Ethereum

Wrapped Ether

WETH

The bouncing ball dynamical system gets more complicated when there are two collision surfaces -- in this case, the balls are sandwiched between two oscillating plates. This is a chaotic hybrid dynamical system.

DYNAMECHS : ball bounce sandwich

Reflections on the barycentric subdivision of a 3-simplex. The linear subdivided 3-simplex is in the foreground; nonlinear reflections lie in the background.

FTDA : Barycentric 3-Simplex

Barycentric subdivision is a simple and automatic method to split simplices of a simplicial complex into a new and more detailed complex. As one iterates the construction, simplices become finer and finer.

FTDA : Barycentric Subdivision

OpenSea

Coal is inspired by the music and beauty of Northern Europe's vast and empty landscapes. Its forms and colors are taken from the minerals deep inside the earth, the rocks, and the formation from cold seas. It welcomes the colors, warmth, and action. It breathes of exploration, visual satisfaction, and a new taste for the art.  

Designed by the folks at 494Jax, a cryptorchid of Transparent House

Coal

Wrapped Ether 

Visit the space at oncyber.io/destinations/openskygallery. Open Sky Gallery, your personal roofless gallery. Showcase your creation or collection in a quiet and cloudy place.

Open Sky Gallery

A point cloud connected via an increasing distance parameter leads to a nested sequence of simplicial complexes called a Vietoris-Rips filtration.

FTDA : VR-Filtration

Every finite simplicial complex is a subcomplex of a single simplex of sufficiently high dimension.

FTDA : SubSimplex

The Sitnikov problem is a restricted 3-body problem in gravitational dynamics. Two equal massive bodies orbit each other on ellipses. A low-mass body is placed on the line of symmetry orthogonal to the orbital plane at the center. By symmetry, this mass remains on the line, bouncing up and down. The dynamics can be chaotic, with seemingly random bounce sequences of varying amplitudes.

DYNAMECHS : sitnikov orbits

a quasisymmetric design based on a penrose quasiperiodic tiling of the plane

QUASI : crystal 13

QUASI : crystal 5

QUASI : crystal 12

QUASI : crystal 9

QUASI : crystal 2

QUASI : crystal 7

QUASI : crystal 3

QUASI : crystal 11

QUASI : crystal 4

QUASI : crystal 6

QUASI : crystal 1

QUASI : crystal 8

QUASI : crystal 10

manifold.xyz

A ball bouncing on an oscillating table gives a simple physical example of a chaotic dynamical system. This can be analyzed as a 2-D discrete-time system on the (impact-velocity, time-between-impacts) plane. The chaotic nature of the dynamics occurs independent of whether the collisions are elastic or inelastic. There are numerous periodic orbits, corresponding to initial conditions that sync with the timing of the table's oscillation.

What happens when you bounce a ball on an oscillating table? Chaos happens. This is demonstrated very well by changing the initial conditions (say, height or radius) and noting how differently the dynamics appear. This sensitive dependence on initial conditions is a hallmark of chaos.

Identical pendula with different initial conditions evolve symmetrically and pleasingly. Note the illustration of equilibrium solutions both stable (at the bottom) and unstable (at the top). Though this is not a chaotic dynamical system, it is still nonetheless, visually compelling.

A pendulum swinging back and forth is not chaotic, but if you add the right amount of forcing, in this case a horizontal shaking, then, depending on the length and mass of the pendulum, the dynamics can become very complicated. This video demonstrates that sensitive dependence. Each forced pendulum is a three-dimensional continuous-time dynamical system.

Repelling magnets embedded in fidget-spinners leads to a coupled system with slightly chaotic dynamics. It sometimes seems periodic, but then veers off to a new direction. The dynamics takes place on the tangent bundle of a torus (two angular variables and two angular velocity variables).

Repelling magnets embedded in fidget-spinners leads to a coupled system with slightly chaotic dynamics. It sometimes seems periodic, but then veers off to a new direction. The dynamics of five coupled spinners takes place on a 10-dimensional space, with 5 angular variables and 5 angular velocity variables.

Repelling magnets embedded in fidget-spinners leads to a coupled system with slightly chaotic dynamics. It sometimes seems periodic, but then veers off to a new direction. The dynamics of these 20 coupled spinners takes place on a 40-dimensional space; nevertheless, the system is not wild or erratic. This is a gently chaotic dynamical system.

An inverted pendulum under gravity is an unstable equilibrium. However, if you shake the pendulum vertically, then, past a certain frequency, the unstable equilibrium becomes stable. This seems counterintuitive, since the addition of the shaking at lower frequencies makes the system chaotic. What do you think happens to the stable equilibrium (at the bottom) when you shake things up?

A single planar pendulum is not chaotic, but a double planar pendulum is, with dynamics taking place on a 4-dimensional phase space (two angles and two angular velocities -- the tangent bundle of a torus). The pendulum's bob traces out a beautiful signature of chaos.

This unusual branched pendulum has four end bobs which trace out very complex paths in the plane. The dynamics of this mechanism are very chaotic, owing to the fact that there are a total of ten angles that determine the configuration of this hydra-like pendulum. Combined with the ten angular velocities, this system produces dynamics on a twenty-dimensional space.

One hallmark of chaotic dynamics is sensitive dependence on initial conditions. In the case of a double planar pendulum, one can see this clearly by starting them out with very close initial configurations. Over time, they evolve to completely uncorrelated states.

ad1968.eth

$140.79

Explore profiles

Company

Learn

Individuals

Businesses

Developers

Support