The Erdős-Bacon Number

A lot of you will be familiar with the Bacon number. It's a play on the "six degrees of separation" concept that states that everyone in the world is six or fewer degrees separated from anyone else, one degree being someone you personally know.

Image: Wikimedia Commons

This graphic shows a minimum spanning tree, which is not what I'm talking about, but I'll take the heat from all the angry graph theorists. 

Someone's 'Bacon number' is this concept played out specifically with Kevin Bacon. Film buffs might play a game trying to link actors to Bacon in as few shared movies as possible.

To give an example, my Bacon number is 4. One of my cousins went to a doctor that had also worked on Micheal Jordan, who appeared in this commercial with Kevin Bacon:

So from me to my cousin (1), to the doctor (2), to Michael (3), to Kevin, there are 4 steps. (I also have a Bacon number of 4 via a family friend, Cozi Zuehlsdorff, Christopher de Stefano to Kevin)

Note: another definition of the Bacon number is that one must use film credits rather than personal connections, and it's this definition we will be using later on.

<sidenote>
Facebook generates a lot of data, and decided to calculate the average connectivity of its users: 3.57 degrees
</sidenote>

Image: Topsy Kretts

Paul Erdős

Another similar number exists, and that is the Erdős number. Paul Erdős was an unusually prolific mathematician and collaborated with a huge number of scientists during his career. Many academics calculate their "Erdős number," which is calculated in a similar fashion as the Bacon Number, but rather than personal connection, you would use academic papers that share the authors' name.

Using these two numbers, you can calculate someone's Erdős-Bacon number, which is the sum of the two numbers. Naturally, this requires both being credited in at least one film as well as author an academic paper, something very few people have done.

A number of famous names have low Erdős-Bacon Numbers:

Richard Feynmann appeared in the film Anti-Clock, which gives him a Bacon number of 3, and an Erdős number of 3 gives him an EBN of 6.

 Carl Sagan had an EBN of 6 as well.

The scientist Stephen Hawking actually has a lower Bacon number (2) than an Erdős number (4), for an ENB of 6.

Natalie Portman, as well as the Big Bang Theory actress Mayim Bialik, have both published papers, giving them Erdős numbers of 5. Both have Bacon numbers of 2, giving both women an EBN of 7.

Perhaps the lowest Erdős Bacon Number belongs to a man named Albert A Chan. He acted alongside Bacon in Patriot Day, giving him a Bacon number of 1, and has published work that achieves an Erdős number of 3, giving him the lowest Erdős-Bacon Number at 4.

Another quick note on interdisciplinary achievement - only one person in history has ever received both an Oscar and a Superbowl Ring - film producer and businessman Steve Tisch. Tisch, as the Executive Vice President of the New York Giants, received Superbowl championship rings for the 2007 and 2011 seasons, as well as an Oscar for producing Forrest Gump.


Cheers,

    - Scott


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How Big is the Death Star?

I recently became curious about the death star. It’s fictional, I know; but I kept asking questions: What would it take to supply such a large space station? What about to build it? Does it have enough gravity to walk along the surface? Come along down this rabbit hole with me.

Image: JMAS

Note: the Death Star I chose to look at in this article is the larger Death Star II (160 km in diameter vs. 120 for Death Star I). Death Star II is the one Lando Calrissian destroys above the forest moon of Endor in Return of the Jedi, not the one that the Luke destroys in A New Hope. 

The Physics

I started this journey by looking up the specs of the Death Star and calculating its volume, mass, surface gravity etc… I soon found an oft-quoted stat online that mentioned the Death Star’s volume was “17.16 quadrillion cubic meters.” This surprised me because the answer I got was an order of magnitude lower. After looking into it, I found out this calculation had been done using 160 km as the radius, not the diameter, yielding an incorrect answer. Below are the correct stats for the Death Star:

Diameter* ………………..……… 160 km

Volume …………………..………. 2.1 quadrillion m³

Mass** ………………….…………1.6 quadrillion kg

Surface gravity …………………. 0.0002 % of Earth

Total Crew* ……………………… 2,471,647 (This is strangely specific)

*According to Wookiepedia

**Assuming 1/10 of the volume is steel

Earth and the Death Star to Scale

The first thing that struck me that the Death Star is small. I know the movie compared the Death Star to a "small moon," but the Death Star is much smaller than I imagined, on the scale of Jupiter's tiny moon Janus, or, appropriately (it looks just like the death star), Saturn's moon Mimas.

The second thing that caught my attention was the very low surface gravity. If I was out walking along the surface of the Death Star, I would weigh about the same as a large grain of sand. If I had one or two friends with me, we could lift the Titanic off the surface. If you could get enough traction, you could simply walk into orbit (1.15 m/s). The Death Star is massive for a space station, but because there are a lot of empty spaces, it just doesn’t have that much pulling force.

The Crew

What about the crew? How much do they need to eat? How much waste needs to be disposed of?

First off, let's look at butter. Butter is the one of the most calorie rich foods out there, so if we assume everyone just eats butter, we'll get our most conservative estimates.

A stick of butter contains about 800 calories, and an adult human needs a little less than three sticks of butter to survive for a day (do not try). By volume, about 10 people need a liter of butter per day. This means that the Death star requires about a quarter-million liters of butter per day to feed its personnel. That's about 7 Boeing 747 cargo planes' worth of butter, which seems manageable. You would need another 2 and a half planes for the water, but overall, that seems pretty reasonable.

According to Wookiepedia, the Death Star has a 3-year store of food (assuming Earth years, even though Earth is in a faraway galaxy in the distant future), which in butter terms is about 1 Hindenburg.

Overall, no real problems arise when stocking the Death Star with food and water, even if the calorie density is much less than that of butter.

What about waste? One average US citizen in 2008 produced about 2 kilos of food per day. On the Death Star, this equates to 5 million kilos over the whole population, or less than half of the daily trash output of New York City.

A good way to think of the Death Star is about one "Chicago" of people (new unit). The Death Star needs about the same amount of food, and generates the same amount of waste as a large city.

The Death Star has about 1,600 Dropships that, if operated around the clock, could supply food and water to the entire Death Star. Waste could simply be expelled into the atmosphere of the forest moon of Endor. Then it's the Ewok's problem.

Image: David Kingham

The Cost

The answer to the question of how much the Death Star would cost to build started with flawed numbers, so I've redone the calculations.

The current (Sept. 2015) cost of steel is 140 US dollars per tonne. To put our weight in tonnes from kilos, we can just knock off three zeros, getting us 1.6 trillion tonnes of steel, coming in at 224 trillion dollars just for the steel. To lift this amount of steel into orbit would cost 35 quintillion dollars and take 70 billion US Shuttle flights. and to avoid collapsing the global steel market (and to make things cheaper), let's just grab an asteroid with that amount of steel and hollow it out to make our death star.

Image: Josef Ďurech, Vojtěch Sidorin

16 Psyche

16 Psyche should do nicely; it's mainly nickel and iron, 200 km across, and we could find some carbon and smelt steel to make our Death Star. That should keep the total bill under a few trillion to get the raw materials together. As far as turning these elements into the Death Star, that's pretty much out of reach with the planet's current resources.

Overall, it seems that most aspects of the Death Star are out of our reach, and without a bent for galactic domination, I think we should leave the construction of Death Stars to galactic empires in the distant past, and we should focus on, say, getting to Mars. That's a good starting point.


     Cheers,
   
          - Scott

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Have We Found Planets Using Math?

Planets Found by Math

Urbain Le Verrier

Ancient peoples have always known about Mercury, Venus, Mars, Jupiter and Saturn because you can easily see them with you eyes. The discovery of Uranus was the first time a planet was found that needed the aid of a telescope to be see. William Herschel discovered this planet in 1781.

In the next century, another planet was discovered, and this one was discovered in a very interesting way. The credit for finding planets is usually thought to go to the first person to point their telescope at it, but this case is a little different. In the 1845, Urbain Le Verrier looked very closely at the orbit of Uranus, and discovered it to be slightly off from what was known from Kepler about planetary motion. Soon after, he had a hypothesis that another planet beyond the orbit of Uranus could account for the perturbations in the orbit he observed. Le Verrier contacted Johann Gottfried Galle at the Berlin Observatory and told him to point his telescope at a particular location at a particular time to look for this eighth planet.

Johann Gottfried Galle

Sure enough, when Galle peered through the telescope lens on September 23^rd in 1846, there sat Neptune, 1° away from Le Varrier's predicted position. Interestingly, the director of the Cambridge observatory, James Challis, later realized he too had seen Neptune on two separate occasions before that, but failed to recognize it as a planet.

This, however, was not the only planet found by math. Later on, small perturbations were noticed in the orbit of the planet Mercury, again by Le Verrier. A small planet was hypothesized, this time inside the orbit of mercury, too close to the sun to see. This theorized planet was given a name – Vulcan.

Yes, the one and same, though the hypothesized planet came before the Star Trek series by more than a century (and was the Roman god of fire, volcanoes, and metalworking well before that). This planet does not actually exist. We have since sent spacecraft close enough to the sun to see any potential Vulcanoids, and to date have found none. So what of the perturbations of the Mercurial orbit? The answer is relativity. Because Mercury is so deep in the sun's gravity well, it experiences relativistic affects, and this accounts perfectly for the precession observed in its orbit.

Image: Terry Virts

LLAP

Cheers,

    - Scott

LINKSTORM:

Can you unlearn how to ride a bike? (yes)

Life on the ISS just got better, there is now coffee, and mugs

We are going to Pluto!

This map is amazing. How cloudy is the Earth?

Videos of space physics. Things behave differently in freefall.

T Handle

Gyroscopic Platform

Symphony of Spheres

Finally here is a volcano erupting! Hail Vulcan





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How Could We Have Broken Germany's Enigma Code?

I initially set out to write about the German Enigma Machine and how it worked, but as I was doing some research, two things happened: I found a video that explains it better than I ever could, with an actual WWII-era Enigma Machine, and I wound up doing some math that I thought was somewhat interesting.

So to start, I think you should watch this video about the Enigma Machine:

Did you watch it? If not, I'll give you one more chance:

IF THE VIDEO DOES NOT SHOW UP CLICK

 >  HERE  <

Alright, so in the video we saw that the Enigma Machine was a device used to encrypt internal messages in the German military, and has an extraordinary number of possible arrangements, each of which will produce a different code. The Enigma code was broken through a regimented and careful application of cleverness on the part of Alan Turing and his team at Bletchley Park. I got to thinking however, “what sort of team would have to be assembled to solve the Enigma code through sheer human power and luck?”

This rabbit hole was quite fun to descend into. I started with a few assumptions for this scenario. Every person who was working on the code had their very own enigma machine to work with, they could try a new arrangement each 100 seconds, and they could work for 16 hours a day. I also assumed that the infrastructure to feed, house and give them water was in place. Alright, here is the number we started with:

158,962,555,217,826,360,000

159 quintillion. That is the number of possible arrangements the enigma machine could be in, each would output a different code, and only one of which would be correct. Oh, and it changed every single day.

In order to try every combination in the space of 24 hours, you would need to try 1,840,000,000,000,000, or 1.8 quadrillion arrangements per second.

If one person can try one arrangement in 100 seconds, that means it will take a force of 184 quadrillion people to try every arrangement in the space of 24 hours, but that only if we work them 24 hours a day. Adjusting for the workforce working 16 hour days, we can multiply 184 quadrillion by (4/3) to get 245 quadrillion people.

So we have ~ 2,350,000,000 people in the world (in 1945)

We need 245,000,000,000,000,000.

Keen-eyed readers will note that the second number is longer, so we are now moving into a very hypothetical world. As long as we're not bound by reality, lets go ahead and pack in our workforce across the entire land area of earth (minus Germany), and lets pack them in at the population density of current-day Tokyo (we are ignoring small facets like food water, adequate shelter, etc...).

Land area of Earth: 57.53 million km²

Land area of Germany: 137,903 km²

Population density of Tokyo: 1800 people per km²

1800 people per km² * ( 57.53 million km² – 137,903 km²)

This gives us a population on our Super-Earth of about 100 billion people.

This means we would need 2.45 million Super-Earths to support our workforce.

It looks like we'll need to drain the oceans to beat the Nazis.

Due to the fact that Earth's surface area is ¾ water, draining the oceans gives us 3x the surface area we had using land alone (minus Germany). We can now fit 400 billion people on each Mega-Super-Earth, and now we need a quarter as many, or a little more than 600,000 Mega-Super-Earths. While that many Earth-Sized objects can orbit many stars, we need to restrict our Enigma team to one star in order to transmit the correct code in time to implement a strategy. The nearest star to us is Proxima Centauri, over 4 light years away. That is not close enough to get information back to central in time to help the war effort.

The question now becomes “Can 600,000 Earth-sized object orbit our sun without bad things happening?” I talked to some astronomers and astrophysicists here at CU Boulder, and the consensus seems to be “no.” When you put a lot of objects in similar orbits, most of the bodies are ejected from the system, most of the rest collide with each other, and quite a few will be engulfed by the star in the middle of the system.

No video?

CLICK HERE

This is the “Nice Model” of the solar system. About 3.8 billion years ago many Kuiper belt objects were ejected from the system and Uranus and Neptune switched places (at 30 seconds in the video). This area was much more sparsely populated than our scenario would be.

This doesn't happen immediately, so to wrap up, if you’re going to use humans alone to solve Enigma, do it fast, and make it worth it.

There are of course a few problems with this plan:

On average, the code will be broken by midday. Sometimes you will get lucky and solve it early on, and sometimes you wont be so lucky, and it will take until late in the evening to break the code. Even with this workforce, there will only be an average of 12 hours to actually use the information obtained.

If you have 245 quadrillion people united to defeat the axis powers, you probably wouldn't even need to bother with the Enigma code. If each person donated one strand of hair (half a milligram), you could bury Germany in 122 billion kilograms of human hair, which would probably deter them. I wanted to figure out how many feet this would be, but there aren't reliable figures for the packing efficiency of average human hair, so I figured a head of hair (a wig) weighs about 100 grams, and I could stuff one in a one liter bottle giving it a density of 0.1 kg/liter. This means 122 billion kilograms of hair could cover Germany to a depth of ~9 meters, give or take a few orders of magnitude.

So now you know that...

If you’re curious about the Enigma Machine itself, as well as the flaw that turned out to be its undoing, there is another video that describes how Turing and his team broke the code:

Cheers,

    - Scott

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LINKSTORM:

Artifacts from the Curiosity landing on Mars are fading, caught by the HiRisecamera on the MRO satellite

A good video describing the Gulf Stream and the little-known effect ofdensity rather than temperature

Some very funny science

An interesting concept laid out by CGP Grey

A great story about how humans are meat, and how that's weird

This is a happy video that will probably make you happy