Physics

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[Laurin Gschnitzer @LGschnitzer]

Acetylcholine

[Julia Gomez @microfusio91945]

Putnam Mathematics Competition from 1980

[ScieVision @scievision369]

Euler's Formula for the Cosine Function ✍️

Mathematics has many branches that seem completely unrelated. Trigonometry deals with angles and waves. Exponential functions focus on growth and decay. Complex numbers involve the square root of negative one. For centuries, these three areas developed independently with no clear connection. Euler's formula showed something remarkable. These three different mathematical worlds are not separate at all; they are different faces of the same underlying reality. Cosine describes anything that oscillates smoothly and repeatedly. This includes sound waves, light, ocean tides, pendulum swings, and electrical current. The easiest way to visualize it is as a point moving around a perfect circle at a steady speed. If you observe only its horizontal shadow on a flat surface, it slides smoothly left and right in a perfect, endless rhythm. That sliding shadow represents the cosine function. Euler's number, approximately 2.718, is the natural constant that appears whenever things grow or decay at a rate proportional to their current size. Normally, raising it to a power leads to growth or decay. But when the power involves the imaginary unit, the square root of negative one, something completely different occurs. Instead of growing or shrinking, the result rotates and traces a perfect circle in the complex number plane. The formula states that cosine is the average of two such rotations moving in opposite directions. Adding a counterclockwise rotation and a clockwise rotation together cancels their vertical components perfectly since they are always equal and opposite. This leaves only the horizontal component, which is exactly the cosine. The smooth back-and-forth oscillation of cosine is literally the shadow created by two points spinning in opposite circles. This reveals a profound truth: oscillation and rotation are the same thing seen from different views. Trigonometry and exponentials represent the same mathematics in different forms. This unification is not only beautiful; it forms the foundation of all modern signal processing, electrical engineering, quantum mechanics, and telecommunications. It works invisibly every time you make a phone call, stream music, or use wireless technology.

[María Lípiz @matessindramas]

Momento angular

[Cosmos Archive @cosmosarcive]

Einstein’s greatest insight was that gravity is geometry.

Massive objects warp the fabric of spacetime, and everything from planets to beams of light follows those curves. This is why starlight bends around galaxies, why black holes trap light, and why gravity shapes the structure of the universe itself.

[Mathematica @mathemetica]

Taylor series expand any smooth function around a point a using its derivatives at that point.

The formula is f(x) = Σ_{n=0}^∞ f^{(n)}(a)/n! (x-a)^n.

The graphic defines the components on the left and shows on the right how the linear (blue, n=1), quadratic (green, n=2), cubic (purple, n=3) and higher (orange) approximations approach the red curve y=f(x) near a. Color-coded terms below match the orders from constant to higher.

In real life, it is used to approximate functions in physics, engineering calculations, and computer algorithms for efficient evaluations.

[idempotent @op_idempotent]

Prove γ is irrational.

[ScieVision @scievision369]

Minkowski light cone diagram ✍️

Einstein discovered that space and time are not separate things... they are woven together into a single fabric called spacetime. Hermann Minkowski turned this idea into a geometric picture, resulting in the light cone diagram. The entire diagram centers around a single point representing an event, anything from a light bulb turning on to two particles colliding. Every question the diagram answers radiates outward from that moment. The shape of the cones is based on one absolute rule: nothing in the universe can travel faster than light. Imagine a flash of light erupting from the central event in every direction at once. As time goes on, that expanding sphere of light traces the surface of a cone through spacetime. The blue cone above the central event is the future light cone. Everything inside it is a place and time that the central event could potentially reach and influence. The red cone below is the past light cone. Everything inside it could have sent a signal or influence to the central event in time. Together, these two cones define the entire boundary of cause and effect for that moment. Everything outside both cones, the flat region spreading horizontally, is called spacelike territory. These locations are so far away in space that even light cannot bridge the gap in time, meaning no signal or influence can connect them to the central event. They are causally invisible to each other. Strangely, observers moving at different speeds will genuinely disagree about the order of events in this region, and all of them are equally correct. There is no universal answer to which happened first when two events have no causal connection. The world line is the path any object traces through spacetime as time goes on. It always threads upward through the interior of the future cone, never tilting beyond its surface. The event horizon marks the boundary where causal contact becomes impossible, most famously at the edge of a black hole. The particle horizon marks the edge of everything we can ever observe. It is not the edge of the universe itself but the edge of the part that has had enough time to send us any light since the Big Bang. In one elegant picture, the light cone diagram maps the entire structure of cause and effect in the universe.

[H.O.(Japan) @fun_Fibonacci]

Fibonacci Quart. Problem B-1198 by H.O.