Different modes of oscillation for a pendulum
The period of a simple pendulum is not a trivial thing, and it depends on the initial conditions.
Shown here are ten different modes of oscillation for the same pendulum. The only difference is the total amount of mechanical energy in the system.
As a result, each one has a completely different period of oscillation, unlike what the small-angle approximation (as taught in high-school) would suggest. They can’t be in sync. You may see some really interesting patterns based on the delay between them in your browser.
The red graph above each pendulum represents the phase portrait for the respective mode of oscillation, with the current state marked as a blue dot. The horizontal axis represents angle (hence why it wraps around the sides) while the vertical axis represents angular velocity.
Pendulums are very interesting dynamical systems, as they are relatively simple to understand but can produce surprisingly complex results in certain cases, such as the chaotic behavior of double pendulums and the odd behavior displayed by coupled pendulums.
The familiar trigonometric functions can be geometrically derived from a circle.
But what if, instead of the circle, we used a regular polygon?
In this animation, we see what the “polygonal sine” looks like for the square and the hexagon. The polygon is such that the inscribed circle has radius 1.
We’ll keep using the angle from the x-axis as the function’s input, instead of the distance along the shape’s boundary. (These are only the same value in the case of a unit circle!) This is why the square does not trace a straight diagonal line, as you might expect, but a segment of the tangent function. In other words, the speed of the dot around the polygon is not constant anymore, but the angle the dot makes changes at a constant rate.
Since these polygons are not perfectly symmetrical like the circle, the function will depend on the orientation of the polygon.
More on this subject and derivations of the functions can be found in this other post
Now you can also listen to what these waves sound like.
This technique is general for any polar curve. Here’s a heart’s sine function, for instance
Work by Jason Padgett, a man with Acquired Savant Syndrome who now sees all of reality as mathematical fractals describable by equations.
The beauty of numbers and their connection to the pure geometry of space time and the universe is shown in his fractal diagrams…He is currently studying how all fractals arise from limits and how E=MC2 is itself a fractal. When he first started drawing he had no traditional math training and could only draw what he saw as math. Eventually a physicist saw his drawings and helped him get traditional mathematics training to be able to describe in equations the complex geometry of his drawings. He is currently a student studying mathematics in Washington state where he is learning traditional mathematics so he can better describe what he sees in a more traditional form. Many of the captions were written before he had any traditional math training. His drawing of E=MC^2 is based on the structure of space time at the quantum level and is based on the concept that there is a physical limit to observation which is the Planck length. It shows how at the smallest level, the structure of space time is a fractal…So sit back and enjoy the beauty of naturally occuring mathematics in pure geometric form connecting E=MC2 (energy) to art. All are HAND DRAWN using only a pencil, ruler and compass.
Hydraulic jumps occur when a fast-moving fluid enters a region of slow-moving fluid and transfers its kinetic energy into potential energy by increasing its elevation. For a steady falling jet, this usually causes the formation of a circular hydraulic jump—that distinctive ring you see in the bottom of your kitchen sink. But circles aren’t the only shape a hydraulic jump can take, particularly in more viscous fluids than water. In these fluids, surface tension instabilities can break the symmetry of the hydraulic jump, leading to an array of polygonal and clover-like shapes. (Photo credits: J. W. M. Bush et al.)
Alejandro Guijarro - Momentum (2010-12)
“The artist travelled to the great quantum mechanics institutions of the world and, using a large-format camera, photographed blackboards as he found them. Momentum displayed the photographs in life-size.
Before he walked into a lecture hall Guijarro had no idea what he might find. He began by recording a blackboard with the minimum of interference. No detail of the lecture hall was included, the blackboard frame was removed and we are left with a surface charged with abstract equations. Effectively these are documents. Yet once removed from their institutional beginnings the meaning evolves. The viewer begins to appreciate the equations for their line and form. Color comes into play and the waves created by the blackboard eraser suggest a vast landscape or galactic setting. The formulas appear to illustrate the worlds of Quantum Mechanics. What began as a precise lecture, a description of the physicist’s thought process, is transformed into a canvas open to any number of possibilities.”
1. Cambridge (2011)
2. Stanford (2012)
3. Berkeley I (2012)
4. Berkeley II (2012)
5. Oxford (2011)
Rubbed on like a temporary tattoo these ultra-thin electronics bend and stretch with the skin. Their development paves the way for sensors that monitor heart and brain activity to take the place of bulky equipment and taped-on electrodes. Electronic components shrunk to the size of tiny bumps on the skin are connected with serpentine wires that meander like rivers, straightening rather than snapping when stretched. The whole thing is mounted on a rubbery sheet that mimics the elastic properties of skin. Known as epidermal electronics, the technology can even control computer games from voice commands. Worn on the gamer’s throat, the patches detect the electrical charges associated with the muscle movements of speech. The potential applications of linking electronics and biology in this way seem boundless.
Written by Mick Warwicker
Charles Jencks is the American landscape architect and designer behind this incredible flight of stairs. Called The Universe Cascade, it has 25 landings that mark the important shifts in cosmic history. Starting at the top, in the present day, and descending down, visitors are moving through 13 billion years of cosmic evolution. The steps finally disappear into the dark water below, which represents the mystery of the origin of the universe.
Now, researchers at the Hamburg University of Technology have created a material that beats out both of these ultralight substances handily. They call it Aerographite, and it has a density of less than 0.2 mg/cm3. The researchers grow the material through a novel twist on a synthesis technique known as chemical vapor deposition, a process which gives rise to a network of ethereal-looking, yet surprisingly resilient, hollow carbon microtubes.
Top image by Tuhh, Karl Schulte/DPA/Press Association Images