Ah, the pentagram, a shape associated with a variety of different ideas, some holy, some less savory. But to me, it's a golden figure, and not just because of how I chose to render it here. The pentagram has a connection with the golden ratio, which is defined as
This number is tied to the Fibonacci sequence and the Lucas numbers and seems to crop up a lot in nature (although how much it crops up is disputed). It turns out that the various line segments present in the pentagram are in golden ratio with one another.
In the image above, the ratio of red:green = green:blue = blue:black is the golden ratio. The reason for this is not immediately obvious and requires a bit of digging, but the proof is fairly straightforward and boils down to a simple statement.
First, let's consider the pentagon at the center of the pentagram. What is the angle at each corner of a pentagon? There's a clever way to deduce this. It's not quite clear what the interior angle is (that is, the angle on the inside of the shape at an individual corner), but it's quite easy to get the exterior angle.
The exterior angle of the pentagon (which is the angle of the base of the triangles that form the points of the pentagram) is equal to 1/5 of a complete revolution around the circle, or 72°. For the moment, let's call this angle 2θ. To get the angle that forms the points of the pentagram, we need to invoke the fact that the sum of all angles in a triangle must equal 180°. Thus, the angle at the top is 180° – 72° – 72° = 36°. This angle I will call θ. While I'm at it, I'm going to label the sides of the triangle x and s (the blue and black line segments from earlier, respectively).
We're nearly there! We just have one more angle to determine, and that's the first angle I mentioned – the interior angle of the pentagon. Well, we know that the interior angle added to the exterior angle must be 180°, since the angles both lie on a straight line, so the interior angle is 180° – 72° = 108° = 3θ. Combining the pentagon and the triangle, we obtain the following picture.
Now you can probably tell why I labeled the angles the way I did; they are all multiples of 36°. What we want to show is that the ratio x/s is the golden ratio. By invoking the Law of sines on the two isosceles triangles in the image above, we can show that
This equation just simplifies to sin 2θ = sin 3θ. With some useful trigonometric identities, we get a quadratic equation which we can solve for cos θ.
Solving this quadratic equation yields
which, when taken together with the equation for x/s, shows that x/s is indeed the golden ratio! Huzzah!
The reason the pentagram and pentagon are so closely tied to the golden ratio has to do with the fact that the angles they contain are multiples of the same angle, 36°, or one-tenth of a full rotation of the circle. Additionally, since the regular dodecahedron (d12) and regular icosahedron (d20) contain pentagons, the golden ratio is abound in them as well.
As a fun bonus fact, the two isosceles triangles are known as the golden triangle (all acute angles) and the golden gnomon (obtuse triangle), and are the two unique isosceles triangles whose sides are in golden ratio with one another.
So the next time you see the star on a Christmas tree, the rank of a military officer, or the geocentric orbit of Venus, think of the number that lurks within those five-pointed shapes.
Yesterday I stumbled across this image (which I recreated and cleaned up a bit). It's a beautiful image. Arranged around the edge is the circle of fifths, which in music is a geometric representation of the twelve tones of the Western scale arranged so the next note is seven semitones up (going clockwise in this figure). The notes are all connected in six different ways to the other notes in the "circle," known as intervals, which are color-coded at the bottom. I thought, "Wow, this is a really cool way to represent this geometrically. How neat!" However, I found the original website that the image came from, and it's a pseudoscience site that talks about the fractal holographic nature of the universe. While fractals do show up in Nature a lot, and there are legitimate theories proposing that the Universe may indeed be a hologram, what their site is proposing is, to put it lightly, utter nonsense. But instead of tearing their website apart (which would be rather cathartic), I instead want to point out the cool math going on here, because that sounds more fun!
Looking at the bottom of the graphic, you'll notice six figures. The first (in red) is a regular dodecagon, a polygon with twelve equal sides and angles. This shape is what forms the circle of fifths. The rest of the shapes in the sequence are dodecagrams, or twelve-pointed stars. The first three are stars made up of simpler regular polygons; the orange star is made up of two hexagons, the yellow is made up of three squares, and the green one is made up of four triangles. The final dodecagram (in purple) can be thought of as made up of six straight-sided digons, or line segments. These shapes point to the fact that twelve is divisible by five unique factors (not including itself): one set of twelve, two sets of six, three sets of four, four sets of three, and six sets of two! You could say that the vertices of the dodecagon finalize the set as twelve sets of one, but they're not illustrated in this image. So really, this image has less to do with musical intervals and more to do with the number 12, which is a rather special number. It is a superior highly composite number, which makes it a good choice as a number base (a reason why feet are divided into twelve inches, for instance, or why our clocks have twelve hours on their faces).
The final dodecagram in cyan is not made up of any simpler regular polygons because the number 12 is not divisible by five. If you pick a note in the circle of fifths to start on, you'll notice that the two cyan lines that emanate from it connect to notes that are five places away on the "circle," hence the connection to the number 5. In fact, it would be far more appropriate to redraw this figure with a clock face.
This new image should shed some more light on what's really going on. The dodecagrams each indicate a different map from one number to another, modulo 12. The only reason this is connected to music at all is due to the fact that a Western scale has twelve tones in it! If we used a different scale, such as a pentatonic scale (with five tones, as the name would suggest), we'd get a pentagon enclosing a pentagram. Really, this diagram can be used to connect any two elements in a set of twelve. The total number of connecting lines in this diagram, then, are
where the notation in parentheses is "n choose 2," and Tn is a triangular number. This figure is known in math as K12, the complete graph with twelve nodes. And it's gorgeous.
So while this doesn't really have anything to do with music or some pseudoscientific argument for some fancy-sounding, but ultimately meaningless, view on the universe, it does exemplify the beauty of the number 12, and has a cool application to the circle of fifths.
where ħ is the reduced Planck constant, μ is the reduced mass of the electron-nucleus system, Z is the number of positive charges in the nucleus that the electron is orbiting, e is the charge of a proton, τ is the circle constant, ε0 is the vacuum permittivity, and ψ is the wavefunction. Solving this equation (which is nontrivial and is usually done after a semester of Advanced Quantum Mechanics) yields a surprisingly simple formula for the energies of the atom,
where h is Planck's constant, c is the speed of light, me is the rest mass of the electron, and n is any integer larger than or equal to 1. The constant R∞ is known as the Rydberg constant, named after Swedish physicist Johannes Rydberg, the scientist who discovered a formula to predict the specific colors of light hydrogen (or any hydrogen-like atom) would absorb or emit. Indeed, the formula I gave, En/hc, is equivalent to the inverse wavelength, or spatial frequency, of light that it takes for the atom in its nth energy state to free the electron of its atomic bond. Indeed, this was a puzzle in the early 20th century. Why was it that hydrogen (and other atoms) only absorbed and emitted specific colors of light? White light, as Isaac Newton showed, is comprised of all visible colors of light, and when you split up that light using a prism or similar device, you get a continuous rainbow. This was not the case for light emitted or absorbed by atoms.
The equation above was first derived by Niels Bohr, who approached solving this problem not from using the Schrödinger equation, but from looking at the electron's angular momentum. If electrons could be considered wavelike, as quantum mechanics treats them, then he figured that the orbits of the electron must be such that an integer number of electron wavelengths fit along the orbit.
Left: Allowed orbit. Right: Disallowed orbit. Image: Wikimedia commons
This condition requires that
The wavelength of the electron is inversely related to its momentum, p = mv, via Planck's constant, λ = h/p. The other relation we need is from the physics of circular motion, which says that the centripetal force on an object moving in a circular path of radius r is mv2/r. Equating this to the Coulomb force holding the proton and electron together, we get
Plugging this into the quantization condition, along with some algebra, yields the energy equation.
What's incredible is that hydrogen's energy spectrum has a closed-form solution, since most problems in physics can't be solved to produce such solutions, and while this equation only works exactly for one-electron atoms, it can be modified to work for so-called Rydberg atoms and molecules, where a single electron is highly excited (large n) and orbits a positive core, which need not be a nucleus, but a non-pointlike structure. In my lab, we consider two types of Rydberg molecules.
The example on the left is an electronic Rydberg molecule, while the one on the right is called an ion-pair Rydberg state, where a negative ion acts as a "heavy electron" co-orbiting a positive ion. To model the energies of these kinds of states, we use a modified energy equation.
where I.P. represents the ionization energy of the electron, and the new quantity δ is known as the quantum defect. It's a number that, for electronic Rydberg states, has a magnitude that's usually less than 1, while for ion-pair states can be quite large (around –60 or so in some cases); it in some sense contains information of how the core ion, e.g. H2+, is oriented, how the electron is spread over space, how its polarized, and so on. It's a vessel into which we funnel our ignorance in using the approximation that the molecule is behaving in a hydrogen-like manner, and it is surprisingly useful in predicting experiments. Currently my research involves studying electronic Rydberg states of molecular nitrogen, N2, and looking at heavy Rydberg states of the hydrogen molecule, H2 to gain a better understanding of the physics of certain states that have been experimentally observed in both systems.
The above image is known as the Pentagram of Venus; it is the shape of Venus' orbit as viewed from a geocentric perspective. This animation shows the orbit unfold, while this one shows the same process from a heliocentric perspective. There are five places in Venus' orbit where it comes closest to the Earth (known as perigee), and this is due to the coincidence that
When two orbital periods can be expressed as a ratio of integers it is known as an orbital resonance (similar to how a string has resonances equal to integer multiples of its fundamental frequency). The reason that there are five lobes in Venus' geocentric orbit is that 13–8=5. Coincidentally, these numbers are all part of the Fibonacci sequence, and as a result many people associate the Earth-Venus resonance with the golden ratio. (Indeed, pentagrams themselves harbor the golden ratio in spades.) However, Venus and Earth do not exhibit a true resonance, as the ratio of their orbital periods is about 0.032% off of the nice fraction 8/13. This causes the above pattern to precess, or drift in alignment. Using the slightly more accurate fraction of orbital periods, 243/395, we can see this precession.
This is the precession after five cycles (40 Earth years). As you can see, the pattern slowly slides around without the curve closing itself, but the original 13:8 resonance pattern is still visible. If we assume that 243/395 is indeed the perfect relationship between Venus and Earth's orbital periods (it's not; it precesses 0.8° per cycle), the resulting pattern after one full cycle (1944 years) is
Which is beautiful. The parametric formulas I used to plot these beauties are
Where t is time in years, r is the ratio of orbital periods (less than one), and τ = 2π is the circle constant.
Should the Earth-Moon system be considered a binary planet? This sounds outlandish at first, since the Moon is a moon, obviously. It orbits the Earth as a natural satellite, just as the Galilean moons (Ganymede, Callisto, Io, and Europa) orbit Jupiter, Titan orbits Saturn, Triton orbits Neptune, and so on, right?
The definition of a moon is vague, and thus there are multiple ways of determining whether or not a planet-moon system is really a binary planet. One way of drawing the line between the two descriptions is by finding the barycenter (or center-of-mass) of the system. The center of mass of a collection of N masses is given by
where M is the total mass of the system, and mi and ri are the mass and position of the ith object, respectively. If the center of mass of a two-body system lies outside the larger object in that system, call it a binary planet. This makes sense, right? This means that the smaller body doesn't orbit the larger body, but instead they both orbit some point in space. For instance, the barycenter of the Pluto-Charon system lies outside Pluto (0.83 Pluto radii above Pluto's surface), the larger of the two bodies, while the Earth-Moon barycenter lies within the Earth (just under 3/4 of an Earth radius from the planet's center). By this definition, the Pluto-Charon system is a binary (dwarf) planet system, while the Earth-Moon system is is a planet-moon system. (Although, we are slowly losing our moon due to tidal acceleration. In a few billion years, the Moon will have drifted far enough away that the barycenter of the Earth-Moon system will leave the interior of our planet.) However, when you plug in values for the Sun-Jupiter system, you find that the center of mass lies outside the Sun! Indeed, Jupiter is the only natural satellite of the Sun for which this is true. (Does this mean Jupiter should have a different classification from the rest of the planets? Not really; the Sun is around 1000 times more massive than Jupiter, so the reason for this is that Jupiter is very distant from the Sun.)
Maybe a different definition is needed to distinguish planet-moons from binary planets, then, since the Sun-Jupiter system is not a binary star (Jupiter is slightly too small to generate nuclear fusion). Another proposition is to look at the so-called tug-of-war value of a body. The tug-of-war value of a moon determines which Solar System object has a stronger gravitational hold, the Sun or the moon's "primary" (the Earth is the Moon's primary). Using Newton's law of gravitation
we can take a ratio of the Sun's pull on a satellite to the primary's pull. The result is the tug-of-war value, proposed by Isaac Asimov.
Here the subscripts s and p refer to the Sun and the primary, respectively; m is the mass of the body referred to by the subscript; and d is the distance between the moon and the body referred to by the subscript. If the tug-of-war value is larger than 1, then the primary has a larger hold on the moon than the Sun, whereas if it's less than 1, the Sun's gravity dominates. For the Earth-Moon system, it turns out this number is 0.46, which means that the Sun pulls on the Moon with more than twice the force of Earth's pull. This is an oddity among moons, but is not unique. It does mean, though, that the Moon, when viewed from the Sun, never undergoes retrograde motion; it moves across the solar sky without changing direction. Another way to put this is that the Moon is always falling toward the Sun (like the planets), and never in its orbit does it fall away from the Sun (unlike most moons). If you look at the orbits of the Earth and Moon from the point of view of the Sun, they dance around each other in careful step, which is unlike most other moons in the Solar System. For Asimov, this was reason enough to consider the Earth and Moon as a binary planet system.
This tug-of-war value does not, however, classify Pluto and Charon as a binary dwarf planet system (they're too far from the Sun for their tug-of-war value to be less than 1). Perhaps the definition of a binary planet is a difficult one to pin down.
Should the Moon be promoted to planet, just as Pluto was renamed as a dwarf planet? I don't know, but it gives us something to think about as we look up at the starry night, watching the dance of all the chunks of rock and gas hurtling through space in our sky, to music written by nature and heard through science.
Chaos is complexity that arises from simplicity. Put in a clearer way, it's when a deterministic process leads to complex results that seem unpredictable. The difference between chaos and randomness is that chaos is determined by a set of rules/equations, while randomness is not deterministic. Everyday applications of chaos include weather, the stock market, and cryptography. Chaos is why everyone (including identical twins who having the same DNA) have different fingerprints. And it's beautiful.
How does simplicity lead to complexity? Let's take, for instance, the physical situation of a pendulum. The equation that describes the motion of a pendulum is
where θ is the angle the pendulum makes with the imaginary line perpendicular to the ground, l is the length of the pendulum, and g is the acceleration due to gravity. This leads to an oscillatory motion; for small angles, the solution of this equation can be approximated as
where A is the amplitude of the swing (in radians). Very predictable. But what happens when we make a double pendulum, where we attach a pendulum to the bottom of the first pendulum?
Can you predict whether the bottom pendulum will flip over the top? (Credit: Wikimedia Commons)
It's very hard to predict when the outer pendulum flips over the inner pendulum mass, however the process is entirely determined by a set of equations governed by the laws of physics. And, depending on the initial angles of the two pendula, the motion will look completely different. This is how complexity derives from simplicity.
Another example of beautiful chaos is fractals. Fractals are structures that exhibit self-similarity, are determined by a simple set of rules, and have infinite complexity. An example of a fractal is the Sierpinski triangle.
Triforce-ception! (Image: Wikipedia)
The rule is simple: start with a triangle, then divide that triangle into four equal triangles. Remove the middle one. Repeat with the new solid triangles you produced. The true fractal is the limit when the number of iterations reaches infinity. Self-similarity happens as you zoom into any corner of the triangle; each corner is a smaller version of the whole (since the iterations continue infinitely). Fractals crop up everywhere, from the shapes of coastlines to plants to frost crystal formation. Basically, they're everywhere, and they're often very cool and beautiful.
Chaos is also used in practical applications, such as encryption. Since chaos is hard to predict unless you know the exact initial conditions of the chaotic process, a chaotic encryption scheme can be told to everyone. One example of a chaotic map to disguise data is the cat map. Each iteration is a simple matrix transformation of the pixels of an image. It's completely deterministic, but it jumbles the image to make it look like garbage. In practice, this map is periodic, so as long as you apply the map repeatedly, you will eventually get the original image back. Another application of chaos is psuedorandom number generators (PRNGs), where a hard-to-predict initial value is manipulated chaotically to generate a "random" number. If you can manipulate the initial input values, you can predict the outcome of the PRNG. In the case of the Pokémon games, the PRNGs have been examined so thoroughly that, using a couple programs, you can capture or breed shininess/perfect stats.
Dat shiny Rayquaza in a Luxury ball, tho.
So that's the beauty of chaos. Next time you look at a bare tree toward the end of autumn or lightning in a thunderstorm, just remember that the seemingly unpredictable branches and forks are created by simple rules of nature, and bask in its complex beauty.
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