Draw a straight line, and then continue it for the same length but deflected by an angle. If you continue doing this you will eventually return to roughly where you started, having drawn out an approximation to a circle. But what happens if you increase the angle of deflection by a fixed amount at each step? The curve will spiral in on itself as the deflection increases, and then spiral out when the deflection exceeds a half-turn. These spiral flourishes are called Euler spirals. [code]
Showing posts tagged with “math”
Congruent Triangles - Bruce Cornwell (1976)
The Fourier transform takes an input function f (in red) in the “time domain” and converts it into a new function f-hat (in blue) in the “frequency domain”.
In other words, the original function can be thought of as being “amplitude given time”, and the Fourier transform of the function is “amplitude given frequency”.
Shown here, a simple 6-component approximation of the square wave is decomposed (exactly, for simplicity) into 6 sine waves. These component frequencies show as very sharp peaks in the frequency domain of the function, shown as the blue graph. In practice, these peaks are never that sharp. That would require infinite precision.
I’m not too happy with this one yet. It’s a bit of a mixture of Fourier series and Fourier transform. The animation could also be a bit smoother in some steps. I may tweak it later.
Graphs of six convex regular 4-polytopes
In geometry, a polychoron or 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells.
The two-dimensional analogue of a polychoron is a polygon, and the three-dimensional analogue is a polyhedron. The term polychoron (plural polychora), from the Greek roots poly (‘many’) and choros (‘room’ or ‘space’) and has been advocated by Norman Johnson and George Olshevsky, but it is little known in general polytope theory. Other names for polychoron include: polyhedroid and polycell.
Topologically 4-polytopes are closely related to the uniform honeycombs, such as the cubic honeycomb, which tessellate 3-space; similarly the 3D cube is related to the infinite 2D square tiling. Convex 4-polytopes can be cut and unfolded as nets in 3-space.
Phase Patterns: Autechre’s Gantz Graf
geometric systems are most important
"Is the greater body of Stanley Kubrick’s films an exposé of a hidden elite obsessed with dark Saturnian sexual rites, paedophilia and the planned ritualistic transmutation of mankind?”
2001: A Space Odyssey: Kubrick’s undoubted masterpiece and at the centre of Jay Weidner’s alchemical take on Kubrick’s films. As discussed above, Weidner claims this film to be Kubrick’s attempt at an initiation rite of passage illustrating humanity’s next planned-for evolutionary step. Weidner believes Kubrick has revealed an occult agenda to launch us as a species into space where we will be able to embrace our destiny as future star children and join the greater cosmos.
Necropolis at Meroe