I’ve still been thinking about this elliptical cone! It has been the main character in my geometric proof on stereographic projection mapping circles to circles. The idea has been to reduce a three-dimensional problem to a two-dimensional one, by noting that something has to be symmetric. A circle on a sphere is mapped to some…
Tag: Proof
Circles to Circles
Using stereographic projection, you create a distorted image of the surface of a sphere, stretched out to cover an infinite plane. Each point on the sphere is mapped to a point in the equatorial plane by a projection ray starting at a pole of the sphere. Draw a circle on the sphere, e.g. by intersecting…
Gödel’s Proof
Gödel’s proof is the (meta-)mathematical counterpart of the paradoxical statement This sentence is false. In his epic 1979 debut book Gödel, Escher, Bach Douglas Hofstadter intertwines computer science, math, art, biology with a simplified version of the proof. In 2007 he revisits these ideas in I Am a Strange Loop. Hofstadter writes: … at age…
Integrating the Delta Function (Again) – Dirac Version
The Delta Function is, roughly speaking, shaped like an infinitely tall and infinitely thin needle. It’s discovery – or invention – is commonly attributed to Paul Dirac[*]. Dirac needed a function like this to work with integrals that are common on quantum mechanics, a generalization of a matrix that has 1’s in the diagonal and…
Delta Function Haiku
I have proved that a Lorentzian bell curve becomes the Dirac Delta Function in the limit. Now I want to look at another representation of the Delta Function. As this is a shorter proof, a haiku will do. ~ Infinite numbers of oscillations added. Need to damp them down Symmetrically attach an exponential for each…
The Improper Function and the Poetry of Proofs
Later the Delta Function was named after their founder. Dirac himself called it an improper function. This time, the poem is not from repurposed snippets of his prose. These are just my own words to describe a proof: ~ In the limit the Lorentzian becomes the improper function. In the limit of tiny epsilons it…
elkemental Force 