The Golden Ratio (φ ≈ 1.618) has a rare self-referential identity: φ² = φ + 1.
It's also the limiting ratio of consecutive Fibonacci numbers - divide any Fibonacci number by its predecessor (13/8, 21/13, 34/21....) and the result converges toward φ, though it never exactly reaches it.
No other positive real number satisfies both properties simultaneously.
Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayes' approach to probability was notably different from the frequentist approach, which dominated statistical methods at the time.
Formation of conic sections ✍️
Circle: When the cutting plane is perpendicular to the axis of the cone, the cross-section is a circle.
Ellipse: When the plane is angled, but still cuts through only one cone (without being parallel to the side of the cone), the result is an ellipse.
Parabola: If the plane is parallel to the side (slant) of the cone, it creates a parabola.
Hyperbola: If the plane cuts both halves of the double cone, it forms a hyperbola - two separate curves.
Quantum Field Theory (QFT) unifies quantum mechanics and special relativity, treating particles as excitations of fundamental fields that permeate space. For example, photons are ripples in the electromagnetic field. QFT provides the framework for the Standard Model, explaining fundamental forces and particle interactions.
The holographic principle is a theory in physics suggesting that all the information contained within a volume of space can be represented on the boundary of that space. In essence, our 3D universe might be like a hologram, with the "real" information encoded on a 2D surface at the edges of the universe.
This idea emerged from black hole physics, where the information about objects falling into a black hole seems to be stored on its 2D event horizon. It has since been extended to the universe as a whole, implying that our perception of three dimensions might be a projection of deeper, 2D laws of physics.
Vector fields help us see invisible forces like gravity, magnetism, and wind currents. They use arrows to show direction and strength at every point in space, like a weather map showing wind flow. From planets orbiting to air moving over wings, vector fields help us understand how forces shape our world!
📷 math.animations
Gabriel's Horn, also known as Torricelli's Trumpet, is a geometric figure that has the paradoxical property of having infinite surface area but finite volume. It is formed by revolving the curve y = ¹⁄ₓ around the x-axis from x = 1 to infinity.
A rotating point traces out a wave:✍️
One full turn (2π = 360°) equals one wavelength λ, completed in time t = 1/f. The wave's speed connects them through v = λf.
In so far as theories of mathematics speak about reality, they are not certain, and in so far as they are certain, they do not speak about reality.
- A. Einstein