Physical Science...In Space!

Each trig function in terms of the other five: color coded. Idk I’m just playin’ around in LaTeX and thought this table on Wikipedia would be worth color coding on its own so I made this little ditty. The reciprocal functions are the lighter colors of their regular trig counterparts. I wouldn’t call this a final version, but, hey, it’s pretty cute. High five for pattern recognition.

Can you flatten a sphere?

The answer is NO, you can not. This is why all map projections are innacurate and distorted, requiring some form of compromise between how accurate the angles, distances and areas in a globe are represented.

This is all due to Gauss’s Theorema Egregium, which dictates that you can only bend surfaces without distortion/stretching if you don’t change their Gaussian curvature.

The Gaussian curvature is an intrinsic and important property of a surface. Planes, cylinders and cones all have zero Gaussian curvature, and this is why you can make a tube or a party hat out of a flat piece of paper. A sphere has a positive Gaussian curvature, and a saddle shape has a negative one, so you cannot make those starting out with something flat.

If you like pizza then you are probably intimately familiar with this theorem. That universal trick of bending a pizza slice so it stiffens up is a direct result of the theorem, as the bend forces the other direction to stay flat as to maintain zero Gaussian curvature on the slice. Here’s a Numberphile video explaining it in more detail.

However, there are several ways to approximate a sphere as a collection of shapes you can flatten. For instance, you can project the surface of the sphere onto an icosahedron, a solid with 20 equal triangular faces, giving you what it is called the Dymaxion projection.

The Dymaxion map projection.

The problem with this technique is that you still have a sphere approximated by flat shapes, and not curved ones.

One of the earliest proofs of the surface area of the sphere (4πr2) came from the great Greek mathematician Archimedes. He realized that he could approximate the surface of the sphere arbitrarily close by stacks of truncated cones. The animation below shows this construction.

The great thing about cones is that not only they are curved surfaces, they also have zero curvature! This means we can flatten each of those conical strips onto a flat sheet of paper, which will then be a good approximation of a sphere.

So what does this flattened sphere approximated by conical strips look like? Check the image below.

But this is not the only way to distribute the strips. We could also align them by a corner, like this:

All of this is not exactly new, of course, but I never saw anyone assembling one of these. I wanted to try it out with paper, and that photo above is the result.

It’s really hard to put together and it doesn’t hold itself up too well, but it’s a nice little reminder that math works after all!

Here’s the PDF to print it out, if you want to try it yourself. Send me a picture if you do!

If you put that ball on that machine while it wasn’t spinning, it would just roll straight down the lower sides. 

The raised edges would keep it in the middle line, but it’s only controlled in one direction. By spinning it, you constantly alternate the position of the tall sides, meaning that the ball is held in the middle, never able to fall off.

Particle accelerators control particles in the same way. Magnetic or electric fields can only direct particles in one plane at a time, so to keep a beam of particles rushing down a particle accelerator in one focused stream, the current gradient must constantly oscillate. This means the particles are constantly held in place, never able to shoot off in one direction.

Here’s the same principle in action: these are tiny pollen grains being held in place by an oscillating field. Rods in the four corners of the beam establish a field that oscillates many times a second to keep the pollen trapped. If it didn’t constantly switch, the pollen would all fly off in one direction.

Watch the full film with Dr Suzie Sheehy for more.

🤐

Ferns

Ferns are more advanced primitive vascular plants. They have true roots, stems, and leaves. Ferns do not have seeds, but produce spores instead. Fern’s lifecycle, unlike primitive non-vascular plants such as bryophytes, is dominated by the sporophyte generation. Some interesting terminology associated with ferns are frond, pinnae, sori, rhizome, and fiddlehead. A frond describes the entire blade of the fern and the smaller individual leaflets are called pinnae. Sori describes clusters of sporangia on the underside of the pinnae (Sori are depicted in the second photo). A rhizome is an underground stem that puts out shoots and adventitious roots. Fiddleheads are furled fronds of a young fern.

Dark Energy

Dark energy is in physical cosmology and astrology as an unknown form of energy that permeates through space making up for 68.3% of the observable universe (dark matter takes up 26.8%, only 4.9% is ordinary matter). Dark energy is currently the most accepted hypothesis to explain why the universe is expanding at an accelerated rate (if the observable universe comprised of only the ordinary matter we see, the universe would be expanding at a much slower rate than it currently is thus meaning dark matter and dark energy make up the mass for the accelerated rate of expansion). On a mass–energy equivalence basis, the density of dark energy (6.91 × 10^−27 kg/m3) is very low, much less than the density of ordinary matter or dark matter within galaxies.

Dark energy is not known to react with any of the fundamental forces other than gravity. It clearly has a large impact on the universe making up for 68.3% of universal density, only because it fills an otherwise empty space. The two current leading models are a cosmological constant and quintessence. Both models conclude that dark energy must have a negative pressure.

The effect of dark energy: a small constant negative pressure of vacuum

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