#sphere

#silhouette #blackandwhitephotography #rsa_bnw #nikonitalia

https://www.instagram.com/p/Da2XKWPM9dt/?igsh=MWhtN2F2emg3M3Riag==

S P H E R E

Have been trying to learn to colour/colour mix properly; I present to you the S P H E R E (it’s a sphere and it’s round that’s basically it) drew it with just red, yellow and blue.

Spherical geometry is highly complex because its coordinate grid warps and changes depending on your exact position. Unlike uniform Cartesian grids, a fixed angular step on a sphere spans vastly different physical distances based on your location—covering miles at the equator but shrinking to nothing near the poles. As an object moves, its reference direction arrows constantly tilt and rotate. This means calculating basic physical behavior like acceleration becomes a major mathematical headache, as you must simultaneously track the object's path and the continuous twisting of the coordinate system itself.

Further, this curvature creates geometric dead ends at the poles where directional coordinates collapse and lose meaning entirely. Because all lines of longitude converge to a single point, standard computer grid simulations can glitch or require intense computational fixes to remain accurate. Because the grid lines are curved and interconnected rather than flat and independent, standard physics equations for waves or fluid flow explode in size. Simple straight-line calculus formulas transform into massive, multi-tiered equations filled with non-linear sines and cosines that require specialized mathematical functions to solve.

Polynomials enter the picture when we try to solve these complex, curved physics equations using a classic mathematical technique called separation of variables. In a flat Cartesian system, this technique splits a big differential equation into separate, simple pieces that can be solved using basic trigonometric polynomials (like sines and cosines). However, because spherical grid lines stretch and warp, separating a spherical equation leaves behind a messy, variable-dependent remnant that standard sines and cosines cannot handle.

To fix this, mathematicians must use highly specialized, higher-order polynomials—specifically Legendre polynomials and Associated Legendre functions—which are uniquely designed to scale and counteract the sphere's non-linear curvature. These polynomials essentially act as the specialized mathematical shape-shifters needed to smooth out the wrinkling of the spherical grid, allowing us to precisely model things like a planet's gravitational field or the electron orbitals of an atom.

In addition, spherically shaped physical plasma probes, for example, exhibit a different behavior than planar or cylindrical ones. When they start collecting charges from the periphery, their "sheath" region can expand without saturation. It is for this reason they are used to probe faint plasma in the outer space to gather as much current as possible. Overall, engineers generally avoid working with spherical geometries in the design of, say, multi-phase reactors, where cylindrical geometry is preferred in the design of such columns.

Watercolor caricature by Julien-Léopold Boilly, the only authenticated portrait of Adrien-Marie Legendre.