness of some nebulae can be attributed to these forbidden spectral lines.
Cool, huh?
This Week’s Science Blog: Balsam Beests
You guys want to see something cool?
BEHOLD!
This is an Animaris Gubernare, a “species” of Strandbeest created by sculptor/inventor/badass Theo Jansen.
Hailing from the Netherlands, Mr. Jansen has been creating these wind-powered, PVC-boned creatures since 1990. He developed a leg system that, from what I can gleam from his explanations, works like a system of gear-like movements that drive the large number of legs up, forward, and down—always with some of the legs keeping in contact with the ground—propelling the creatures across the beach. Once he had an idea of how he wanted the legs to function, he developed a computer model that assessed various rod ratios in the legs to determine which ratios would be best for the “ideal walking curve”, as he put it. Each iteration of attempts resulted in a new generation of beests. He calls this “evolution.” How freaking cool is this guy?
Read about Mr. Jansen, the beests, and his methods here. I really like how this guy describes his process:
“Usually there are only one or two animals living at one time. As soon as the development of an animal is at its end, I declare it extinct and I push it onto the bone yard. The animals there can be seen as the fossils of extinct species. Exposure to sun and rain causes the tubes to fade, making these appear more bonelike with time. The sandpit is the pre-heaven for the beach animals. They are not yet ready to survive the real beach. I still have to train them. Usually I take them out once a year to the real beach to let them get a taste of their natural environment.”
He’s also got kits for sale from which you can build your own “mini beest.” I think it’d be cool to get one, just to see in person how the leg system is constructed.
And people say art and science don’t need one another.
This Week’s Science Blog: Shuttle Show
I’m sure a bunch of you have seen this already, but it’s a pretty spectacular thing so I’d like to put it up here.
Shuttle program stats and info:
- 135 flights total between the five shuttles (Columbia, Challenger, Discovery, Atlantis, Endeavour)
- Longest flight: 17 days, 15 hours, 53 minutes, 18 seconds (Columbia)
- First lunch: April 12, 1981 (Columbia)
- Last launch: May 16, 2011
- Total earth orbits: 21,158
- Says Wiki: “Each vehicle was designed with a projected lifespan of 100 launches, or 10 years’ operational life.”
Now sit back and watch some launches, if you haven’t already done so.
TWSB: And here’s some eye surgery to creep you out for the rest of your life
WARNING: not for the squeamish! Turn back now and don’t watch the video I link to if eye surgery/surgery in general/eye stabbing/stabbing in general grosses you out or offends you. You’ve been warned!
Anyway, I had no idea this was a thing: an iridectomy is a procedure where surgeons go into a person’s eye and remove part of the iris (the colored ring around the pupil). According to Wikipedia, it’s most commonly performed in the treatment of glaucoma and iris melanoma. Sector iridectomis leave patients with something known as a keyhole pupil, which looks like this:
Total iridectomies, as you would guess, leave patients with NO iris and a GIANT pupil.
This stuff is freaky to me.
In order to freak everyone else out, I found a video of doctors performing a peripheral iridectomy. Again, don’t watch if needles in eyes freaks you out.
I like how they have to waterjet the iris back into the eye.
Actually, I’m surprised at how little info there is on this procedure, at least on the internet. I would think freaky eye stuff would get a lot of attention.
Okay, that’s all.
TWSB: Waiter! There’s a Higgs-Boson in my Universe!
(Happy birthday, United States!)
So I’m bringing back my This Week’s Science Blogs. Because.
HIGGS BOSON, GUYS!!
Why this news isn’t being vomited from every front page of every news organization is beyond me. Actually it’s not—people want Kardashians, not science—but still.
HIGGS. BOSON.
Even if it does turn out that the scientists just thought they found the H.B. instead of actually finding it, this is a big deal, people.
I think things like this are most easily explained using pictures/animation, so here are two of the clearest/simplest explanation videos I could unearth. Pretty snazzy!
TWSB: Five Minutes to Midnight
I suppose this really isn’t technically a “science” news story, but I think the reason behind today’s topic is scientific enough.
The big news? The Doomsday Clock was pushed another minute forward, putting us at 11:55 PM, five minutes from certain annihilation.
OH AND IT’S HAPPENING IN 2012 OMGWTFBBQ CONSPIRACY THEORISTS UNITE BLAH BLAH BLAH!
While the clock had been pushed back a minute to 11:54 in 2010 due to worldwide cooperation to reduce the effects of climate change and to chill out about the possibility of nuclear war, The Bulletin of Atomic Scientists (those in charge of the Clock) took another look at our current situation, saw that we’re really sucking at the whole “halt human-based climate change” and “hey, we probably shouldn’t be tossing nukes around” goals and have moved the clock back to 11:55.
Bummer, yo.
Read more of The Bulletin’s concerns and recommendations here.
TWSB: This Smell Tastes Funny
From the site: “The Smell & Taste Treatment and Research Foundation is dedicated to advancing research and knowledge on the effects of smell and taste on human emotion, mood, behavior and disease states.”
Just have a look around, especially at their research studies. Pretty cool stuff.
I wish I could smell, man. That’s another reason why I want to try out the 23andMe DNA thingy…they have an odor detection test of which I’d like to see my results. I’d just like to know at what level my sense is missing. Genetic issue? Brain structure issue? Olfactory bulb issues? “Olfactory bulb to brain” link issue? Something else?
Ah, the mysteries of life!
Maybe I’m meant to work at a dump or a skunk breeding farm or something.
TWSB: And This is Why I Couldn’t Fly Out To Seattle Today
Lewiston’s under there, somewhere.
These pictures, taken yesterday around 3 PM, show the result of an inversion, or “a deviation from the normal atmospheric property with altitude” (thanks, Wiki). In the case of Lewiston here, the inversion involved temperature: Lewiston was uncharacteristically colder than the surrounding higher elevations (it was 21 degrees in Lewiston, according to my dad’s car, whereas it was about 32/33 in Moscow).
For those readers (like none of you, but whatever) who aren’t familiar with the surrounding geography of Northern Idaho, it’s like this: Moscow and the majority of the surrounding towns sit at about 2,500 feet above sea level. Lewiston, on the other hand, is situated at the confluence of the Snake and Clearwater rivers and is thus at about 745 feet above sea level. To get to Lewiston from Moscow you have to drive down the Lewiston Grade, a 10-mile winding spiral of highway that wraps in and out of the surrounding hills and drops you a total of 2,000 vertical feet.
It sounds a lot more fun than it actually is.
Anyway, on any given day Lewiston is usually about 9,000 degrees hotter than Moscow, so to experience below freezing temperatures (and snow! Break out the panicked drivers) down there is weird.
There are several causes of inversions so I don’t know if I picked the right one to explain here, but it seems most likely given the conditions and surrounding geography. An inversion can occur when warmer, less dense air moves over a cooler and denser mass of air. This usually occurs around warm fronts (or areas of oceanic upwelling, but that’s not happening in Lewiston, haha). If the lower dense area is sufficiently humid, then a layer of thick fog can often be found at the inversion cap.
Other consequences of thermal inversions I learned about when researching this on Wiki include:
- Still, murky air
- Mirages
- a green flash (no, not the love child of the Green Lantern and The Flash, sorry)
- the total reflection of sound waves off the cap of the inversion, which causes major problems in the case of explosions.
Cool, huh?
But now I can’t fly out until Monday morning, at best.
Oh, also this:
Master’s degree, bitches!
TWSB: Santa Ana is Coming to Town
I don’t know how closely any of you dudes follow the news, especially weather-related news, but as of right now the Southwest US is being battered by what are known as the Santa Ana winds.
The Santa Ana winds, which arise in Southwest California in late fall or early winter, are the strongest they’ve been in nearly a decade according to CNN. Gusting at 60 MPH in some places, they’ve already knocked out the power to a large portion of people living in Southern California.
According to the Wiki article, the Santa Ana winds arise when the surface air in the Great Basin and Mojave Desert becomes cool and begins to fall to lower elevations. Sinking into the valleys and canyons of the lower desert, the air is often pushed through the land’s channels at near hurricane forces. In addition, its sinking in elevation causes it to heats up again (a process called adiabatic heating; as the air sinks the pressure around it increases, thus causing the air to heat as it is compressed). This, when combined with the fact that the air has already been dried by orographic lift before reaching the Great Basin, makes for extremely dry winds, often with relative humidity levels below 10%, gusting at extremely high forces. These winds push across Southern California and, as evident by the recent news, cause havoc.
Edit: video!
Woohoo, crazy weather!
TWSB: Moo Cow, Don’t Bother Me
So apparently cows line themselves up with the earth’s magnetic field.
Maybe. Probably. The evidence is somewhat inconclusive.
A 2008 study by German zoologist Hynek Burda showed that cattle, like birds, lobsters, and turtles (among others), use magneto-reception, the ability to detect the magnetic field and to perceive direction and location based on it. Burda and his team used data from Google Earth in the form of images of cattle to determine whether the cattle tend to align themselves with the magnetic field or not.
A follow-up study showed that the alignment disappeared near and around power lights, which further supports Burda’s original findings as it is suggested that power lines disrupt magneto-sensing.
However, a 2011 study by a group of Czech researchers was unable to replicate Burda’s Google Earth-supported findings using different images. In addition, after looking at Burda’s original images, the Czech team’s lead researcher, Lukas Jelinek, stated that the original phenomenon “[was] not as fantastic as it was presented.”
One key difference, however, between the original and the Czech studies is that Burda analyzed the herds as a whole, while Jelinek examined individual cows. Researcher Sönke Johnsen adds support to Burda’s original results by stating that not only should some of Jelinek’s images not have been analyzed, but that the proper unit of measurement is more likely at the level of the herd rather than the individual cow.
So why there appears to be a bit of contradictory evidence, it seems like there’s a fair number of scientists who group cattle with other animals who utilize magneto-reception.
Cool, huh?
TWSB: Hydrogen: Putting the ‘H’ in “Holy Crap, the Universe is So Empty”
Crap!
So I found this page the other day and bookmarked it for a TWSB post…the page was a demonstration of how ‘empty’ we all are at the atomic level. On the current replacement page, the author states, “The page had a picture of a proton that was one thousand pixels wide, and a little electron that was only one pixel wide, and they were separated by fifty million pixels of empty space – I worked it out that that was eleven miles if your monitor displayed 72 pixels per inch, not uncommon at the time. You could try to scroll between them and it would take a long time. It was kind of neat.”
It was neat. But because of browser issues and issues surrounding the model of the atom the author used (he used the model Bohr developed), he took down the page.
In its place, though, he offers a similar study of scale and emptiness: the solar system.
To me, his atomic demonstration is more powerful since its fascinating how “solid” beings such as ourselves are composed of so much space, but the solar system demo is pretty snazzy, too.
Related: I’m assuming some of you Moscow people who stumble across this have taken the Moscow-Pullman trail…have you seen the little solar system distance exercise set up by a bunch of elementary school kids? It’s pretty cool. Pay attention at the head of the Moscow end of the trail to see the pics.
This Week’s Science Blog: Prime Suspects
Okay, so this is a result of my efforts to complete “Partying with the Primes: Part II” (see this blog for explanation. Or just scroll down a few days). Because I knew trying to get R to output some sort of number spiral would be quite an arduous task, I first decided to do a few more elementary visualizations of the primes. My first attempt led to today’s science blog.
Question: is there any sort of pattern to the spacing of prime numbers? That is, is there any sort of predictive sequence that demonstrates that the primes are “evenly spaced” (or not) amongst the other numbers?
I’d done a little bit of research on this topic prior to today, due to my 2009 NaNo (haha, we keep coming back to that, don’t we?), but it had been awhile, so I did a little bit more reading and came up with a few good sources to check out: here, here, and here.
Specifically, Zagier’s comment stood out to me: “there are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision.”
So what’s a good way to visualize this stuff? My first attempt involved coding all prime numbers as “1” and all non-prime numbers as “0” and then plotting the results with 0 and 1 on the y-axis and the actual numbers (1 through whatever the highest number I chose was…I think it was 1,000), but that was a horrible mess of jagged lines and insanity, so I scrapped that and tried to think of a better way of looking at it.
In the end, I decided the best way to examine the instances of prime numbers amongst the non-primes was to plot the numbers by the numbers themselves. That is, for a given sequence of numbers (say, 1 through 10, just to make the explanation simpler) I would repeat each number by that number itself, create a new vector containing these numbers, and then plot the result.
Defunct code for better understanding:
RepPrime=function(j) {
return (rep(j,j))
}
This function says that for any number j in a given set of numbers (again, let’s say 1:10), output that number j times. So if I had the number 7, this function would give me a vector [7 7 7 7 7 7 7]’, or 7 repeated seven times. And if I ran it for all numbers 1 through 10, I’d get the vector
[1 2 2 3 3 3 4 4 4 4 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10]’.
Of course, I couldn’t get this function to work but after screwing around a little bit more I finally figured out how to get this to work for larger sets of numbers, including sets just containing primes.
But what would plotting vectors like this reveal about any prevalence patterns for the primes? Well, let’s look at the plot for all numbers, shall we?
This plot is for all numbers from 1 to 1,000.
It’s pretty! Nice and smooth. So this can be said to be a plot for numbers that have a uniform or consistent pattern (all instances in this case occur one number apart, just because there’s one number difference between each instance; such is the nature of just listing the numbers 1 through 1,000).
Okay, that’s cool. So how about we look at a case where instances occur more “randomly?” In this case, I took a list of the numbers 1 through 1,000 and then went through and haphazardly deleted single numbers or large chunks of numbers so that I was left with a list that appeared to have numbers omitted at random.
Plot!
Much choppier, eh? This can be said, then, to be a plot pattern for numbers that have an inconsistent or random pattern of deletion.
So what would a plot of the primes—say, all the primes below 5,000—look like?
THIS!
So it’s obvious that this plot looks a lot more like the plot for numbers 1:1,000 and less like the plot involving random deletion. Interesting…I’d like to see what goes on with much larger primes, but unfortunately I can’t do that due to how huge the resulting vectors would be. R + large datasets = trouble.
Woo!
TWSB: The Sound of a Solar Re (and a Do, a Mi, a Fa, a So, a La, a Ti, and More Do)
The authors of the Ask a Mathematician/Ask a Physicist blog received and answered this question not too long ago: If you could hear through space as though it were filled with air, what would you hear?
The answer is as follows: the sun.
Yes, our big showy center of the universe is also the loudest thing around, at least to us. The Physicist explains: both the loudness and brightness of an object is exactly proportional to how big it is. The sun’s brightness, therefore, is a function of its temperature and size. If a small ball of the same temperature as the sun were to be held up so that it appeared to be the same size as the sun, it would feel exactly as warm and seem exactly as bright as the sun.
Taking this comparison of a small bright ball = distant, huge sun with respect to the amount of heat omitted, The Physicist states that the sun, if we could hear it, “would be exactly as loud as any other large-marble-sized nuclear explosion held at arm’s length.”
So we’d pretty much be deaf. Or dead.
Insanity! Article here.
30-Day Meme – Day 19: A talent of yours.
Rewriting song lyrics. I swear this is my single talent in life. Like I said in my “100 Things” list, I credit my mom playing a lot of Weird Al when I was a kid. It comes very naturally to me. Here are a few examples:
- Justin Timberlake’s Sexy Back rewritten as Easy Mac.
- Boston’s Peace of Mind rewritten as Piece of Pi.
- MC Hammer’s U Can’t Touch This rewritten as U Can’t Prove This (it’s about logic class and how long it took me to wrap my head around proofs).
TWSB: Do the Hustle! Er, Shuffle. Do The Shuffle. iPods. This post is about iPods.
So I found this NPR transcript this afternoon discussing a listener’s question about the “shuffle” feature of iPods. Specifically, what the mathematics behind the shuffle feature are. After all, the listener notes, it sometimes seems like the same songs keep coming up while many others are never played.
Keith Devlin, executive director of the Center for the Study of Language and Information at Stanford, answers the question. The shuffle, he says, is meant to be random and in fact actually behaves as a random event. Mr. Devlin explains that people have difficulty recognizing randomness because of the simple fact that one of the features of truly random selection is the repetition of patterns. He uses the example of coin tossing. If you toss a coin twenty or so times, you’re very likely, he says, to get streaks of the same side coming up, like a streak of five heads or something. We tend to see such streaks as non-randomness, though, because such patterns surprise us (after all, it does seem intuitively weird to see five heads in a row when you’re tossing a coin when you know that both sides of the coin have an equal probability of occurring. The magic of independent probability!).
Mr. Devlin concludes, “so, assuming that Apple have designed a really good randomizer in the iPod, then you are going to start getting repeats of songs and you are going to find that some songs don’t come up seemingly for a long time. That’s the way random behavior is.”
Cool, huh?
30-Day Meme – Day 15: A poem you wrote.
I wrote this poem for my grandpa the night before he died of a rare cancer. I remember reading it to him and watching him slowly fade in and out of consciousness. Everything in this poem is totally true of him. You rocked, grandpa!
A Grandpa is someone you can count on in a mess
Though with woodworking and painting he is easily obsessed
Not a job too tough
Not a place too rough
For this daring man of a ripe old age
Who has lived in the past and seen the future.
Can re-fix a shoe with a single suture.
No need to ask for directions
Full of funny imperfections
This daring man of a ripe old age.
Whether zooming to the store for his lottery ticket
Or checking the numbers—did he win it?
Doesn’t complain
Need I explain?
The daring man of a ripe old age.
The daring man of a ripe old age.
This Week’s Science Blog: She Deafened Me with Science!
Super cool! Says a related article, “The Audio Spotlight system uses nonlinearly propagating ultrasound to create highly directional beams of sound in mid-air, which can be “shone” and “directed” much like light.”
This could be used to freak so many people out.
30-Day Meme – Day 8: Describe the style you had 10 years ago.
Haha, what is this “style” you speak of? Let’s see…ten years ago was 2001, so I was in 7th grade. I wasn’t quite as colorful as I am nowadays, but I still wore a lot more color than my peers. I remember being compared to Phoebe from Friends with respect to my clothing…though I wouldn’t really know how accurate this statement is, ‘cause I’ve never seen Friends. So I guess my style is the same as it is now, only subdued.
TWSB: Who Let the Dogs Out? Boston Dynamics, Apparently
OH GOD THEY’RE BACK. And with their scariest rendition of “dog” robots yet.
Bigger.
Stronger.
Quieter.
Ready to STOMP YOUR SOUL!
“When fully developed the system will carry 400 lbs of payload on 20-mile missions in rough terrain. […] AlphaDog is designed to be over 10x quieter than BigDog.”
Every time those guys push or kick it I expected ROBOT RETALIATION! Then I saw it right itself and I felt the need to run. Fast.
Here’s an article that expresses a similar degree of “OMFGWE’REALLGOINGTODIE” as me.
30-Day Meme – Day 1: Your favorite song.
Hahaha, I think we all know my answer to this one. Sleepyhead, by Passion Pit, is what I take to be the epitome of musical awesomeness. Here are the reasons I like it:
1. It’s short. I don’t know why, but I’ve always been a fan of shorter songs over longer ones. Sleepyhead clocks in at just under three minutes. I think I tend to like shorter songs because they’re harder for me to get sick of.
2. Good tempo + good beat. Any song with a steady, unwavering time signature is always a winner for me. I don’t like songs that have a great momentum going on and then break things down by either losing the back beat, slowing down, or having the singer stop their singing to rap/talk/go to the bathroom. Sleepyhead stays the same through its duration. That makes me happy.
3. That FREAKING CHORUS. I admit that I didn’t automatically like Sleepyhead when I first downloaded it as an iTunes freebie back in February of 2009. I remember my first listen through and thinking “okay, this song has a nice beat, but there’s nothing too special about it.” Then it hit 1:21 and I fell in love with the chorus. I still love the chorus. I NEED the chorus. I would make sweet, sweet love 24/7 to the chorus.
I would also make sweet, sweet love to the following remixes, which heavily utilize warping the chorus into new and fantastically shocking eargasm-giving sex toys: Cillo, Jazzsteppa, Neo Tokyo.
Haha, okay, I’m done.
TWSB: Weebles Wobble (But They Wouldn’t if They Had Three Legs)
You know those “duh” phenomena or experiences that you come across every day? You know, the ones that you would think would have a simple explanation for why they occur but have never actually asked yourself? Those make the best TWSBs.
Take today’s topic, for example: why a three-legged stool is stable, while a four-legged stool (or any higher-legged stool, though I’m pretty sure octuple-legged stools are quite rare) often wobbles.
This article here talks about the fundamentals of why this is. It uses the analogy of a cane: suppose you’re holding a cane unrestrained in the air. You can twirl it in any direction possible, in all three dimensions. Now suppose you set one end of the cane on the ground. You’ve now constrained its motion to two dimensions (you can’t lift or rotate it). Next? Take a pair of canes, connect the tops, and place the other two ends on the ground in a little triangle of cane, cane, ground. The tops are still movable, but only along an arc. The motion is now constrained to one dimension. If you do the same thing with three canes? Can’t move the top at all. Now motion is constrained in all three dimensions, meaning the canes cannot be moved at all. As the article puts it, “each time you add a cane, you remove one dimension in which the top can move freely – that is, each new cane removes one ‘degree of freedom.”
At this point I had to stop and have a little stats freak out, ‘cause this means that stool-leggedness is almost perfectly analogous to model identification in structural equation modeling.
So let’s check that out, shall we?
A structural equation model is made up of variables and parameters (paths between variables, either dependent or independent, and paths between variables and errors). Parameters are, in other words, covariances between the variables. In this example, parameters are analogous to the dimensions in which the stool’s legs can rotate (so # of parameters = 3).
The number of covariances in any given model is called the number of “known values.” In the case of the stool, the number of legs the stool has is analogous to the number of known parameters in a model.
A just-identified (or saturated) structural equation model is one for which the number of parameters is equal to the number of known values. Such a model has zero degrees of freedom (since df = number of known values minus the number of parameters). A three-legged stool is like such a model, since df: 3 parameters – 3 known values = 0). A just-identified model has only one unique solution. The stool, analogously, has only one “solution” in the sense that there are exactly three legs used to stabilize the stool on the plane that is the ground.
Give the stool any more legs, though, and it becomes like an over-identified model, or one for which there are more known values (four in the case of a four-legged stool) than parameters (still only three dimensions in which the stool’s legs can rotate). An over-identified model, unlike a just-identified model, does not have a single unique solution, owing to the non-zero number of degrees of freedom. As the article puts it, “…now you have too many constraints. This means that there are multiple ways that the stool can “solve” the problem of which legs to use for support.”
Statistics analogies FTW!
Haha, sorry, this was a longer blog than originally planned. It got me excited.
This Week’s Science Blog: What are the Odds?
A recent study has shown that babies as young as twelve months old have a basic understanding of probability.
How did researchers come to this conclusion? They ran a bunch of twelve- and fourteen-month-olds through a series of tests. First, they gave the babies both a black and a pink lollipop and recorded the preference of the baby. Once the preference was recorded, the babies were let go and then brought back at a later date for a second test.
The second test involved showing the babies two clear jars—one of which contained many pink lollipops and only a few black ones, the other containing many black lollipops but few pink ones. With the babies still watching, a researcher reached a hand into each jar and pulled out a lollipop, but concealed with their hand the color of the candy. They dropped the lollipops into separate opaque cups so that the babies still couldn’t see the color, then asked them to choose one of the cups.
The cool thing? The babies consistently chose the cup belonging to the more “likely” jar. That is, they selected the cup that belonged to the candy drawn from the jar containing the greatest number of the babies’ preferred lollipop color. From the article, “this indicates that many kids, even when they’re very young, are able to make the mental connection that a random lollipop picked from a jar that had more pink lollipops, is more likely to be pink than one picked from a jar containing mostly black lollipops.”
How cool is that?
TWSB: Death By Jupiter
Kind of a cop-out science blog entry, but what’re you going to do, eh? I haven’t had much time to actually research science articles.
Haha, Jupiter would eat us alive. Comments from others:
“Glad I’m not the only one that found Jupiter fucking terrifying.”
“if jupiter was that close we should all shit our pants in terror every morning”
“That was damn scary, the idea that if we were that close we would be sucked into Jupiter almost made me shit bricks.”
“Jupiter actually made my heart jump a bit.”
“Jupiter shows up- HOLY FUCK.”
“In Soviet Jupiter, Earth orbits you”
TWSB: Holes and the Pigeons that Occupy Them
Today I’m going to talk about the pigeonhole principle and why it’s so freaking awesome.
The principle can be best introduced using an example. Suppose you had (for whatever weird reason) three pigeons that you wanted to put into holes. However, though you have three pigeons, you only have two holes. Can you still put the pigeons in the holes? Yes, of course you can, but there’s a catch: one of the holes will have to contain two pigeons, regardless of how you arrange the birds.
This awesome article entitled “16 Fun Applications of the Pigeonhole Principle” shows some examples of how this idea can be extended to larger numbers—that for any n number of “pigeonholes,” if there exist >n “pigeons,” then there has to be one hole with more than one pigeon in it.
The easiest demonstration of this principle (at least to me) was this one: “For every 27 word sequence in the US constitution, at least two words will start will the same letter.” In this case, the words are the “pigeons” and the letters of the English alphabet are the “pigeonholes.” There are 26 pigeonholes (the 26 letters of the alphabet, obviously) but 27 pigeons. Thus, the n, >n principle applies and therefore you have to have a pigeonhole (letter) that has more than one pigeon (word) “in” it.
Cool, huh? Or rather, FREAKING AWESOME.
The article also talks about another way to understand the principle. It involves math! Go back to the fact that the principle rests on their being n pigeonholes and >n pigeons. For any dataset that doesn’t consist of each datum having the same value, the average is (very) loosely defined as a “middle” value. That is, it’s a value larger than the minimum but also smaller than the maximum.
This is mathematically the same as the n, >n thingy we’re talking about. If we were to have the same number of pigeons and pigeonholes (n and n, respectively), the average value of (pigeons/pigeonholes) would be (n/n) or 1, meaning that on average, 1 pigeon would belong to 1 pigeonhole. However, in cases where there are more pigeons than pigeonholes, the average value of (pigeons/pigeonholes) would be (>n/n), or a value greater than one. What does that mean? It means that on average, more than 1 pigeon belongs to each hole. And because the average is (in general!) smaller than the maximum, it means that there has to be at least two pigeons for one hole.
SNAZZY.
I leave you with one of the examples that’s still a bit crazy for me, even though the way he breaks it down makes total sense.
On New Years at New York’s Time Square, over 820 people will have the same birthday.
There are roughly 300,000 attendees on New Years split over a possible 366 birthdays. The average is 300,000 / 366 = 819.7 people per birthday. The maximum must at least be the average, so there must be a birthday that at least 820 people share.
TWSB: Pokes and Tweets and Tech, Oh My!
I’m using the thin veil of “technology involves science” to disguise today’s blog as This Week’s Science Blog. This is mainly because I’ve had trouble finding any super-exciting science articles as of late. Apologies.
Anyway.
This is something I’ve always been pretty interested in: how different generations view and interact with technology. This article, written by Susan Weinschenk, reviews the similarities and differences regarding how three generations, Baby Boomers (born between 1943 and 1961), Generation X-ers (born between 1961 and 1981), and Millenials (or Generation Y, born between 1982 and 2002), interact with technology. The article is super interesting and I recommend reading it, but here are a few highlights with CLAUDIA COMMENTS!
Dualism vs. Ubiquitous: Boomers view technology as something separate, e.g., they “make a call on their cell phone,” while Millenials “make a call” and assume that said call is occurring on a cell phone as implicit information.
I don’t know if this one is inherent to generational differences or just availability differences. It seems like for Millenials (particularly the youngest ones), cell phones have existed forever and pretty much rule the world of telephone communication. To them, a landline phone is a pretty weird thing. Therefore, they make a call and assume that everyone knows the call is occurring via a cell phone because cell phones are the main available tool for making calls. For Boomers, on the other hand, they’ve had to deal with the transition from landline phones as being the main medium of voice communication to cell phones. Therefore, they still feel the need to qualify the means by which they are making a call.
Twitter and Facebook: Millenials prefer Facebook, Generation X-ers prefer Twitter, and Baby Boomers are kind of in the middle with the two.
I thought this one was pretty interesting. I would think the Millenials would be all over Twitter (‘cause, well, they are), but maybe the Generation X-ers like it because it’s not as complicated (read: mutable) as Facebook and therefore is consistently easy to use.
Is Technology Trapping Us? Baby Boomers grew up without the recent technology and are most able of the three generations discussed here to “let it go” and live without it. Generation X-ers are simultaneously most enamored with and feel most trapped by technology. Millenials incorporate technology in their lives pretty much 24/7, but are the most likely to eschew it in favor of more personal communication, such as speaking face-to-face.
It’s interesting that the “middle” generation of X-ers feel the most trapped by technology, but it also make sense. For Boomers, they can leave it ‘cause they remember life without it. Millenials seek to free themselves from it because it’s so engrained in their lives that it can become a hassle. It seems like X-ers have had to make the most effort to incorporate new technology in their lives and therefore have the worst time trying to break free of it again.
Once we’ve all aged another 20-30 years, it’ll be interesting to compare these data with those from Generation Z.
By the way, I found this article via a list entitled “47 Mind-Blowing Psychology-Proven Facts You Should Know About Yourself.” Usually I stay away from lists like these on the internet because they all seem to rehash the same things that everyone already knows over and over again, but this one is wonderfully written and actually involves interesting stuff.
TWSB: Neptune’s Birthday
So apparently tomorrow will mark the first time Neptune will have made a complete revolution around the sun since it was discovered back in 1846.
TWSB: Portals, Portals, Everywhere and Not a Cake to Eat
Matt, This Week’s Science Blog is for you, ‘cause I know you really dig Portal.
Here on the Steam forums is a list of Portal physics answers (warning: not devoid of childish internet arguing and name-calling).
I like the crushing one (the second scenario). This stuff makes me want to play around with Garry’s Mod.
Speaking of Garry’s Mod…
Just a little tidbit of the internet to lighten your day. My day’s pretty much sucked, at least, so I hope yours is going better.
Eigenblogger 