Guys, I am freaking out right now

IT WORKS

What does?

My division algorithm! it works perfectly, and even for rational inputs!

I fixed it :D

Great!

I'm just cleaning up the workspace before I post it

@robjohn I've pinged you alot--sorry--but it's finally complete. I've fixed the sum. It works. The most expensive function here is the integer coefficient function $I(y, n_{term})$ requiring modulus, so the sum for $\frac{x}{y}$ is not suitable for implementing division, but modulus can, using the identities in "Noteworthy Identities", one can derive an efficient modulus function for $2^x \bmod y$ for naturals x and y just as I did for quotients here. desmos.com/calculator/bncsqi8qmk

@PM2Ring I did not know it at the time, but I wrote an answer about Ted's avatar.

@AMDG I won't have time to look at it for a while. We are sitting down to dinner in a bit.

I can wait.

@robjohn Before you even knew Ted. This shape appears as an exercise in two of my books!

An interesting shape. Top and bottom are a double cone with apices extending inwards and the inside is a catenoid.

@robjohn Ah. Nice. It's a lot easier to see what's going on in your anim than in Ted's image. ;) But to be fair, you don't have a lot of pixels to play with in an avatar.

Not a catenoid. Read robjohn's answer.

What isn't a catenoid? The center of the object?

That shape considered geometrically is the outline of a line as it rotates in time with an angle less than pi/2 relative to a disk and also provides a visual interpretation of the complex identities for $\cos(ix)$ and $\sin(ix)$ to their hyperbolic counterparts.

My avatar's a rhombic dodecahedron. This low quality JPEG shows how it's related to the cube & octahedron.

my avatar is a simulated lower half of a drowned person in a kiddie pool

Nope. Hyoerboloid of one sheet. Follows abstractly from classification of doubly-ruled surfaces!

@leslietownes munchkin projection?

Ohhh, that's what I was thinking of. Seems I still mix up catenoid with hyperboloid.

ted: would you believe she conned us into a second trip to the big duck pond in two days.

Hey, people mis-identify catenaries as parabolas all the time (and occasionally do the reverse). Confusing hyperboloids & catenoids is less common. :)

Oh right, catenary $\to$ catenoid

Chains, am I right

The shape made by hyperbolic cosine

Very beautiful function

So... it looks like it might be possible to generalize the sum I found for computing roots given that $x^{-1}$ is a type of root which this can compute.

Oh dear. The top answer here confuses the hyperboloid with a parabola. But the other answers are correct. physics.stackexchange.com/q/221339/123208

Yikes

If I integrate the sum, I should get an infinite sum for $\ln(x)$ which would allow for efficiently computing logarithms lol

@AMDG catena = chain

I should have known...

@AMDG Yes. Uniform chains. Also good for arches. But some things that you'd think ought to be catenaries are actually parabolas, due to the load they're supporting. I did rather nicely on the HNQ with this catenary answer: physics.stackexchange.com/a/421965/123208

Everything's a uniform chain if you're brave enough.

Like concatenate!

$f:g\mapsto \cosh(x)$ :)

projections amirite

Carlson's modified AGM (arithmetic-geometric mean) algorithm for log is really fast. It uses no divisions (apart from bitshifts), but it does need a few square roots, of course. But it's so fast that for big arithmetic you can actually use it with Newton-Raphson to compute exponentials.

Well I imagine the speed is due to the convergence of AGM

Well, it's accelerated on top of the natural speed of the AGM.

Now that there is cool and all, but what I have here is an infinite sum for $\frac{x}{y}$, so naturally it can approximate $\frac{1}{x}$ to arbitrary precision. Integrating the terms should therefore approximate $\ln(x)$. That's what I'm meaning.

It seems to converge pretty quickly, too.

Just set b to 4 and increase it from there. Compare the error to $\frac{1}{x}$ and zoom in.

Ok. Assuming your integration is fast, and doesn't accumulate errors

Who said anything about integral approximation? :)

I look for closed forms where possible and then approximate the closed form. That gives the highest quality approximation (and usually the cheapest).

I still wish I could make the convergence more uniform, though.

BTW, using Carlson's algo with a different initialization constant gives you arctan. And IIRC, there are minor variants for hyperbolic functions.

Is it valid for complex arguments?

Hi all, looking for inspiration here: is there anything that behaves analogous to light?

I.e., $\theta_i=\theta_r$?

What, like dipping your finger in a puddle of water?

Yup, like reflection.

For instance, billiard balls do:

Anything else?

I am confused. What behavior are you looking for an analogy of?

Light's tendency to reflect such that $\theta_i=\theta_r$

I don't know what $\theta_i$ and $\theta_r$ are.

Angle of incidence = angle of reflection

Ah ok

Well that's described by just throwing something at a wall even...

Like tossing a very bouncy ball at a wall

Or a dodgeball as it bounces off of your face.

Perhaps something nontrivial ...

wired.com/2016/05/…

Not sure I understand. Under idealized conditions, that slope there is 1.0, such as in a mathematical environment.

Idealized conditions being zero friction, zero mass, infinitely small period of time ("moment"), etc.

Any other physical phenomenon that obeys the law of reflection ... other than light, balls, and billiards?

Radiation in general. Atoms. Any projectile whatsoever. Anything that can interact with matter and in principle is a moving object that is to make contact with a stationary object.

A bullet

Hm ... that gives me an idea, although not a very good one.

See, I'm trying to reframe the triangular billiards problem in a different way, hoping that it may lend some additional insight into the problem.

@rb3652 At the quantum level, the reflection of light is a bit more complex (pun intended). Feynman gives a nice account in his popular QED book, and explains how the classical behaviour emerges from the quantum. Because it's a pop-science it keeps the maths to a minimum, so he explains stuff using arrows instead of complex numbers. But maybe this conversation is better suited to the physics chat.

@PM2Ring Perhaps, but I'm not really going for a thorough, physical description of light.

Anything is on-topic for math if you're brave enough.

I'm trying to reframe an open mathematical problem (periodic orbits in triangular billiards) as a physical one, hoping it may give me some fres insight.

No luck though.

Huh, what is a periodic orbit?

A path that repeats itself, kinda like:

(Just got this from Google)

The open problem is whether there are periodic orbits in any triangular table.

Greg Egan did some stuff related to that. I'll try to find it. In the mean time, here's another one of his articles involving reflection: gregegan.net/SCIENCE/Catacaustics/Catacaustics.html

@PM2Ring Perhaps one way to imagine it is that the energy of the photon is flung around the atom and then exits near the entry via the medium of the electron cloud like a sling shot being primed and rotated then released.

@PM2Ring Interesting, thank for the reference. I'll take a read into it.

The way it is described with an electron's orbit being temporarily elevated would seem to coincide and for which I imagine that the orbit would appear as a flower petal.

Another example of a periodic orbit:

Well, come to think of it, could chasing this lead be productive?:

If a path is periodic, launching multiple balls along that trajectory would mean that balls would eventually collide at some point.

So a periodic orbit is just a graph whose vertices are the contact points of a projectile and a surface and the edges the path that the projectile follows?

@AMDG A periodic orbit is just a complicated way of saying a "repeating path," so that the ball eventually comes back to where it started and starts following the same path again.

Here's an idea I've had as concerns orbits because I've thought about how one might get a more efficient algorithm that analytically computes the amount of light a point on a surface receives by taking the surface described by the entire environment.

Not sure if it's relevant to periodic orbits as given here, but consider that the orbit is a straight line with $n$ distinct rotations at varying points in time.

(Assuming of course that the orbit is linear)

@AMDG Hm, you may be interested in the "Illumination Problem". Sounds quite similar to the lighting problem you're describing.

It also has to do with billiards:

Probably. I want photorealistic graphics in real time, and I'm not satisfied with existing technology.

Technology as in algorithms

@rb3652 This is the one I was thinking of earlier. gregegan.net/SCIENCE/Lattices/Lattices.html

@PM2Ring Thanks, looking into it!

Crazy to think such a simple problem has been unsolved for so long.

In any case, assuming we want to compute the position of a single photon tracked from some initial condition as the point of emission, then if one knows the possible surfaces by which it could bounce off of (all of them), then one could in theory compute the position in constant time assuming that the surface is stationary and does not change over time (that would require more work).

Perhaps an obvious use case here is brownian motion

Indeed, certain variations of the billiards problem have been used to model bronwian motion.

@AMDG Electrons don't have planet-like orbits. Classical trajectories don't apply when Heisenberg Uncertainty effects are significant. Yes, there's harmonic motion going on, but it has no simple classical counterpart. But an electron orbital is more like a vibrating surface than a planetary orbit. This page has some nice anims, but using a 2D disk instead of a sphere. en.wikipedia.org/wiki/…

Yeah, well that's why my theory would claim that such things as photons, for instance, are not wave-particles so-called but moving fields of energy that have a massive component or nature applied to them, and an electron cloud might be described as bodies moving so fast and rapidly that they cause what appears to be a solid surface that you could touch in much the same way that something can appear to be in two places at once if it oscillates between two locations fast enough.

It would be hopeless to simulate illumination at the level of individual photons. You'd need too many, and photons are even worse than electrons. ;) A travelling photon doesn't have a well-defined location. More formally, the photon doesn't have a position observable, so you can't even get a probability that the photon is in a specified volume.

Well that's why I say a sort of field with mass where said field is a sort of continuum capable of holding energy and the mass is incredibly close to zero. The continuum allows preserving the field while allowing energy to spread like a wave through space while the mass allows for certain physical phenomena to take place (if that makes sense) such as being affected by gravity.

Much like the idea of the electron cloud

Photons have zero rest mass. All of their energy is kinetic. They don't need mass to be affected by gravity. And if photons did have mass, the inverse square law wouldn't work properly, and the range of the electromagnetic force wouldn't be infinite. Of course, it's possible that photons do have a really tiny mass, but if that were the case, we'd have to rewrite large chunks of modern physics. You need really good evidence to justify such a radical change.

I could have sworn there were more recent developments which at least suggested that photons do in fact have said tiny mass, so that is why I said so.

good evening

is anybody around?

no

Alright, I have to go to bed. @robjohn perhaps we can discuss my algorithm tomorrow whenever you're available?

i think you have scared him away

Nah, he had to eat dinner, and it's late now.

nah, he's hiding from you

can we talk about Covid for a bit?

Last thing I'll say before I go to bed: iirc from when I looked at it earlier, the requirement of modulus in the function is actually dependent on the input itself. If that is so, then for 64-bit divides, we only need to compute $\bmod y$ for $y$ in the range $[1, 64]$.

what do you want to say about covid Bob?

it seems out of control

I fear we will soon be at 1000K new cases per day

it is milder and will become endemic shortly. smart governments will recognise this an act appropriately

what do you think the US will do?

i am more concerned about the enormous impact on education

Personally, I favor a long hard shutdown

what's the point?

I fear we have a big problem

which is?

we need to beat this thing

people are dying

were you concerned about flu, colds?

people died from them

and there are other issues including education

are you a college professor?

no

I have done some tutoring

college students mostly

and the general trend is towards lower standards

the usa seems afraid of enforcing academic standards

but my concern above was the about the impact of covid on education

well, I had a friend who was an adjunct professor

one semester he failed his entire class

they did not ask him to teach next semster

there was a time when I thought I wanted to be a high school math techer

I was told without tenor

well, obviously i do not know all the facts, but that sounds irresponsible

I was told without tenure

I better give everybody at least a B

i do not think high school teachers should have tenure

How do you feel about all high school students getting at least a B in every math class?

that is irresponsible too

well, if a high school teacher does not have tenure

and they give low grades, they are not rehired

I have another story for you

A friend of mine was teaching second grade

she had a student named John

John was not doing his homework

almost all of my grandparents were teachers, some principals.

She sent a note home to the parents

they parents sent a note back to Kim and her boss saying that they were unhappy with the school and that the school should make sure that John does his homework.

Any comments?

parents have some responsibility

not the teacher's responsiblity to enforce homework

yep but many schools feel when the parent's are not doing their job, the school should do it for the parent

I agree with you, that it is not right

i am not in that camp,

but it is

is your PhD in math?

people have a hard time accepting that some fraction will be poorly performing. it is very sad and resources should be spend to give them an opportunity to succeed, but tearing down standards in some well intentioned pc effort devalues everyone.

no, eecs

minor in maths

but i am not a mathematician

thanks

you are a mathematician, copper.

my degrees are in CS

with that

it was nice chatting

i have cs degree somewhere

noce chatting!

good night

@Koro i know enough to get into trouble

When I was a kid, parents always backed the school and teacher. The last 25 years I had parents interfering in their college kids’ lives, harassing professors. The modern way … He who pays the bill decides curriculum and standards. F*** that.

ted: your asterisks confused me momentarily, but i too am fond [of] that

that was the way when we were growing up.

even at berkeley, profs have parents calling about their kids

@leslietownes check your neck

i'd rather check my back, which is sore from carrying around munchkin

Yup. The US is all about mediocrity. Just look at the stupidity we elect.

kids ruined me physically

every country is like that Ted

More and more, yes.

The world may not be here for all of your children.

had a wonderful cycle to west point inn on tamalpais this morning.

a bit tired from my vollmer peak ride yesterday.