Category: DIY

Throughout history, humanity’s view of the universe has changed drastically. For a long time, people believed that the Earth was all there was, with a dome high above it that had pinpricks they called stars. Later, they believed the solar system was the extent of the universe. Then that view expanded to the galaxy as we realized the pinpricks were actually other suns. Now, we believe there is a universe beyond even what we can observe, with other galaxies than our own populating space. When our understanding of the universe allowed us to understand that other stars exist, there began to be questions of extraterrestrial life around other stars and whether we on earth could somehow communicate with it. In the 1960’s, a man named Frank Drake wrote a book exploring the chances of life on other planets that was intelligent and capable of communicating with Earth in our galaxy. The book lead to an equation that expresses this chance numerically:

N = R * f(g) * f(p) * n(e) * f(l) * f(i) * f(c) * L

This equation has many terms that are not able to be calculated exactly, but they do lay extremely good groundwork for an estimate of intelligent, communicative life. This equation got experts talking about what they thought the odds for life in the universe could be and helped encourage the search for extraterrestrial life. This equation is also really easy to use to develop your own guess on the chances of extraterrestrial life. Breaking down this equation, there are three categories the values can be put into based on what factors they relate to. 

Astronomically Based Values

The first three values in the equation are values that are gained from observing the galaxy. They are astrophysical values that, with our current scientific capabilities, are reasonable to calculate somewhat accurately. The first value, R, is the rate of star formation in the galaxy. It can alternatively be the number of stars estimated to be in the galaxy. It is the starting point of the equation, as most of the other terms are fractions that whittle this value smaller and smaller as the criteria gets more specific. The current estimate of stars in the galaxy is between 100 and 400 billion stars. The second value, f(g), is an estimate of how many stars in the galaxy are suitable for hosting planets with life. Stars that can host planets with life must be long living and steady stars and they cannot have large fluctuations in temperatures and intensity. This value is currently estimated to be between 5 and 10 percent, meaning .05 to .1 for use in the equation The third value, f(p), is a measure of how many of those suitable stars actually have planetary systems. For example, stars that are in binary systems are thought to very rarely host planetary systems. This value is often estimated at about .5, as about 50% of star systems are binaries or multistar systems and the other 50% are single stars. 

Biologically Based Values

The next three values of the Drake equation are based on biological factors with life. The first one, n(e), is how many planets per star system are suitable for life. This mainly means how many planets are in the habitable zone of a star. The habitable zone is the distance from a star where planets can have water in all three phases, solid, liquid, and gas. The value to plug into the question can be found by taking your estimate of the number of planets in the habitable zone and dividing it by the estimated total number of planets in the system. For a system like our solar system, this would be .125. The next value, f(l), is the fraction of planets in the habitable zone that will actually have life form. This is where the values in the equation begin to vary greatly from person to person. Some people argue that if life can form it will and put this value at 1. Others argue that life is extremely rare to happen, putting the value closer to .001, or even smaller. Since we have so little understanding of how life started on Earth, it is hard to estimate its rarity in other places.  The final value, f(i), is the fraction of cases in which life develops intelligence. Once again, this is a controversial value to choose. Currently, we don’t know if life always evolves to intelligence. It can be argued that we did, so why wouldn’t other planets? However there have also been millions of species on our planet, and only one has evolved to our level of intelligence: humans. With those odds, intelligence seems rare. 

Sociological Based Values

The last two values of the Drake equation are sociological factors. The first one, f(c), is the number of intelligent races that have the technological capability and desire to try and communicate with other species. An example of this coming into play may be with an intelligent species that lives underwater. It would be difficult, if not impossible, for them to invent electronic devices like ours, and things like radio telescopes to communicate with might never be designed by them. Experts tend to estimate this as a fairly large value because it is thought by many that upon developing intelligence, a species will continually improve their technology. This is believed to always lead to communication technology in the process. The exception to this are species that face rare environmental factors that dissuade the invention of specific types of technologies, such as the underwater species example. The last value, L, is the lifetime of a civilization. This is as simple as it sounds. This value is necessary because we live so far from other planets that it would take a very long time to communicate with them. The very long time it takes to send a message means there needs to be long standing civilizations to continue any conversations, otherwise they would not be able to truly communicate with us. This value is often estimated to be between 1,000 and 100,000,000 years. 

Concluding Thoughts

Current estimates of the value calculated from the Drake equation range from as low as .00000000000091 to 15,600,000 planets that host intelligent, communicative life. The value can be much larger or lower than this as well, but there is no “correct” answer. You can find your own solution by plugging in values that you think are true. All you have to do is start with the number of stars in the galaxy and fill in what fractions you believe fit for the other values, and choose a lifetime to use. If your answer is greater than 1, your chosen parameters suggest that life exists elsewhere in our galaxy. However, if your answer is less than 1, your parameters suggest that we are likely alone in the galaxy. As long as you can give a reason for the values you chose, you have created your own theory for life in the universe.

Burchell, M. (2006). W(h)ither the Drake equation? International Journal of Astrobiology, 5(3), 243-250. doi:10.1017/S1473550406003107

Drake, F.; Sobel, D. (1992). Is Anyone Out There? The Scientific Search for Extraterrestrial Intelligence. Delta. pp. 55–62. ISBN 0-385-31122-2.

Whether you’re browsing TikTok, Instagram, or scrolling through random Wikipedia articles, do you ever wonder if your internet browsing is secure? Probably not – but that’s okay! The world can peacefully rely on the wonders of mathematics to keep their data and internet browsing safe from peeking eyes. In fact, the entire safety of the internet primarily relies on a single mathematical algorithm to secure it all, ranging from state secrets in secret government databases – which are probably all about UFOs – to your grandmother’s email account. This algorithm was first developed in 1977; it’s so powerful that it still remains the primary public cryptographic algorithm today! It’s called the Rivest-Shamir-Adleman (RSA) cryptosystem.

Background

Developed in 1977 from a paper released by Ron Rivest, Adi Shamir, and Leonard Adleman, while they were at the Massachusetts Institute of Technology, RSA quickly became adapted to securing digital communication because of its robust mathematical properties that makes it virtually impossible to break.

Interestingly, the actual mathematics behind the RSA cryptosystem is rather simple. It’s based on the idea of prime factorization – it’s really easy to find the prime factors for a small number but becomes much harder as the numbers grow. For example, if I ask you to give me the two prime factors that multiply together to give the number 15, you would probably answer really quickly with “3 and 5”. Easy, right? Now, without a calculator, what if I asked you to give me the prime factorization for the number 414,863? You would answer with “577 and 719” right away, right? Unless you’re a super math genius or a human calculator, probably not.

Implementation Idea

The actual RSA algorithm is used in public-key cryptography, which is the process that secures data over the internet. It’s a complicated process but the basic idea is this: if Alice wants to send Bob a secret gift, Bob can send her a padlock that only he has the key to. Then, Alice can put her gift in a box and lock it with Bob’s padlock. Then, when Alice sends her package, if anyone tries to steal the gift, they can’t because only Bob has the key. Once Bob receives his gift, all he has to do is unlock the padlock that he gave to Alice. The padlock that Bob gave to Alice is called Bob’s public key, while the key that only Bob has is his private key.

In the real world, if my friend wants to send me a message, I will give them my public key (which is basically just a really large number). They can then use my public key to encrypt, or hide, the message. Then, in transit, the message is completely safe because I’m the only person that holds the key that decrypts, or unlocks, the message. The only tricky part about the process is making keys that are not able to be broken.

The Magic Behind It All

The RSA algorithm is responsible for creating the public and private keys. To create the actual keys, RSA relies on some basic math: modular arithmetic, Euler’s Theorem, and prime numbers.

To create my public key that I can give to anyone who wants to send me secrets, I do two things:

• First, I will choose two very large prime numbers, p and q. The multiplication of these two numbers, x = pq, is the first half of the key. The key take-away in this first portion is that the longer that x is, the more secure your key will be. As of 2020 practices, a length of 4096 bits (or characters) is recommended!
• Then, I calculate another number y by multiplying from the following: y = (p-1)(q-1). From here, I choose a number that is relatively prime to y and call it e. If you don’t recall what it means for two numbers to be relatively prime, it just means that two numbers have a greatest common divisor of 1. In practice, e is typically chosen to be 65,537 – e then becomes the other half of my public key.

Now, to create my private key that no one else knows about, I use modular arithmetic. If you’re not familiar with modular arithmetic, think of it as the remainder produced after dividing a small number into a big number – for example, 10 modulo 3 = 1 because 3*3 is 9, which leaves us with 1 left over. To create my private key, I calculate the modular inverse of the second half of my public key with the first half of my public key, e modulo x. That’s it! I have my private key! Now I can share my public key with friends, and they can share juicy, secret memes with me without the fear of someone else seeing!

Convinced?

What makes RSA so robust and secure is that, even as computers gain more processing power and become faster, all we have to do is increase the length of the key! Think about how fast and powerful our current computers are: our computers are solving problems in astrophysics, mathematics, and are beginning to become ubiquitous in controlling cars. Even with how advanced technology currently is, if someone uses a 4096-bit RSA key, it would take tens of thousands of years for computer to break the code! And even when that’s broken, all we have to do is increase the number of bits in the RSA key and it’s secure for another few thousand years or so!

If you’re not convinced by how amazing (and even basic) cryptography is, then I don’t think you’re someone who can be easily impressed!

The Process in Pictures

If you prefer pictures, here is a simple illustration of the process above:

Step 1:

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This year, continuing the theme from the Taste of Science, the Georgia Tech STEMcomm team decided to host a Sober Science Speakeasy, a party all about the science of drinks. While looking around for something to prepare, the concept of “clear milk” was brought up, and I was intrigued. Clear milk? That’s practically oxymoronic. And yet, these drinks, called “clarified cocktails” or “milk punches”,  added milk to end up with clear, colorless products. Naturally, we had to investigate and try for ourselves. This is what we learned:

The Science:

Milk is a colloid, with fat and proteins suspended in the aqueous solution. Since these are fairly evenly distributed, they usually scatter the light beam around in all directions in the Tyndall effect, making the liquid appear white and opaque.    Proteins in the milk, particularly caseins, repel each other at the normal pH of milk, which is what maintains that even distribution throughout. This is because the isoelectric point of casein, or the point at which it it is electrically neutral (uncharged) is around 4.6, but milk’s normal pH is closer to 6.6. The casein proteins therefore have a negative charge that makes them repel one another and spread out as far apart as possible.

When pH of milk is lowered however, by adding acid, the casein proteins will lose their negative charge, and stop repelling one another. This, along with the protein degradation, or unfolding of their shape, at the lower pH, causes many of them to clump together in solid blobs we call “curds”. They will separate out to the bottom, and the liquid left behind is called the “whey”. But these proteins don’t just separate from the rest of the drink- they take compounds with them, including much of the color! The proteins bind to molecules from the drink, especially polyphenols, which are associated with bitter/astringent tastes, like tannins in black tea or red wine.

Therefore, when you strain out the curds, you also remove compounds bound to the caseins. This “clarifies” the drink, stripping away most of its original color, and giving it a less harsh flavor. Some also claim that the addition of the remaining milk whey also gives the drinks a sliker texture, making them even more desirable.

The Drinks

This property of milk has been used to make clarified cocktails for many years, with the first written recipe dating all the way back to 1711. Even our favorite kite-flying founding father, Benjamin Franklin, was a fan, recording his own recipe for brandy-filled milk punch in 1763. You can still see the original document online in the Massachusetts Historical Society archives: https://www.masshist.org/database/263.

In our case, we went for a non-alcoholic version, with a base of black tea, grape juice, lemon juice, and of course, milk.

This concoction, lovely nicknamed “clarified purple cow punch” or CPCP, was initially dark reddish-purple, and while sweet, was also tad bitter from the black tea flavoring. The addition of the juice to milk resulted in a (rather unappetizing) mixture of curdled milk and juice in a clumpy, brown…milkshake? Ick.

At that stage, I was pretty doubtful that this was going to work. With a bit of patience, however, and several rounds of straining through cheese clothes and coffee filters, we ultimately ended up with a pale pink, ultra-sweet drink that proved pretty popular. There was no hint of milk flavoring or bitter taste, and all together, CPCP was a surprising success.

If you want to try making clear drinks yourself, all you need is some juice mix, an acid, and milk- and a lot of time to sit and watch the straining process. Make sure to add the acid slowly to the milk, and not the other way around, to ensure the proteins have time to bind to the polyphenols before curdling, and you’ll be all set to impress with your friends with this magically clear milk.