Soap Films and the Minimal Surface
One subject of particular interest to me is soap films, while they hold wonder for children they’re also amazing in scientific terms. The shape and structure of a soap films is determined by what configuration minimizes surface area, this is why bubbles are round. However other interesting shapes known as minimal surfaces arise such as the catenoid and helicoid. The catenoid is the shape formed by rotating a caternary around it’s axis of symmetry, the catenary in turn is the shape formed by a hanging chain. The helicoid is a minimal surface that can be formed from a catenoid without any deformation or stretching. Both of these shapes (along with the plane) have zero mean curvature and also minimize surface area and as such are energetically favorable shapes for soap films (with boundaries) to exist in.
Image: 1, 2, 3, 4
An Overview of Chaos Theory
In its simplest form chaos theory simply describes a system that is very sensitive to initial conditions. That’s not so hard is it? The term is often used in reference to the butterfly effect which postulates that something so seemingly insignificant as a butterfly flapping its wings can generate a storm on the other side of the planet. While that may be a bit of an overstatement, many real life systems, including the weather, display chaotic behavior. The effect is best seen in a double pendulum system (a pendulum attached to the end of another pendulum) which generates the image above (a long exposure image tracked with LEDS). Small changes, such as changing the angle between the two pendulums when released, would generate a gratuitous amount of widely different pathways.
Image source
The Platonic Solids
A Platonic solid is a convex geometric shape made entirely of congruent, regular 2D sides. This may seem like it leaves a lot of room for experimentation, but no matter what you try there are only 5: the tetrahedron, hexahedron (cube), octohedron, dodecahedron and the icosahedron (shown in order). The name comes from the famous Greek philosopher Plato, who described the classical elements in terms of these polyhedra with cubes representing earth, tetrahedra as fire, air by octahedra and the icosahedron represented water. The dodecahedron was said to be the material that made the Heavens, an idea that would go on to become that of the aether.
The German astronomer Johannes Kepler also held a fascination with them, believing that the 5 known planets some how corresponded to the 5 platonic solids. As such he became obsessed with forming a model of the solar system where the orbits of the planets were defined according to the geometries of these shapes. Kepler was however forced to abandon this idea but these concepts gave rise to the discovery of elliptical orbits.
The other cool thing about Platonic solids is that each has a dual pair that is another Platonic solid. This means that the vertices of one shape correspond to the faces of another, for example the cube and the octahedron. Some Platonic solids (the cube, octahedron and tetrahedron) also form the basis of crystal structures along with being the body shape of several species of radiolarian (smaller protozoa that form their own mineral skeletons).
The Hexagon
Hexagons have to be one of my favorite shapes, it’s hard to say why but I just think they’re neat. They have internal angles that add up to 720 or 4 lots of pi for those who prefer radians (everyone). The other neat thing is that the length between a vertex and the one opposite is twice the length of one of the sides in a regular hexagon. This in turn means that hexagons can be constructed from equilateral triangles. The area for a hexagon is given by the formula A=((3√3)/2)t^2 where t is the length of one of the sides, or alternatively simply A=1.5dt where d is the length between parallel sides. The elegant construction discovered by Euclid is also pictured.
Hexagons also pop up a lot in nature, being the shape of honey combs (for it’s space optimizing ability), igneous basalt columns, crystal structures, benzene rings, snow flakes and the clouds on Saturn’s north pole.
Mathematical Mollusks
To me it’s always incredibly amazing and beautiful when something rooted firmly in mathematics shows up in nature. When that mathematical entity is also incredibly obscure and complex it gets even better. The two images above show similar (but not quite identical) patterns, one is known as Rule 30 while the other is the pattern on the shell of Conus textile, a marine gastropod. Rule 30 is a cellular automaton which is a mathematical tool used to see the change in a system over time. In these a cell can exist in several states dependent on certain variables, in rule 30 there are two states a cell can be in that is determined by the state of adjacent cells. Over time rows are added and a pattern begins to emerge that exhibits chaotic behavior. A similar pattern can also be seen here on the shell of C. textile, a venomous marine gastropod.
The Inherent Beauty of Geometry
To me geometry is absolutely beautiful, I find small mathematical quirks such as the relation of volume to surface area of a sphere (simple derivation of the formula) to be astounding but the following is just extraordinary.
The above pentagram is pretty cool, it’s set inside a pentagon and makes up a series of isosceles triangles using sets of parallel lines (marked b) all the while making a second, inverted regular pentagon in the middle, but it gets better. The pentagram includes the golden ratio in every aspect of its design: the ratio of the length of a to b is equal to the golden ratio ((1+√5)/2 ≈ 1.618). However it doesn’t stop there, similarly the ratios b:c and c:d are also golden.
To summarize: a:b = b:c = c:d = (1+√5)/2, the golden ratio.
Real graphs have curves.
Molecular Fractals
Many of you will be familiar with the concept of fractals: shapes that have the same form at many different levels. Whilst being common in mathematics and the macroscopic world, nanoscale ones are harder to make and virtually nonexistent. This large molecule made from many ruthenium and iron complexes however is one such molecular fractal. It comes in the hexagonal equivalent to the Sierpinski triangle and was made at the University of Akron.
Super-Heavy Hydrogen Atoms
Who ever said that alchemy wasn’t possible? Well, on the atomic scale - it’s been done.
A Helium atom consists of a nucleus and two electrons orbiting said nucleus. The nucleus is composed of two protons and two neutrons. On the other hand, a Hydrogen atom has only one proton and one electron. Typically, it is pretty simple to distinguish the two from each other!
Meet the muon. The Greek letter mu, where the particle derives its name, also serves as it’s symbol - μ. Muons are very similar to electrons since they also have a negative charge - but muons are far more massive than electrons. If a muon is replaced with one of the electrons in a helium nucleus, the muon sits about 200 times closer to the helium nucleus than an electron would (due to all of its mass.) Since it is so close to the center, the negative charge of the Muon effectively cancels out the positive charge of one of the protons in the nucleus. Thus, the remaining electron continues orbiting the nucleus and since the positive charge in the nucleus is now only one, the atom behaves just like a regular hydrogen atom - even though it is 4.1 times heavier than normal!
For a video description of this, check this out!
Richard Feynman: Physics is Fun to Imagine.
If you have some time to spare, I can’t recommend this series enough. Richard Feynman was nothing short of a genius, and these videos are enthralling to say the absolute least. The short chats of a few minutes each show Feynman at his best. His wit and charm shine through as he forms fascinating new pictures of the world around us.
I recommend these also. Feynman is incredibly underrated…
I’ve never really been asked “what is your favorite shape?” but when someone finally does I am sure as hell going to have the most awesome reply: this thing. This thing is in fact known as an umbilic torus and it totally one ups the good old Möbius strip. First off a regular torus for those of you who don’t know is basically a donut shape, but the umbilic torus is a hell of a lot more interesting. The thing that’s really cool about this thing is that like the Möbius strip it really only has one side. If you start at one point and move along the side you will eventually get back to the starting point after “going around” the torus three times.
Gibb’s Free Energy
Decided to do some more of that post-on-things-I-need-to-study. And what I really need to study is chemistry, something I absolutely abhor. So decided to start with Gibb’s free energy. Yay…
Okay anyways, Gibb’s free energy is a way of calculating whether or not a reaction is spontaneous or not. This means whether or not the reaction will occur without addition of energy or catalysts or anything else. The general formula for it is: ΔG = ΔH - TΔS Where G is the actual Gibb’s free energy value, H is enthalpy (Which can be thought of as heat or energy), T which is temperature and S that is entropy which is really quite confusing but is basically a measure of all the states particles in a system can be in. Now if all the values are inserted and ΔG = a negative number the reaction is spontaneous, meaning it will occur without any addition of energy or other things. Conversely if the number is positive it simply will not occur.
It sounds a lot simpler when I do that. I’m probably missing something crucial, oh well.
