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
Elliptical Orbits
Just about every popular depiction I’ve ever seen of the solar system has the planets either in a straight line or in circular paths. In fact one humorous (but not satirical) book I own entitled LIFE HOW DID IT GET HERE? By evolution or by creation? goes so far as to compare the orbits of planets to the “orbit” of electrons around around a nucleus (the Bohr model no less) as evidence of God’s existence and apparent intelligence. In truth the planets do not follow these nice circular orbits instead they’re drawn out into ellipses. While on Earth this isn’t noticeable (our orbit only has an eccentricity of about 0.02) with comets and protoplanets such as Pluto this is hugely noticeable.
The idea of an elliptical orbit was first put forward by Johannes Kepler and was a major shock to astronomy at the time as it was believed that circular orbits were “perfect” and infallible. Within the orbit itself the larger of the bodies is situated on one of the ellipse’s foci. The other foci is empty space, but is the center of gravity determined by other objects such as planets or even distant galaxies, in fact by everything in the universe. The elliptical orbit also has the interesting affect of causing the orbiting body to change speed depending on where it is in its orbit. This means that when the planet or comet is closest to the sun (a point known as the perihelion) it is traveling faster than it is at the point farthest away (the aphelion).
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).
When Science and Art Collide
I don’t always post about art, but when I do it’s pretty damn awesome. This kite is called “Three Cubes Colliding” and was conceived and designed by Ivan Morison and Sash Reading. The kite itself is made predominantly from aerospace fabric and more than 1700 connectors made using a 3D printer. Its design was inspired by the structure of pyrite along with Alexander Graham Bell’s (the inventor of the telephone) experiments and has a certain “bucky quality” to it.
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.
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.
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.
Foucault’s Pendulum
It’s always nice to be reminded that we’re on a giant ellipsoid floating through space that is spinning at 465 m/s and what better way to do this with a bit of classical physics involving a pendulum. At the base level a Foucault’s Pendulum is no different from an ordinary pendulum but serves more of an illustrative purpose. The bob at the end of course has it’s own momentum which causes it to oscillate back and forth. The thing is that if left for long enough with no decay in amplitude (done by using a very massive pendulum) we notice something peculiar. As the day goes on it appears that the angle of the pendulum’s swing changes (as can be seen in this animation). In reality the momentum of the pendulum keeps it oscillating back and forth relative to the universe, the change in angle is actually due to the Earth rotating while the pendulum remains swinging in more or less the same plane. As it does this it traces out what is known as a Rose Curve (the type of shape you get using spirographs) when viewed from above.
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 it 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.
(via capntrips)
The Logarithmic Spiral
Now all you guys who are like “Yeah man the Fibonacci spiral is awesome” can just take a back seat here, because here we have the coolest of all spirals: the logarithmic spiral. Truth be told just about every time you’ve heard someone talk about the Fibonacci (or more accurately known Golden Spiral) they’ve been talking about this guy and just not realized it. The logarithmic spiral is given by the equation r=ae^(bθ) where r is the radius, a & b are positive constants and θ is the angle around the origin.
The logarithmic spiral also pops up quite often in nature, being the mathematical pattern behind such things as nautilus shells, Romanesco broccoli, spiral galaxies, the Mandelbrot set, storms, ferns and even sea horses.
Helicenes
Taking some time to talk about some molecular architecture. Here we have a particularly cool kind of organic molecule known as a helicene. A helicine is basically a broken ring of benzene molecules and because of geometric constraints and electrostatic repulsion it forms a helix. The other really cool thing about helicenes is that they display chirality despite not having an asymmetric carbons or a chiral center.
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.
