The Simple Harmonic Oscillator

Note to self: debugging

always remember to remove the Log.d(”Well fuck you too”) lines and the variables named “wtf” and “whhhyyyyyyy” before pushing anything to git

Regarding Fractals and Non-Integral Dimensionality

Alright, I know it’s past midnight (at least it is where I am), but let’s talk about fractal geometry.

Fractals

If you don’t know what fractals are, they’re essentially just any shape that gets rougher (or has more detail) as you zoom in, rather than getting smoother. Non-fractals include easy geometric shapes like squares, circles, and triangles, while fractals include more complex or natural shapes like the coast of Great Britain, Sierpinski’s Triangle, or a Koch Snowflake.

Fractals, in turn, can be broken down further. Some fractals are the product of an iterative process and repeat smaller versions of themselves throughout them. Others are more natural and just happen to be more jagged.

Fractals and Non-Integral Dimensionality

Now that we’ve gotten the actual explanation of what fractals are out of the way, let’s talk about their most interesting property: non-integral dimensionality. The idea that fractals do not actually have an integral dimension was originally thought up by this guy, Benoit Mandelbrot.

He studied fractals a lot, even finding one of his own: the Mandelbrot Set. The important thing about this guy is that he realized that fractals are interesting when it comes to defining their dimension. Most regular shapes can have their dimension found easily: lines with their finite length but no width or height; squares with their finite length and width but no height; and cubes with their finite length, width, and height. Take note that each dimension has its own measure. The deal with many fractals is that they can’t be measured very easily at all using these terms. Take Sierpinski’s triangle as an example.

Is this shape one- or two-dimensional? Many would say two-dimensional from first glance, but the same shape can be created using a line rather than a triangle.

So now it seems a bit more tricky. Is it one-dimensional since it can be made out of a line, or is it two-dimensional since it can be made out of a triangle? The answer is neither. The problem is that, if we were to treat it like a two-dimensional object, the measure of its dimension (area) would be zero. This is because we’ve technically taken away all of its area by taking out smaller and smaller triangles in every available space. On the other hand, if we were to treat it like a one-dimensional object, the measure of its dimension (length) would be infinity. This is because the line keeps getting longer and longer to stretch around each and every hole, of which there are an infinite number. So now we run into a problem: if it’s neither one- nor two-dimensional, then what is its dimensionality? To find out, we can use non-fractals

Measuring Integral Dimensions and Applying to Fractals

Let’s start with a one-dimensional line. The measure for a one-dimensional object is length. If we were to scale the line down by one-half, what is the fraction of the new length compared to the original length?

The new length of each line is one-half the original length.

Now let’s try the same thing for squares. The measure for a two-dimensional object is area. If we were to scale down a square by one-half (that is to say, if we were to divide the square’s length in half and divide its width in half), what is the fraction of the new area compared to the original area?

The new area of each square is one-quarter the original area.

If we were to try the same with cubes, the volume of each new cube would be one-eighth the original volume of a cube. These fractions provide us with a pattern we can work with.

In one dimension, the new length (one-half) is equal to the scaling factor (one-half) put to the first power (given by it being one-dimensional).

In two dimensions, the new area (one-quarter) is equal to the scaling factor (one-half) put to the second power (given by it being two-dimensional).

In three dimensions, the same pattern follows suit, in which the new volume (one-eighth) is equivalent to the scaling factor (one-half) put to the third power.

We can infer from this trend that the dimension of an object could be (not is) defined as the exponent fixed to the scaling factor of an object that determines the new measure of the object. To put it in mathematical terms:

Examples of this equation would include the one-dimensional line, the two-dimensional square, and the three-dimensional cube:

½ = ½^1

¼ = ½^2

1/8 = ½^3

Now this equation can be used to define the dimensionality of a given fractal. Let’s try Sierpinski’s Triangle again.

Here we can see that the triangle as a whole is made from three smaller versions of itself, each of which is scaled down by half of the original (this is proven by each side of the smaller triangles being half the length of the side of the whole triangle). So now we can just plug in the numbers to our equation and leave the dimension slot blank.

1/3 = ½^D

To solve for D, we need to know what power ½ must be put to in order to get 1/3. To do this, we can use logarithms (quick note: in this case, we can replace ½ with 2 and 1/3 with 3).

log_2(3) = roughly 1.585

So we can conclude that Sierpinski’s triangle is 1.585-dimensional. Now we can repeat this process with many other fractals. For example, this Sierpinski-esque square:

It’s made up of eight smaller versions of itself, each of which is scaled down by one-third. Plugging this into the equation, we get

1/8 = 1/3^D

log_3(8) = roughly 1.893

So we can conclude that this square fractal is 1.893-dimensional.

We can do this on this cubic version of it, too:

This cube is made up of 20 smaller versions of itself, each of which is scaled down by 1/3.

1/20 = 1/3^D

log_3(20) = roughly 2.727

So we can conclude that this fractal is 2.727-dimensional.

New Research Heading to Earth’s Orbiting Laboratory

It’s a bird! It’s a plane! It’s a…dragon? A SpaceX Dragon spacecraft is set to launch into orbit atop the Falcon 9 rocket toward the International Space Station for its 12th commercial resupply (CRS-12) mission August 14 from our Kennedy Space Center in Florida.

It won’t breathe fire, but it will carry science that studies cosmic rays, protein crystal growth, bioengineered lung tissue.

Here are some highlights of research that will be delivered:

I scream, you scream, we all scream for ISS-CREAM! 

Cosmic Rays, Energetics and Mass, that is! Cosmic rays reach Earth from far outside the solar system with energies well beyond what man-made accelerators can achieve. The Cosmic Ray Energetics and Mass (ISS-CREAM) instrument measures the charges of cosmic rays ranging from hydrogen to iron nuclei. Cosmic rays are pieces of atoms that move through space at nearly the speed of light

The data collected from the instrument will help address fundamental science questions such as:

Do supernovae supply the bulk of cosmic rays?

What is the history of cosmic rays in the galaxy?

Can the energy spectra of cosmic rays result from a single mechanism?

ISS-CREAM’s three-year mission will help the scientific community to build a stronger understanding of the fundamental structure of the universe.

Space-grown crystals aid in understanding of Parkinson’s disease

The microgravity environment of the space station allows protein crystals to grow larger and in more perfect shapes than earth-grown crystals, allowing them to be better analyzed on Earth. 

Developed by the Michael J. Fox Foundation, Anatrace and Com-Pac International, the Crystallization of Leucine-rich repeat kinase 2 (LRRK2) under Microgravity Conditions (CASIS PCG 7) investigation will utilize the orbiting laboratory’s microgravity environment to grow larger versions of this important protein, implicated in Parkinson’s disease.

Defining the exact shape and morphology of LRRK2 would help scientists to better understand the pathology of Parkinson’s and could aid in the development of therapies against this target.

Mice Help Us Keep an Eye on Long-term Health Impacts of Spaceflight

Our eyes have a whole network of blood vessels, like the ones in the image below, in the retina—the back part of the eye that transforms light into information for your brain. We are sending mice to the space station (RR-9) to study how the fluids that move through these vessels shift their flow in microgravity, which can lead to impaired vision in astronauts.

By looking at how spaceflight affects not only the eyes, but other parts of the body such as joints, like hips and knees, in mice over a short period of time, we can develop countermeasures to protect astronauts over longer periods of space exploration, and help humans with visual impairments or arthritis on Earth.

Telescope-hosting nanosatellite tests new concept

The Kestrel Eye (NanoRacks-KE IIM) investigation is a microsatellite carrying an optical imaging system payload, including an off-the-shelf telescope. This investigation validates the concept of using microsatellites in low-Earth orbit to support critical operations, such as providing lower-cost Earth imagery in time-sensitive situations, such as tracking severe weather and detecting natural disasters.

Sponsored by the ISS National Laboratory, the overall mission goal for this investigation is to demonstrate that small satellites are viable platforms for providing critical path support to operations and hosting advanced payloads.

Growth of lung tissue in space could provide information about diseases

The Effect of Microgravity on Stem Cell Mediated Recellularization (Lung Tissue) uses the microgravity environment of space to test strategies for growing new lung tissue. The cells are grown in a specialized framework that supplies them with critical growth factors so that scientists can observe how gravity affects growth and specialization as cells become new lung tissue.

The goal of this investigation is to produce bioengineered human lung tissue that can be used as a predictive model of human responses allowing for the study of lung development, lung physiology or disease pathology.

These crazy-cool investigations and others launching aboard the next SpaceX #Dragon cargo spacecraft on August 14. They will join many other investigations currently happening aboard the space station. Follow @ISS_Research on Twitter for more information about the science happening on 250 miles above Earth on the space station.  

Watch the launch live HERE starting at 12:20 p.m. EDT on Monday, Aug. 14!

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Cassini Spacecraft: Top Discoveries

Our Cassini spacecraft has been exploring Saturn, its stunning rings and its strange and beautiful moons for more than a decade.

Having expended almost every bit of the rocket propellant it carried to Saturn, operators are deliberately plunging Cassini into the planet to ensure Saturn’s moons will remain pristine for future exploration – in particular, the ice-covered, ocean-bearing moon Enceladus, but also Titan, with its intriguing pre-biotic chemistry.

Let’s take a look back at some of Cassini’s top discoveries:  

Titan

Under its shroud of haze, Saturn’s planet-sized moon Titan hides dunes, mountains of water ice and rivers and seas of liquid methane. Of the hundreds of moons in our solar system, Titan is the only one with a dense atmosphere and large liquid reservoirs on its surface, making it in some ways more like a terrestrial planet.

Both Earth and Titan have nitrogen-dominated atmospheres – over 95% nitrogen in Titan’s case. However, unlike Earth, Titan has very little oxygen; the rest of the atmosphere is mostly methane and traced amounts of other gases, including ethane.

There are three large seas, all located close to the moon’s north pole, surrounded by numerous smaller lakes in the northern hemisphere. Just one large lake has been found in the southern hemisphere.

Enceladus

The moon Enceladus conceals a global ocean of salty liquid water beneath its icy surface. Some of that water even shoots out into space, creating an immense plume!

For decades, scientists didn’t know why Enceladus was the brightest world in the solar system, or how it related to Saturn’s E ring. Cassini found that both the fresh coating on its surface, and icy material in the E ring originate from vents connected to a global subsurface saltwater ocean that might host hydrothermal vents.

With its global ocean, unique chemistry and internal heat, Enceladus has become a promising lead in our search for worlds where life could exist.

Iapetus

Saturn’s two-toned moon Iapetus gets its odd coloring from reddish dust in its orbital path that is swept up and lands on the leading face of the moon.

The most unique, and perhaps most remarkable feature discovered on Iapetus in Cassini images is a topographic ridge that coincides almost exactly with the geographic equator. The physical origin of the ridge has yet to be explained…

It is not yet year whether the ridge is a mountain belt that has folded upward, or an extensional crack in the surface through which material from inside Iapetus erupted onto the surface and accumulated locally.

Saturn’s Rings

Saturn’s rings are made of countless particles of ice and dust, which Saturn’s moons push and tug, creating gaps and waves.

Scientists have never before studied the size, temperature, composition and distribution of Saturn’s rings from Saturn obit. Cassini has captured extraordinary ring-moon interactions, observed the lowest ring-temperature ever recorded at Saturn, discovered that the moon Enceladus is the source for Saturn’s E ring, and viewed the rings at equinox when sunlight strikes the rings edge-on, revealing never-before-seen ring features and details.

Cassini also studied features in Saturn’s rings called “spokes,” which can be longer than the diameter of Earth. Scientists think they’re made of thin icy particles that are lifted by an electrostatic charge and only last a few hours.  

Auroras

The powerful magnetic field that permeates Saturn is strange because it lines up with the planet’s poles. But just like Earth’s field, it all creates shimmering auroras.

Auroras on Saturn occur in a process similar to Earth’s northern and southern lights. Particles from the solar wind are channeled by Saturn’s magnetic field toward the planet’s poles, where they interact with electrically charged gas (plasma) in the upper atmosphere and emit light.  

Turbulent Atmosphere

Saturn’s turbulent atmosphere churns with immense storms and a striking, six-sided jet stream near its north pole.

Saturn’s north and south poles are also each beautifully (and violently) decorated by a colossal swirling storm. Cassini got an up-close look at the north polar storm and scientists found that the storm’s eye was about 50 times wider than an Earth hurricane’s eye.

Unlike the Earth hurricanes that are driven by warm ocean waters, Saturn’s polar vortexes aren’t actually hurricanes. They’re hurricane-like though, and even contain lightning. Cassini’s instruments have ‘heard’ lightning ever since entering Saturn orbit in 2004, in the form of radio waves. But it wasn’t until 2009 that Cassini’s cameras captured images of Saturnian lighting for the first time.

Cassini scientists assembled a short video of it, the first video of lightning discharging on a planet other than Earth.

Cassini’s adventure will end soon because it’s almost out of fuel. So to avoid possibly ever contaminating moons like Enceladus or Titan, on Sept. 15 it will intentionally dive into Saturn’s atmosphere.

The spacecraft is expected to lose radio contact with Earth within about one to two minutes after beginning its decent into Saturn’s upper atmosphere. But on the way down, before contact is lost, eight of Cassini’s 12 science instruments will be operating! More details on the spacecraft’s final decent can be found HERE.

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Every. Single. One. Of. You.

That’s the secret of programming.