Flat Earth “Research”

You no doubt heard about this fellow in the last week with the steampunk rocket with “Flat Earth Research” written on the side. In my opinion, he was pretty clearly trolling the media; not much likelihood of resolving any issues about the shape of the Earth if the peak altitude of your rocket is only a fraction of the altitude of a commercial airline jet. He said a number of antiscience things and sort of repurposed mathematical formulae for aeronautics and fluid mechanics as “not science” as if physics is anything other than physics. The guy claimed he was using the flight as a test bed for a bigger rocket and wanted to create a media circus to announce his run for a seat in the California legislature. Not bad for a limo driver, I give him that.

Further in the background, I think it’s clear he was just after a publicity stunt; his do-it-yourself rocket cost a great deal of money, and his conversion to flat eartherism obviously helped to pay the bill. It really did make me wonder what exactly flat earthers think “research” is given that they were apparently willing to pony up a ton of money for this rocket, which won’t go high enough to resolve anything an airline ticket won’t resolve better.

My general feelings about flat earth nonsense are well recorded here and here.

A part of why I decided to write anything about this is that the guy wants to run for congress in California. This should be concerning to everyone: someone who is trusted to make decisions for a whole community had better be doing so based on a sound understanding of reality. Higher positions currently filled in the Federal government not withstanding, a disconnect seems to be forming in our self-governance which is allowing people to unhinge their decision-making processes from what is actually known about the world. I think that’s profoundly dangerous.

In my opinion also, this is not to heap blame on those who actually hold office now, but on everybody who elected to put them there. Our government is both by the people and for the people: anybody in power is at some level representative of the electorate, possessing all the same potentially fatal flaws. If you want to bitch about the government, the place to start is society itself.

Now, Flat Eartherism is one of those pastimes that is truly incredibly past its time. There are two reasons it subsists; the first is people trolling other people for kicks online, while the second is that some people are so distrusting and conspiracy-minded that they’re willing to believe just about anything if it feeds into their biases. There are some people who truly believe it. A part of why people have the ability to believe the conspiracy theories is that what they consider visual evidence of the Earth’s roundness comes through sources that they define as questionable because of their connection to ostensibly corrupt power –NASA, for all its earnest effort to keep space science accessible to the common man, has not been perfect. Further, not just anybody can go to a place where the roundness of the Earth is unambiguously visible given exactly how hard it is to get to very high altitudes over Earth in the first place. For all of SpaceX’s success, space flight still isn’t a commodity that everyone can sample. Travel into space is held under lock and key by the few and powerful.

Knowing and having worked a bit around scientists associated with space flight projects, I understand the mindset of the scientists, and it offends me very deeply to see their trustworthiness questioned when I know that many of them value honesty very highly. Part of why the conspiracy garbage circulates at all is because our society is so big that “these people” never meet “those people” and the two sides have little chance of bumping into one another. It’s easy to malign people who are faceless and its really easy to accuse someone of lying if they aren’t present to defend themselves. That doesn’t mean that either is due. This comes back to my old argument about the constitutionally defended right to spout lies in the form of “Freedom of Speech” being a very dangerous social norm.

Now, that said, another of the primary reasons I decided to write this post is because I saw a Youtube video of Eddie Bravo facing down two scientists and more or less humiliating them over their inability to defend “round eartherism.”

You may or may not know of him, but Eddie Bravo is a modern hero to the teenage boy; he’s another of these podcaster/micro-celebrity types who is widely accessible with a few keystrokes in an environment with basically zero editorial content control. He’s a visible face of the UFC (Ultimate Fighting Challenge) movement along with Joe Rogan. He’s attained wide acclaim for being a “Gracie Killer,” which is a big thing if you know anything about UFC… the Gracies being the renown Brazilian Jiu-Jutsu family who dominated the grappling world early in the UFC and brought the art of Jiu-Jutsu in its Brazilian form to the whole world. From this little history, you can easily guess why Bravo is a teenage boy hero: he’s a brash, cocky bad ass. He’s a world class Jiu-Jutsu fighter, hands down. Unfortunately, as with many celebrities, his Jiu-Jutsu street cred affords him the opportunity to open his mouth about whatever he feels like. Turns out he’s a bit of a crank magnet too, including being a flat earther.

To begin with, I don’t believe Mr. Bravo –or any other crank, for that matter– is stupid. I’ve long since seen that great intelligence can exist in people who for one reason or another don’t know better or choose not to “believe” in something for whatever reason. If he weren’t talented at some level, he wouldn’t be a hard enough worker to develop the acclaim he has attained. But, he conflates being able to shout over whoever he feels like to being able to beat them, which absolutely isn’t true in an intellectual debate.

In the Youtube clip I saw, Mr. Bravo confronts two scientists in a room full of people friendly to him. The first scientist is brought to the forefront where he introduces himself as an “Earth Scientist”… much to the rolling eyes and derision of the audience. Eddie Bravo then demands that he give the one bit of evidence which proves that the “Earth is round.” Put on the spot, this poor fellow then makes the mistake of trying to tell Mr. Bravo that science is a group of people who specialize in many different disciplines, across many different lines of research, and fails to provide Mr. Bravo with a direct answer to his question. It’s true that science is distributed, but by not answering the question, he gives the appearance of not having the answer and Eddie Bravo was completely aware that he’d said nothing to the point! When the second scientist comes forward, Eddie Bravo demands (a poorly worded demand at that, in my opinion) that since most people hold the disappearance of a ship’s mast over the horizon as the “proof” that the world is round, “why was it that people are able to take pictures of ships after they’re supposedly over the horizon?” This second scientist really did step up, I think: he tried to explain that light doesn’t necessarily travel in straight lines (which is true) and that the atmosphere can work like a fiber optic to bring images around the curve of the earth. Mr. Bravo derided this explanation, basically saying “Oh, please, that’s garbage, everybody knows you can’t see around corners.” And, at a superficial level, this will be regarded as a true response, despite the fact that the numbers always fall out the bottom of the strainer in a rhetorical confrontation. The second scientist ended up sounding like he was talking over everybody’s head with his too intricate explanation, and Eddie Bravo was able to use that to make him out as “other,” winning the popular argument at that point. Combine these incidents with a lot of shouting over the other guy, and Eddie Bravo came off well…. the video is listed as a “debate,” never mind that it was anything but.

If you are a science educator, I would recommend watching that video. Scientist #1 comes off as stupid and scientist #2 comes off as pompous.

You’ll love me for saying this, but that was all preface to the purpose of this blog post. Most modern flat earthers are Youtube trolls; they castrate their opposition by relying on the fact that evidence of the Earth’s roundness is provided by a source that is intrinsically tainted and questionable. And, the truth is that many people who believe the Earth is round really only understand this fact based on a line of evidence that people like Eddie Bravo will not accept. How do you straighten out a guy who will not accept the satellite images?

Well, how is it that we know the earth is round? We knew it before there were satellites, computer graphics and photoshop. With globalism and information society, these knowable, observable things are amplified. Flat earthers prove they are incompetent researchers every time they open their mouths and say “Well, have you researched it? I did and the earth is flat!”

Now, suppose I was a flat earth researcher, how would I go about the science of establishing the shape of the earth using a series of modern, readily available, cheap tools?

Hypothesis: The Earth is flat! It’s the stable, unmoving center of the universe and the sun and sky move over it.

1 flat earth model

One thing that we can immediately see about this model is a simple thing. When the sun is in the sky, every point on the plane can see it at the same time since there is nothing to obstruct the line of sight anywhere. In the 1800s, nobody could really travel fast enough to be able to tell whether or not this was the case: for every person in that time, it was enough to suppose that everybody on Earth wakes up from the night at the same time and goes about their day. For this flat earth modeled when seen from the side, the phenomenon of sunrise (a phenomenon as old as the beginning of the Earth, by the way) would look like this:

2 simple sunrise model

We have all seen this: the sun starts below the edge of the Eastern Horizon and pops up above it. For a majority of people on Earth, this is what the sun seems to do in the morning.

There are a number of simple tests of this model, but the simplest question to ask is this: Does everybody on Earth see the sun appear at the same time? Everybody is standing on that flat plane: when the sun comes up from below the horizon, does everybody on Earth see it at once?

3 simple sunrise model at sunrise

Notice, this is a requirement: if the Earth is flat, people all across the plane of the Earth will be able to see something big coming over the edge of that plane almost simultaneously, depending on nearby impediments, like mountains for instance.

So, here’s the experiment! If you live in California, grab your smart phone, buy an airplane ticket and fly to New York. The government has no control at all over where you fly in the continental US of A and they really won’t care if you take this trip. New York, New York is actually a kind of fun place to visit, so I recommend going and maybe catching a Broadway show while you’re there. When you get to New York, find someplace along the waterline where you can look east over the ocean and go there in the morning before sunrise. After the sun rises, wait 30 minutes and then place a phone call back to one of your buddies in California and ask him if the sun is up.

This experiment can be repeated with any two east-west related locations on Earth, though the time delays will depend on the separation so that maybe a half hour is long enough for the sun to rise in both places. Any real flat earth “researcher” should be running this experiment.

For the set-up written above, the sun comes up in New York four hours before it actually comes up in California! A California view of the sun is blocked below the horizon of the Earth for four hours after it has become visible in New York.

Now, you might argue, New York is on the east side of the US and is much closer to where the sun comes up on our hypothetical plane, so maybe the Rocky Mountains are obstructing some view of the sun in LA.

4 mountain occlusion

And that this blocking effect lasts 4 hours.

So, here’s the new experiment. Drive your car from LA to NY and watch the odometer; you can even get a mechanic you trust to assure you that the government hasn’t fiddled with it. You now know the approximate distance from LA to NY by the odometer read-out. Next, you buy a barometer and use the pressure change of the air to measure how high the Rocky Mountains are… or, you could just use a surveying scope to measure the angular height of the mountains and your car to check distances, then work a bit of trig to estimate the height of the mountains.

5 measure mountain height

The Rockies are well understood to be just a bit taller than 14,000 ft.

With these distances available, you do the following experiment with surveying scopes. When the sun appears above the horizon in LA, your friend measures the angle above ground level where it is visible (surveying scopes have bubble levels for leveling the scope). You measure the angle above the horizon at the same time using a survey scope of your own in New York. Remember, you’ve got smartphones, you can talk to each other and coordinate these measurements.

For the flat earth, the position of the sun in the sky should obey the following simple triangular model:

6 flat earth trig model

This technique is as old as the hills and is called “triangulation.” Notice, I’ve used three measurements made with cheap modern equipment: angle at LA, angle at NY and the distance from LA to NY (approximate from the odometer). What I have in hand from this is the ability to determine the approximate altitude of the sun using a bit of high school level trig. Use law of sines and it’s easy to forecast the altitude of the sun from these measurements:

7 height of sun

I won’t do the derivation this once, but you just plug in the distance and the angles, then voila, the height of the sun over the flat earth. (I’m not being snide here: Flat Earthers don’t even seem to try to use trig.)

What we know so far is that the sun comes up four hours earlier in New York than LA and that we would expect that the sun should be visible everywhere on the flat earth at the same time as it comes over the horizon. Maybe the Rockies are blocking LA from seeing the sun for four hours. This would give rise to the following situation:

9 mountain triangle

You end up with similar triangles formed by the triangle of LA to the Rocky Mountains and the triangle of LA to the sun. Knowing the height of the mountains and the distance from LA to the mountains, you get the angle that the sun must be at when it appears in LA. This gives us a relation where the angle from LA to the top of the mountains must be the same as the angle from LA to the sun when it appears. We would expect the angle to be very small since the Rockies are really not that high, so finding it nearly zero to within the noise of the instrument would be expected.

Now, LA to New York is about 2,800 miles and the distance from LA to Denver is 1,020 miles. The mountains are 14,000 feet tall. In four hours of morning, from New York, the sun will appear to be at an angle of ~60 degrees over the horizon (neglecting latitude effects… leave that for later). If you start plugging these figures into equations, the altitude of the sun must be 7.3 miles up in the sky, or 38,500 ft.

Huh.

You can fly at 40,000 ft in an airliner. Easy hypothesis to test. If the sun is only 7.3 miles up and visible at 60 degrees inclination in New York, you could go fly around it with an airplane.

Has anybody ever done that?

A good scientist would keep looking at the sun through the whole day and might notice that the angular difference of the sun’s inclination observed in the spotting scopes at New York and in LA does not change. Both inclinations increase at the same rate. There is always something like 60 degree difference in inclination in the sky from where the sun rose between these two places (again, neglecting latitude effects; this argument will appear a tiny bit janky since New York and Los Angeles are not at the same latitude, but the effect should be very close to what I described).

For this flat earth model to be true, the sun would need to radically and aphysically change altitude from one part of the day to the next in order for the reported angles to be real. We know with pretty good accuracy that the sun does not just pop out of the Atlantic ocean several dozen miles off the coast every morning when it rises over the United States, whatever the flat earthers want to tell you. And, this is pretty much observable without any NASA satellites. Grab yourself a boat and go see! The other possibility is that the sun is much further away than 7 miles and that the physical obstruction between LA and New York is much larger than just the height the Rocky Mountains over sea level –and also maybe that the angles on the levels of the spotting scopes somehow don’t agree with each other.

For this alone, the vanilla flat earth model must be discarded. You cannot validate any of the predictions in the model above: LA and New York do not see the sunrise at the same time and the sun clearly is not only 7 miles high in New York. To give them some credit, most modern flat earthers, including Eddie Bravo, do not subscribe directly to this model.

For a point, I would mention that every flat earth model struggles with the observable phenomenon of time zones and jet lag. If any flat earther ever asks you what convinced you of a round Earth, just say “Time Zones” in order to forestall him or her and to not look like you’re avoiding the question. Generally speaking, time zones exist because the curve of the Earth (something that flat earthers claim shouldn’t exist) obstructs the sun from lighting every point on the surface of the Earth at the same time.

So then, now that we’ve made basically two tests of a flat earther hypothesis and seen that it fails rather dramatically in the face of simple modern do-it-yourself measurements, what model do these people actually believe in?

flat_earther_believers_explain_their_theory_on_australien_television__234804

Most modern flat earthers believe in some version of the model above (one of the major purveyors of this is Eric Dubay. I won’t link his site because I won’t give him traffic.) In this model, you can think about the Earth as a big disc centered on an axle that passes through the north pole. The sun, the moon and the night sky spin around this axle over the Earth (or maybe the Earth spins like a record beneath the sky). The southern tips of South America, Africa and Australia are placed at extreme distances from one another and Antarctica is expanded into an ice wall that surrounds the whole disc. The model here is actually not a new one and originated some time in the 1800s.

For the image depicted here, I would point out once again that if the sun is an emissive sphere, projecting light in all directions, the model above gives a clear line of sight for every location on Earth to see the sun at all times. For this reason, the flat earthers usually insist that the sun is more like a flashlight or a street lamp which projects light in a preferred direction so that light from it can’t be seen at locations other than where the light is being projected (never mind that this prospect immediately begins to suffer for trying to generate the appropriate phases of the moon).

To generate this model, the flat earthers have actually cherry-picked a few rather interesting observations about the sky. You can find a Youtube video where Eddie Bravo tries to articulate these observations to Joe Rogan. Central among them is that the North Star, Polaris, seems to not move in the night sky and that all the stars and even the sun seem to pivot around this point. In particular, during the season of white nights above the arctic circle, the sun seems to travel around the horizon without really setting (never mind that during the winter months, the sun disappears below the horizon for weeks on end… again with that pesky horizon thing; on the flat earth, the sun is not allowed to drop below the horizon and still be visible elsewhere on the same longitude since that intrinsically implies that the Earth’s surface must curve to accomplish said feat).

sun-path-arctic-circle-large

Taken from Scijinks.gov, this image demonstrates the real observation of what the sun does during the season of white nights as viewed at the arctic circle. The flat earth model amplifies this into the depiction given above.

If this is our hypothetical model, we could say that the sun is suspended over the flat Earth so that it sits on a ring at the radius of the equator in its revolution around the pole.

10 disc model

This image shows you right away the first thing to test. As seen at a distance of 3/4 of the disc’s diameter away, the sun cannot ever be seen in the sky at a lower angle of inclination than is allowed by its altitude over the surface. In other words, it can never go down below the horizon or come up over it.

11 min angle of inclination

Here, theta is the minimum angle of inclination that the sun will visit in the sky. I’ve heard flat earthers quote ~3,000 miles for the height of the sun and the absolute length of the longitude would be (3/4)*24,000 miles = 18,000 miles, which gives a minimum inclination angle of about 9 degrees over the horizon. And, that’s seen from the maximum possible distance across the width of the disc, where the flat earthers claim the sunlight can’t be seen. As a result, the sun will always have to *appear* in the sky at some inclination greater than 9 degrees –just suddenly start making light– at the time when the sun supposedly rises.

The truth of that is directly observable: do you ever see the sun just appear in the sky when day breaks? I certainly haven’t.

This failure to ever reach the horizon mixed with the requirement for time zones is enough to kill the flat earth model above: it can’t produce the observations available from the world around us that can be obtained with just the tiniest bit of leg work! The model can’t handle sunrises (period). There’s a reason that the round earth was postulated in 2,500 BC; it’s based on a series of clever but damn easy measurements. And I reiterate, those measurements are easier to make with modern technology.

It is inevitable that this logic won’t satisfy someone. The altitude number for the sun, 3,000 miles, was cribbed from flat earth chatter. Suppose that this number is actually different and that they don’t actually know what it is (surprise, surprise, I don’t think I’ve ever seen evidence of any one of them doing something other than making YouTube videos or staring through big cameras trying to see ships disappear over the horizon and not understanding why they don’t. Time to get to work, guys, you need to measure the altitude of the sun over the flat earth or you’ll all just keep looking like a bunch of dumbasses staring at tea leaves!)

Now, then, in some attempt to justify this model, a measurement needs to be made of the altitude of the sun (again). You can do it basically in the same way you did it before; you mark out a base length along the surface of the Earth and station two guys with surveying scopes at either end: you count “1,2,3” over the smartphone and then both of you report the angle you measure for the inclination of the sun. In this case, I recommend that one guy be stationed south of the equator and the other guy stationed north, both off the equator by the same distance along a longitude line. The measurement should be made on either the Vernal or Autumnal equinox and it should be made at noon during the day when the sun is at its highest point in the sky. This should make calculations easier by producing an isosceles triangle. How do you know you’re on the same longitude line? The sun should rise at the same time for both of you on the equinox. And, I specify equinox because I would rather not get into effects caused by the Earth’s axial tilt, like the significance of the tropics of Cancer and Capricorn (you want to know about those, go learn about them yourself).

12 height of sun ver 2

From this measurement how do you get the height of the sun? You use the following piece of very easy trig:

13 trig height

And, note, this trig will not work unless both angles measured above are the same… but you can orchestrate this with a couple spotters, an accurate clock and a couple surveying scopes.

If you do this very close to the equator, where d is small, you will find that the sun is at some crazily high altitude. You may not be able to distinguish it because of the sizeable angular width of the sun, but it will be very high… in the millions of miles. This by itself will push the minimum allowed angular height of the sun up, not down, because it’s larger than what was taken for the calculation above. To handle the horizon problem where the sun can only appear to be higher than about 9 degrees in the sky and never cross the horizon, the height of the sun must be lower than 3,000 miles, not higher. Humans were unable to do this calculation in prehistory and used a different set of triangles to try to estimate the height of the sun.

If you are a good scientist, you will repeat this measurement a number of times with different base distances between the spotters. If the Earth is flat, every base length you choose between the spotters should produce the same height for the sun (this is an example of the scientific concept of Replication).

Here’s what you will actually find:

14 three measurements

At a latitude close to the equator, during the first measurement, the sun will appear to be very far away at a really high altitude. With the second measurement, at mid latitudes on either side of the equator, the sun will appear to be at a significantly lower altitude. During the final measurement, at distant latitudes, as far north and south as you can get, the sun will appear to actually sit down on the face of the Earth. If you coordinate this experiment with six people on group chat all at once, this is what they will all see simultaneously. Could I coordinate the measurement locations so that the sun appears to be 3,000 miles high? Sure, but who in the hell would ever take that as honest? Flat earthers blame scientists for being dishonest… what if the flat earthers are the ones being dishonest? Does it not count for them somehow?

Since the sun suddenly appears to be speeding toward the Earth, does this mean that it’s about to crash down onto the experimenters you have stationed at the equator? No. It just means that your model is completely wrong because it hasn’t produced a self-consistent measurement. A mature scientist would consider the flat earth a dead hypothesis at this point.

Why does the round earth manage to succeed at explaining this series of observations? For one thing, the round earth doesn’t assume that the spotting scopes are stationed at the same angular level.

15 round earth contrast

The leveling bubble on the spotting scope can only assume the local level. And, the angle that you end up measuring is the one between the local horizon and the sight line. On the equinox (very important) the sun will only appear to be directly overhead at noon on the equator.

If you’re still unconvinced that the flat earth is a dead hypothesis which doesn’t live up to testing and continue to focus on strange mirages seen over the surface of the ocean on warm days as evidence that the round earth can’t be right, consider the following observations.

Flat earthers use Polaris as the pivot around which the sky spins. Why is it that Polaris is not visible in the sky from latitudes south of the equator? Why is it that the Southern Cross star constellation is not visible from the northern hemisphere? Eddie Bravo, as a Gracie hunter, surely must have visited Brazil: did he ever go outside and look for the north star during a visit? Pending that, did he look for the Southern Cross from Las Vegas?

Flat earthers use the observation that the stars in the sky rotate counterclockwise around Polaris as evidence that the sky is rotating around the disc of the Earth. Have they ever gone and observed at night from the tip of Argentina in South America that the sky seems to rotate clockwise around some axis to the south? How can the sky rotate both clockwise and counterclockwise at the same time? In the flat earth model, it can’t, but in reality, it does! As an extension, why in the hell does the sun come straight up from the east and set straight in the west on equinox at the equator? When seen at the North Pole, on equinox day, simultaneously, the sun rolls around the horizon at the level of the ground and never quite rises. Use your smartphone and take the trip to see! Send a friend to Panama while you go to Juneau Alaska and talk on the smartphone to see that it happens this way in both places at once.

Don’t take my word for it, go and make the observations yourself!

How is this all possible?

I’ll tell you why.

It’s because flat earthers never test the models they put forward with the tools that are at their flipping fingertips. “Flat Earth ‘Research'” my ass.

Do I need NASA satellite pictures or rocket launches to know that the Earth is round? Pardon my french, but Fucking hell, no! Give me the combination of time zones with the fact that the sun actually pops up over the horizon when it rises and your ass is grass. Flat earth models can’t explain these observations simultaneously, they can only do one or the other.

Edit 11-28-17

Yeah, I have a tiny bit more to say.

If all of what I’ve said still does not convince you, likely you’re hopeless. But, here’s a comparison between what the sun does in the sky over the disc shaped flat earth and what it actually does.

Here’s how the sun travels across the sky on the disc-shaped earth:

16 flat earth sun track

Here’s what the sun really does depending on latitude:

17 earth sun track

This particular set of sun behaviors in the sky is actually visible year round, but the latitude where the sun travels from East, straight over the apex, to West varies North to South depending on the season when you look. At equinox, the observation is symmetric at the equator, but it shifts north and south of there as the months move on, producing the same general pattern above. In the winter, the axial tilt of the Earth prevents the sun from rising over the north pole –ever– while the same is true at the south pole during the summer of the northern hemisphere. Flat earthers seem to never make any observations about what happens in the sky to the sun south of the equator. Do they not go to Australia or South America to take a look?

As an extra, I have made the mistake of rooting through Eric Dubay’s “200 proofs” gallop. I once even thought about writing a blog post about the experience, but decided it was too exhausting. For one thing, quantity does not assure quality. Many of the 200 proofs are taken from accounts of 19th century navigation errors, and one must wonder whether such accounts hold as valid in the 21st century world. Further, some of the proofs are simple, flat out lies: among the proofs is an exhaustive observation of the lack of airline flight routes in the southern hemisphere, twisting route information to show that flights must pass through the northern hemisphere to reach destinations as far separated as the tip of South America and the tip of South Africa, which simply ignores the fact that flight routes exist for these destinations that do not go to the northern hemisphere. Are there more flight routes in the Northern hemisphere than in the southern hemisphere? Yes, most of the human population lives at or north of the equator… most of the places anybody would want to go are in the northern hemisphere. If you doubt that such a flight route exists, go to the Southern hemisphere and take an airline flight from Argentina to South Africa and use a stopwatch during the flight to see if it’s a fraction of the length Dubay would claim –commerical airline jets have a known flight profile that would be impossible to hide; the rate at which they cross distance is well-characterized. Did Dubay do this experiment? Nope. What should stun a person about Dubay is that he does not merely make wrong claims, it’s that he repeats the same wrong claims 60 times in a row to an audience that not only fawns over it, but fails to point out the giant logical gaps that are detailed above. How hard is it to see that you not only need to cope with time zones, but with sunrises too?

Pointing out a tiny detail, like not understanding how mirages work on the surface of the ocean, does not somehow validate a model that can’t handle the big ticket items, like time zones and sunrises. It only shows that you can’t understand how the small details work. I can also sort of understand that people are losing touch with the world around them as they grow more and more entrenched in the online world, but if you fail to understand that the online world does not dictate the physics of the real world, you are in big trouble.

(Edit 3-26-18:)

The steam rocket dude finally shot himself 1,800 ft into the air. Oh yeah, and “flat earth and stuff.” Tell me again how his little stunt was supposed to test anything. His interest was in launching himself in a steam powered rocket, it had nothing to do with finding out the roundness (or lack thereof) of the Earth.

If you vote for him for Governor, you deserve what you get.

For anybody actually interested in a test that did something, check this out. For the record, there are aberrations to the lenses here which do effect exactly what you see along the edges of the image, but ask yourself how the rocket can appear straight while the background appears curved. Further, if you doubt it, that test is something that can be done by someone with the limo driver’s means.

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Interaction Picture

It’s not always about the cat. Here, I will show how to hop from the time dependent Schrodinger equation to the Interaction picture form.

This post is intended to help recover a tiny fraction of the since-destroyed post I originally entitled “NMR and Spin Flipping part II.” I have every intention to reconstruct that post when I have time, but I decided to do it in fragments because the original loss was 5,000 words. I don’t have time to bust my head against that whole mess for the moment, but I can do it in bits, I think.

One section of that post which stands pretty well as a separate entity from the NMR theme was the fraction of work where I spent time deriving a version of the time dependent Schrodinger equation in the interaction picture.

I thought I would go ahead and expand this a little bit and talk generally about some of the basic structural features of non-relativistic quantum mechanics. Likely, this will mostly not be very mathematical, except for the derivation at the end. I’ll warn you when the real derivation is about to start if you are math averse…

Everybody has heard about Schrodinger’s cat. Poor cat is dragged out and flogged semi-dead, semi-alive pretty much any time anybody wants to speak as if they know something about “quantum physics.” The cat might be the one great mascot of quantum in popular culture. The kitty drags with it a name that you no doubt have heard: Erwin Schrodinger, the guy who first coined the anecdote of feline torture as an abstraction to describe some features of quantum mechanics on a level that laymen can embrace, if not totally understand. This name is immediately synonymous with the spine of quantum mechanics as the Schrodinger equation. This equation is not so simple as E = mc^2 or F = ma, but it is a popular equation…

Schrodinger equation

I’ve included it here in its full-on psi-baiting time-dependent form with Planck’s constant uncompressed from ħ.

You hardly ever see it written this way anymore.

All this equation says is that the sum of kinetic and potential energy is total energy, which is tied implicitly to the evolution of the system with time. This equation is popular enough that I found it scrawled on a wall along with some Special Relativity inside the game “Portal 2” once. Admittedly, the game designers used ħ instead of h for Planck’s constant. It may not look that way, but the statement of this equation is no more complicated than F = ma or E = mc^2. It just says “conservation of energy” and that’s pretty much it.

Schrodinger’s equation is the source of wave mechanics, where Psi “ψ” is the notorious quantum mechanical wave function. If you care nothing more about Quantum mechanics, I could say that you’ve seen it all and we could stop here.

The structure of basic quantum mechanics has a great deal to it. Schrodinger’s equation tells you how dynamics happens in quantum physics. It says that the way the wave equation changes in time is tied to some characteristics related to the momentum of the object in question and to where it’s located. Structurally, this is the foundation of all non-relativistic quantum mechanics (I say “non-relativistic” because the more complete form of the Schrodinger equation competent to special relativistic energy is the Klein-Gordon Equation, which I will not touch anywhere in this post.) Pretty much all of quantum mechanics is about manipulating this basic relation in some manner or another in order to get what you want to know out of it. Here, the connection between position and momentum as well as between energy and time hides the famous “uncertainty relations,” all built directly into the Schrodinger equation and implicit to its solutions.

One thing you may not immediately know about Schrodinger’s equation is that it’s actually a member of a family of similar equations. In this case, the equation written above tells about the motion of an object in some volume of space, where the space in question in literally only one dimensional, along an effective line. Another Schrodinger equation (as the one written in this post) expands space into three dimensions. Still other Schrodinger equation-like forms are needed to understand how an object tumbles or rotates, or even how it might turn itself inside out or how it might play hopscotch on a crystalline lattice or bend and twist in a magnetic field. There are many different ways that the functional form above might be repurposed to express some permutation of the same set of general ideas.

This tremendous diversity is accomplished by a mathematical structure called “operator formalism.” Operators are small parcels of mathematical operation that transform the entity of the wave function in particular ways. An operator is sort of like a box of gears that hides what’s going on. You might fold down the gull-wing door in the equation above and hide the gears in an operator called the “Hamiltonian.”

Schrodinger equation 2

This just shuffles everything you don’t care about at a given time under the rug and lets you work overarching operations on the outside. Operators can encode most everything you might want. There are a ton of rules that go into the manipulation of operators, which I won’t spend time on here because it distracts from where I’m headed. A hundred types of Schrodinger equation can be written by swapping out the inside of the Hamiltonian.

An additional simplification of operators comes from what’s called “representation formalism.” The first Schrodinger equation I wrote above is within a representation of position. Knowing about the structure of the representation places many requirements which help to define the form of the Hamiltonian. I could as easily have written the same Schrodinger equation in a representation of momentum, where the position variable becomes some strange differential equation… momentum is in that equation above, but you would never know it to look at because it’s in a form related to velocity, which is connected back to position, so that position and time are the only variables relevant to the representation. By backing out of a representation, into a representation free, “abstract form,” operators lose their bells and whistles while wave functions are converted to a structure called a “ket.”

Ket is short for “Bra-Ket,” which is a representation free notation developed by Paul Dirac, another quantum luminary working in Schrodinger’s time. A “bra” is related to a “ket” by an operation called a “conjugate transform,” but you need only know that it’s a way to talk about the wave equation when you are not saying how the wave equation is represented. If you’ve dealt with kets, you’ve probably been in a quantum mechanics class… “wave function” has a place in popular culture, “ket” does not.

Most quantum mechanics is performed with operators and kets. The operators act on kets to transform them.

One place where this general structure becomes slightly upset is when you start talking about time. And, of course time is needed if you’re going to talk about how things in the real world interact or behave. The variable of time is very special in quantum mechanics because of how it enters into Schrodinger’s equation… this may not be apparent from what I’ve written above, but time is treated as its own thing. Schrodinger’s equation can be rewritten to form what’s called a time displacement operation.

You might take a breath, derivation begins here….

hamiltonian time dependence

This is just a way to completely twist around Schrodinger’s time dependent equation into a ket form where the ket now has its time dependence expressed by a time displacement modulated by the Hamiltonian. I’ve even broken up the Hamiltonian into static and time dependent parts (as this will be important to the Interaction Picture, down below). The time displacement operation just acts on the ket to push it forward in time. The thing inside the exponential is a form of quantum phase.

This ket is an example of a “state ket.” It is the abstract representation of a generalized wave function that solves Schrodinger’s time dependent equation. A second form of ket, called an “eigen ket,” emerges from a series of special solutions to the Schrodinger equation that have no time dependence. An eigen ket (I often write “eigenket”) remains the same at all times and is considered a “stationary solution” to the Schrodinger equation. “Eigen solutions” tend to be very special solutions in many other forms of physics: the notes on your flute or piano are eigen solutions, or stationary wave solutions, for the oscillatory physics in that particular instrument. In quantum mechanics, eigen modes are exceptionally useful because any general time dependent solution to the Schrodinger equation can be fabricated out of a linear sum of eigenkets. This math is connected intimately to Fourier series. The collection of all possible eigenket solutions to a particular Schrodinger equation forms a complete description of a given representation of that Schrodinger equation, which is called a Hilbert space. You can write any general solution for one particular Schrodinger equation using the Hilbert space of that equation. A particular eigenket solves the Hamiltonian of a Schrodinger equation with a constant, called an eigenvalue, which is the same as saying that an eigenket doesn’t change with time (producing Schrodinger’s time-independent wave equation).

eigen function equation 2

This is just the eigenvalue equation for the stationary part of the Hamiltonian written above, which could be expanded into Schrodinger’s time independent equation.

Deep breath now, this dives into Interaction Picture quickly.

How quantum mechanics treats time can be reduced in its extrema to two paradigms which are called “Pictures.” The first picture is called the “Schrodinger Picture,” while the second is called the “Heisenberg Picture” for Werner Heisenberg. Heisenberg and Schrodinger developed the basics of non-relativistic quantum mechanics in parallel from two separate directions; Schrodinger gave us wave mechanics while Heisenberg gave us operator formalism. They are essentially the same thing and work extremely well when used together. Schrodinger and Heisenberg pictures are connected to each other from the time displacement operator. In Schrodinger picture, the time displacement operation acts on the state ket, causing the state to evolve forward in time. In Heisenberg picture, the time displacement is shifted onto the operators and the eigenkets, while the state ket remains constant in time. Schrodinger picture is like sitting on a curbside and watching a car drive past, while Heisenberg picture is like sitting inside the car and watching the world drive past. Both pictures agree that the car is traveling the same speed, but they are looking at the situation from different vantage points. The Schrodinger time dependent equation is balanced by the Heisenberg equation of motion.

Where time dependence starts to become really interesting is if the Hamiltonian is not completely constant. As I wrote above, you might have a part of the Hamiltonian which contains some dependence on time. One way in which quantum mechanics addresses this is by a construction called the “Interaction Picture”… Sakurai also calls it the “Dirac Picture.” The interaction picture is sort of like driving along in your car and wondering at the car you’re passing; the world outside appears to be moving, as is the car you’re looking at, if only at different speeds and maybe in different directions.

I’ve likened this notion to switching frames of reference, but I caution you from pushing that analogy too far. The transformation between one picture and the next is by quantum mechanical phase, not by some sort transformation of frame of reference. Switching pictures is simply changing where time dependent phase is accumulated. As the Schrodinger picture places all this phase in the ket, Heisenberg picture places it all on the operator. Interaction picture splits the difference: the stationary phase is stuck to the operator while the time dependent phase is accumulated by the ket. In all three pictures, the same observables result (rather, the same expectation values) but the phases are broken up. Here is how the phases can be split inside a state ket.

State function in schrodinger picture 3

I’ve written the state ket as a sum of eigenkets |n>. The time dependence from a time varying potential “V” is hidden in the eigenket coefficient while the stationary phase remains behind. The “n” index of the sum allows you to step through the entire Hilbert space of eigenkets without writing any but the one. Often, the coefficient Cn(t) is what we’re ultimately interested in, so it helps to remember that it has the following form when represented in bra-ket notation:

expanding the ket 5

I’ve skipped ahead a little by writing that ket in the Interaction picture (these images were created for the NMR post that died, so they’re not quite in sequence now), but the effect is consistent. The usage of “1” here just a way to move into a Hilbert space representation of eigenkets… with probability normalized eigenkets, “spanning the space” means that you can construct a linear projection operator that is the same as identity. The 1 = sum is all that says. This is just a way to write the coefficient above in a bra-ket form.

The actual transformation to the Interaction picture is accomplished by canceling out the stationary phase…

Transformation to interaction picture 4

By multiplying through with the conjugate of the stationary phase, only the time varying phase in the coefficient remains. This extra phase will then show up on operators translated into the interaction picture…

Potential in interaction picture 7

This takes the potential as it appears in the Schrodinger picture and converts it to a form consistent with the Interaction picture.

You can then start passing these relationships through the time dependent Schrodinger equation. One must only keep in mind that every derivative of time must be accounted for and that there are several…

Time dependent Schrodinger in interaction pic 5

(edit 5-22-18: The image right here contains a bit of wrong math, see the end of the post for a more comprehensive and correct version. I made a mistake and I won’t try to hide it: see if you can find it!;-)

This little bit of algebra creates a new form for the time dependent Schrodinger equation where the time dependence is only due to the time varying potential “V”. You can then basically just drop into a representation and use all the equalities I’ve justified above…

Time dep in Interaction pic diff eq 8

The last result here has eliminated all the ket notation and created a version of the time dependent Schrodinger equation where the differential equation is for the coefficients describing how much of each eigenket shows up in the state ket. The dot over the coefficient is a shorthand to mean “time derivative.”

This form of the time dependent Schrodinger equation gives an interesting story. The interaction represented by the time dependent potential “V” scrambles eigenket m into eigenket n. As you might have guessed, this is one in the huge family of different equations related to the Schrodinger equation and this particular version has an apt use in describing interactions. Background quantum mechanical phase accumulated only by the forward passage of time is ignored in order to look at phase accumulated by an interaction.

I will ultimately use this to talk a bit more about the two state problem and NMR, as from the post that died. Much of this particular derivation appears in the Sakurai Quantum Mechanics text.

edit 5-22-18:

There is a quirk in this derivation for the interaction picture that continues to bother me. I didn’t really see it at first, but it bothers me having thought some time about it. The full Hamiltonian is defined to be some basic part plus some separable time-dependent potential. In the derivative that produces the evolution from the time-dependent potential, there is a basic assumption that this time-dependent potential does not contain time explicitly, meaning that no time derivative is taken on the potential. This seems like a self-contradiction to me: the potential is defined as time dependent, but must be the same form as the basic part of the Hamiltonian and not contain explicit time dependence in order for the derivation to work as shown above. I’m still thinking about it.

Here is a better version of the derivative that gives the time dependent Schrodinger form involving only the potential within the interaction picture:

time dependent schrodinger

Powerball Probabilities

If you’ve read anything else in this blog, you’ll know I write frequently about my playing around with Quantum Mechanics. As a digression away from a natural system that is all about probabilities, an interesting little toy problem I decided to tackle is figuring out how the “win” probabilities are determined in the lottery game Powerball.

Powerball is actually quite intriguing to me. They have a website here which details by level all the winners across the whole country who have won a Powerball prize in any given drawing. You may have looked at this chart at some point while trying to figure out if your ticket won something useful. A part of what intrigues me about this chart is that it tells you in a given drawing exactly how much money was spent on Powerball and how many people bought tickets. How does it tell you this? Because probability is an incredibly reliable gauge of behavior with big samples sizes. And, Powerball quite willingly lays all the numbers out for you to do their book keeping for them by telling you exactly how many people won… particularly at the high-probability-to-win levels which push into the regime of Gaussian statistics. For big samples, like millions of people buying powerball tickets, where N=big, the errors on average values become relatively insignificant since they go as sqrt(N). And, the probabilities reveal what those average values are.

The game is doubly intriguing to me because of the psychological component that drives it. As the pot becomes big, people’s willingness to play becomes big even though the probabilities never change. It suddenly leaps into the national consciousness every time the size of the pot becomes big and people play more aggressively as if they had a greater chance of winning said money. It is true that somebody ultimately walks away with the big pot, but what’s the likelihood that somebody is you?

But, as a starter, what are the probabilities that you win anything when you buy a ticket? To understand this, it helps to know how the game is set up.

As everybody knows, powerball is one of these games where they draw a bunch of little balls printed with numbers out of a machine with a spinning basket and you, as the player, simply match the numbers on your ticket to the numbers on the balls. If your ticket matches all the numbers, you win big! And, as an incentive to make people feel like they’re getting something out of playing, the powerball company awards various combinations of matching numbers and adds in multipliers which increase the size of the award if you do get any sort of match. You might only match a number or two, but they reward you a couple bucks for your effort. If you really want, you can pick the numbers yourself, but most people simply grab random numbers spat out of a computer… not like I’m telling you anything you don’t already know at this point.

One of the interesting qualities of the game is that the probabilities of prizes are very easy to adjust. The whole apparatus stays the same; they just add or subtract balls from the basket. In powerball, as currently run, there are two baskets: the first basket contains 69 balls while the second contains 26. Five balls are drawn from the first basket while only one, the Powerball, is drawn from the second. There is actually an entire record available of how the game has been run in the past, how many balls were in either the first or second baskets and when balls were added or subtracted from each. As the game has crossed state lines and the number of players has grown, the number of balls has also steadily swelled. I think the choice in numbering has been pretty careful to make the smallest prize attainably easy to get while pushing the chances for the grand prize to grow enticingly larger and larger. Prizes are mainly regulated by the presence of the Powerball: if your ticket manages to match the Powerball and nothing else, you win a small prize, no matter what. Prizes get bigger as a larger number of the other five balls are matched on your ticket.

The probabilities at a low level work almost exactly as you would expect: if there are 26 balls in the powerball basket, at any given drawing, you have 1 chance in 26 of matching the powerball. This means that you have 1 chance in 26 of winning some prize as determined by the presence of the powerball. There are also prizes for runs of larger than three matching balls drawn from the main basket, which tends to push the probabilities of winning anything to a slightly higher frequency than 1 in 26.

For the number savvy this begins to reveal the economics of powerball: an assured win by these means requires you to spend, on average, $48. That’s 26 tickets where you are likely to have one that matches the powerball. Note, the prize for matching that number is $4. $44 dollars spent to net only $4 is a big overall loss. But, this 26 ticket buy-in is actually hiding the fact that you have a small chance of matching some sequence of other numbers and obtaining a bigger prize… and it would certainly not be an economic loss if you matched the powerball and then the 5 other balls, yielding you a profit in the hundreds of millions of dollars (and this is usually what people tell themselves as they spend $2 for each number).

The probability to win the matched powerball prize only, that is to match just the powerball number, is actually somewhat worse than 1 in 26. The probability is attenuated by the requirement that you hit no matches on any other of the five possible numbers drawn.

Finding the actual probability is as follows: (1/26)*(64/69)*(63/68)*(62/67)*(61/66)*(60/65). If you multiply that out and invert it, you get 1 hit in 38.32 tries. The first number is, of course, the chances of hitting the powerball, while the other five are the chance of hitting numbers that aren’t picked… most of these probabilities are naturally quite close to 1, so you are likely to hit them, but they are probabilities that count toward hitting the powerball only.

This number may not be that interesting to you, but lots of people play the game and that means that the likelihood of hitting just the powerball is close to Gaussian. This is useful to a physicist because it reveals something about the structure of the Powerball playing audience on any given week: that site I gave tells you how many people won with only the powerball, meaning that by multiplying that number by 38.32, you know how many tickets were purchased prior to the drawing in question. For example, as of the August 12 2017 drawing, 1,176,672 numbers won the powerball-only prize, meaning that very nearly 38.32*1,176,672 numbers were purchased: ~45,090,071 numbers +/- 6,715, including error (notice that the error here is well below 1%).

How many people are playing? If people mostly purchase maybe two or three numbers, around 15-20 million people played. Of course, I’m not accounting for the slavering masses who went whole hog and dropped $20 on numbers; if everybody did this, 4.5 million people played… truly, I can’t really know people’s purchasing habits for certain, but I can with certainty say that only a couple tens of millions of people played.

The number there reveals quite clearly the economics of the game for the period between the 8/12 drawing and the one a couple days prior: $90 million was spent on tickets! This is really quite easy arithmetic since it’s all in factors of 2 over the number of ticket numbers sold. If you look at the total prize pay-out, also on that page I provided, $19.4 million was won. This means that the Powerball company kept ~$70 million made over about three days, of which some got dumped into the grand prize and some went to whatever overhead they keep (I hear at least some of that extra is supposed to go into public works and maybe some also ends up in the Godfather’s pocket). Lucrative business.

If you look at the prize payouts for the game, most of the lower level prizes pay off between $4 and $7. You can’t get a prize that exceeds $100 until you match at least 4 balls. Note, here, that the probability of matching 4 balls (including the powerball) is about 1 in 14,494. This means, that to assure yourself a prize of $100, you have to spend ~$29,000. You might argue that in 14,494 tickets, you’ll win a couple smaller prizes ($4 prizes are 1 in 38, 1 in 91, and $7 prizes are 1 in 700 and 1 in 580) and maybe break even. Here’s the calculation for how much you’ll likely make for that buy-in: $4*(14,494*(1/38 + 1/91)) + $7*(14,494*(1/700 + 1/580))… I’ve rounded the probabilities a bit… =$2482.65. For $29,000 spent to assure a single $100 win, you are assured to win at most $2500 from lesser winnings for a total loss of $27,500. Notice, $4 on a $44 loss is about 10%, while $2500 on $27,500 is also about 10%… the payoff does not improve at attainable levels! Granted, there’s a chance at a couple hundred million, but the probability of the bigger prize is still pretty well against you.

Suppose you are a big spender and you managed to rake up $29,000 in cash to dump into tickets, how likely is it that you will win just the $1 million prize? That’s five matched balls excluding the powerball. The probability is 1 in 11,688,053. By pushing the numbers, your odds of this prize have become 14,500/11,688,053, or about 1 chance in 800. Your odds are substantially improved here, but 1 in 800 is still not a wonderful bet despite the fact that you assured yourself a fourth tier prize of $100! The grand prize is still a much harder bet with odds running at about 1 in 20,000, despite the amount you just dropped on it. Do you just happen to have $30,000 burning a hole in your pocket? Lucky you! Lots of people live on that salary for a year.

Most of this is simple arithmetic and I’ve been bandying about probabilities gleaned from the Powerball website. If you’re as curious about it as me, you might be wondering exactly how all those probabilities were calculated. I gave an example above of the mechanical calculation of the lowest level probability, but I also went and figured out a pair of formulae that calculate any of the powerball prize probabilities. It reminded me a bit of stat mech…

prob without powerball

prob with powerball

number for hits

I’ve colored the main equations and annotated the the parts to make them a little clearer. The final relation just shows how you can see the number of tries needed in order to hit one success, given a probability as calculated with the other two equations. The first equation differs from the second in that it refers to probabilities where you have matched numbers without managing to match the powerball, while the second is the complement, where you match numbers having hit the powerball. Between these two equations, you can calculate all the probabilities for the powerball prizes. Since probabilities were always hard for me, I’ll try to explain the parts of these equations. If you’re not familiar with the factorial operation, this is what is denoted by the exclamation point “!” and it denotes a product string counting up from one to the number of the factorial… for example 5! means 1x2x3x4x5. The special case 0! should be read as 1. The first part, in blue, is the probability relating to either hitting on missing the powerball, where K = 26, the number of balls in the powerball basket. The second part (purple) is the multiplicity and tells you how many ways that you can draw a certain number of matches (Y) to fill a number of open slots (X), while drawing a number of mismatches (Z) in the process, where X=Y+Z. In powerball, you draw five balls, so X=5 and Y is the number of matches (anywhere from 0 to 5), while Z is the number of misses. Multiplicity shows up in stat mech and is intimately related to entropy. The totals drawn (green) is perhaps mislabeled… here I’m referring to the number of possible choices in the main basket, N=69, and the number of those that will not be drawn M = N – X, or 64. I should probably have called it “Main basket balls” or something. The last two parts determine the probabilities related to the given number of hits (Y) (orange) and the given number of misses (Z) (red) and I have applied the product operator to spiffy up the notation. Product operator is another iterand much like the summation operator and means that you repeatedly multiply successive values, much like a factorial, but where the value you are multiplying is produced from a particular range and given a set form. In these, the small script m and n start at zero (my bad, this should be under the Pi) and iterate until they are just less than the number up top (Y – 1 or Z – 1 and not equal to). At the extreme cases of either all hits or all misses, the relevant product operator (either Miss or Hit respectively) must be set equal to one in order to not count it.

This is one of those rare situations where the American public does a probability experiment with the values all well recorded where it’s possible to see the outcomes. How hard is it to win the grand prize? Well, the odds are one in 292 million. Consider that the population of the United States is 323 million. That means that if everybody in the United States bought one powerball number, about one person would win.

Only one.

Thanks to the power of the media, everybody has the opportunity to know that somebody won. Or not. That this person exists, nobody wants to doubt, but consider that the odds of winning are so scant that you not only won’t win, but you pretty likely will never meet anyone who did. Sort of surreal… everything is above board, you would think, but the rarity is so rare that there’s no assurance that it ever actually happens. You can suppose that maybe it does happen because people do win those dinky $4 prizes, but maybe this is just a red herring and nobody really actually wins! Those winner testimonials could be from actors!

Yeah, I’m not much of a conspiracy theorist, but it is true that a founding tenant of the idea of a ‘limit’ in math is that 99.99999% is effectively 100%. Going to the limit where the discrepancy is so small as to be infinitesimal is what calculus is all about. It is fair to say that it very nearly never happens! Everybody wants to be the one who beats the odds, which is why Powerball tickets are sold, but the extraordinarily vast majority never will win anything useful… I say “useful” because winning $4 or $7 is always a net loss. You have to win one of the top three prizes for it to be anywhere near worth anything, which you likely never will.

One final fairly interesting feature of the probability is that you can make some rough predictions about how frequently the grand prize is won based on how frequently the first prize is won. First prize is matching all five of the balls, but not the powerball. This frequency is about once per 12 million numbers, which is about 26 times more likely than all 5 plus the Powerball. In the report on winnings, a typical frequency is about 2 to 3 winners per drawing. About 1 time in 26 a person with all five manages to get the powerball too, so, with two drawings per week and about 2.5 first prize winners per drawing, that’s five winners per week… which implies that the grand prize should be won at a frequency of about once every five to six weeks –every month and a half or so. The average here will have a very large standard deviation because the number of winners is compact, meaning that the error is an appreciable portion of the measurement, which is why there is a great deal of variation in period between times when the grand prize is won. The incidence becomes much more Poissonian and stochastic, and allows some prizes to get quite big compared to others and causes their values to disperse across a fairly broad range. Uncertainty tends to dominate, making the game a bit more exciting.

While the grand prize is small, the number of people winning the first prize in a given week is small (maybe none or one), but this number grows in proportion to the size of the grand prize (maybe 5 or 6 or as high as 9). When the prize grows large enough to catch the public consciousness, the likelihood that somebody will win goes up simply because more people are playing it and this can be witnessed in the fluctuating frequency of the wins of lower level prizes. It breathes around the pulse of maybe 200 million dollars, lubbing at 40 million (maybe 0 to 1 person winning the first prize) and dubbing at 250 million (with 5 people or more winning the first prize).

Quite a story is told if you’re boring and as easily amused as me.

In my opinion, if you do feel inclined to play the game, be aware that when I say you probably won’t win, I mean that the numbers are so strongly against you that you do not appreciably improve your odds by throwing down $100 or even $1,000. The little $4 wins do happen, but they never pay and $1,000 spent will likely not get you more than $100 in total of winnings. It might as well be a voluntary tax. Cherish the dream your $2 buys, but do not stake your well-being on it. There’s nothing wrong with dreaming as long as you understand where to wake up.

(edit 8-24-17)

There was a grand prize winner last night (Wednesday 8-23-17). The outcomes are almost completely as should be expected: the winner is in Massachusetts… the majority of the country’s population is located in states on either the east or west coast, so this is unsurprising. There were 40 match 5 winners, so you would anticipate at least one to be a grand prize winner, which is exactly what happened (1 in 26 difference between 5 with powerball and 5 without). There were about 5.9 million powerball-only winners, so 38.32*5.9 is 226 million total powerball numbers sold in the run-up to last night’s drawing… with grand prize odds of 1 in 292 million, this is approaching parity. This means that more than $452 million was spent since Saturday on powerball lottery numbers (calculation excludes the extra dollar spent on multipliers). About five times as many ticket numbers were sold for this drawing as when I made my original analysis a week ago. With that many tickets sold, there was almost assuredly going to be a winner last night. This is not to say there shouldn’t have been a winner before this –probability is a fickle mistress– but the numbers are such that it was unlikely, but not impossible, for the prize to grow bigger. The last time the powerball was won was on 6-10-17, about two months and thirteen days ago… you can know that this is an unusually large jackpot because this period is longer than the usual period between wins (I had generously estimated 6 weeks based on the guess of 2 match 5 winners per drawing, but I think this might actually be a bit too high).

There was only one grand prize winning number out of 226 million tickets sold (not counting all the drawings that failed to yield a grand prize winner prior to this.) Think on that for a moment.