Math
Intersecting Chord Theorem
This video sets out a puzzle that uses the intersecting chord theorem to solve.
Which led me to a couple of basic geometric properties that I did not explicitly know.
Math
Dandelin spheres -Conic Sections Proof of Circle Ellipse Parabola Hyperbola
Research Links
Ellipse Proof: The sum of distances d(F1, P) + d(F2, P) remain constant as the point P moves along the curve.
- A line passing through P and the vertex S of the cone intersects the two circles at points P1 and P2.
- As P moves along the curve, P1 and P2 move along the two circles.
- The distance from F1 to P is the same as the distance from P1 to P, because these distances are the lengths of the line segments PF1 and PP1, which have a common endpoint P and are both tangent to the same sphere (G1) at their other endpoints.
- Likewise, by a symmetrical argument, the distance from F2 to P is the same as the distance from P2 to P.
- Consequently, the sum of distances d(F1, P) + d(F2, P) must be constant as P moves along the curve because the sum of distances d(P1, P) + d(P2, P) also remains constant. Therefore, the curve is an ellipse. This follows from the fact that P lies on the straight line from P1 to P2, and the distance from P1 to P2 remains constant.
Math
Lune Of Hippocrates – Famous Ancient Math Problem
Hippocrates of Chios solved this, nearly 100 years before Euclid even wrote The Elements, and before we knew the exact formula for the area of a circle. This is said to be the first precise mathematical of the area between two curved lines. Learn how he solved for the area of a crescent-shaped lune (same Latin root as luna for moon). And I hope this will inspire you to solve a problem today that students thousands of years from now will learn in grade school.
Math
Visualizing the Riemann zeta function and analytic continuation
This function is famously confusing outside its "domain of convergence", but a certain visualization sheds light on how it extends.
If I were awoken after having slept for a thousand years my first question would be: Has the Riemann hypothesis been proved? – David Hilbert
History
Freeman Dyson – Web of Stories – Life Stories of Remarkable People
American-British physicist and mathematician, Freeman Dyson, was born in 1923. He is known for his achievements in solid state physics, nuclear engineering and quantum field theory.
Research Links
