Trisecting An Angle W/Ruler And Compass

[acropolis]

An easy way to do it occurred to me the other day. I'm sure this is trivially obvious to some people, and has been done a million times by others over the past few centuries, but I wanted to work on my logic etc so I worked it out in ~functional form.

Assume it's possible to bisect an angle <ABC. We create a bisection function B2(A,B,C) = D where D is such that <ABD = <DBC

fair enough?

If there were a trisect function, it would look like:
T3(<ABC) = (D,E) where D and E are such that <ABD = <DBE = <EBC

so that's the function we want to define.

I need to define more functions now.

F1(X) = B2 (<ABX)
F2(X) = B2 (<XBC)

using those we create Jn and Kn.

for n > 0
Jn = F1(Kn-1)
Kn = F2(Jn-1)

and when n=0
Jn = B2(<AB(B2(<ABC)))
Kn = B2(<(B2(<ABC))BC)

now, with Jn and Kn defined, we are set.

the trisection function can be defined simply:
T3(<ABC) = lim n=>infinity (Jn,Kn)

yeah. obviously my notation is ~poo, and constructive criticism would be appreciated.

(yes I'm that bored)

[Nusselt]

alleviate your boredom on your wife (what else is she there for?)

[acropolis]

Originally Posted by Nusselt

alleviate your boredom on your wife (what else is she there for?)

doesn't have security clearance to visit me at work.

[QazokRouge5]

thank you for making a thread that noone cares about

[acropolis]

Originally Posted by Nantoz

I'm sure it was a good try though, I cant be bothered to read your notations...

what they proved is that you can't do it in a finite number of steps.

but often i hear the question phrased as 'can you trisect an angle with only an (unmarked) straight edge and compass?'

to which the answer is obviously 'yes'

[Nusselt]

send your wife some really dirty txt msgs.

[NEWSBOT3]

Originally Posted by Nusselt

send your wife some really dirty txt msgs.

if she needs security clearance to get in, i'd be suprised if he's allowed a mobile.

[acropolis]

Originally Posted by NEWSBOT3

if she needs security clearance to get in, i'd be suprised if he's allowed a mobile.

i would be allowed a mobile if it didn't have a camera on it.

it does

they won't allow those evul PRC types even a mobile w/o camera tho.

[NEWSBOT3]

Originally Posted by acropolis

i would be allowed a mobile if it didn't have a camera on it.

it does

they won't allow those evul PRC types even a mobile w/o camera tho.

Peoples Republic of China ?

[acropolis]

Originally Posted by NEWSBOT3

Peoples Republic of China ?

yeah.

theres a chinese kid on my project too. we tried to get him clearance.

emails went back and forth, and finally we got a message from some beaurocrat that kinda ended our hopes.

there's about 30 or so people that work at the facility, but tons more people come out to do short research projects from one of the universities involved or industry. plus they have lots of tours etc. but apparently, over the past year 6 of these people have been Chinese! So the guy says "I think we are being targetted by the PRC..." and he gave a few conditions in which we could get our guy clearance.

1. He'd need to turn his planned hours in in advance.
2. He'd need to be escorted.
3. that includes the bathroom.
4. He could only use specific project related computers.
5. Any computers he used would be regularly purged of all non-project info
6. No camera. Or mobile. Or pocket pal. Or basically anything electronicesque.
7. Random visits would be made to check and make sure all protocols were being followed.

Given that I have minesweeper high scores on all those project-related computers, this was not happening.

Anyway, this place is no Area 51. There's like two projects there that are funded in part by the defense department, and if they contract out to have a stoplight designed, they are damn well classifying it as 'secret.'

[Nusselt]

where the **** do you work? is it a for real shizzle top secret facility (which would make you ) or is it a standard run of the mill place but with zealous civil servants, i encountered the latter once

is this commercially sensitive stuff or military sensitive stuff?

[Judge]

Originally Posted by Nusselt

where the **** do you work? is it a for real shizzle top secret facility (which would make you ) or is it a standard run of the mill place but with zealous civil servants, i encountered the latter once

is this commercially sensitive stuff or military sensitive stuff?

If he told you, he would have to kill you.......

[acropolis]

Originally Posted by Nusselt

where the **** do you work? is it a for real shizzle top secret facility (which would make you ) or is it a standard run of the mill place but with zealous civil servants, i encountered the latter once

is this commercially sensitive stuff or military sensitive stuff?

the place does a lot of work on wide band gap semiconductors (basically, growing them by cvd) such as SiC and GaN. WBG semis have potential for laser applications as well as high frequency apps in general (radar).

they have an mbe system too tho, and i'm doing mbe growth of oxide dieletrics. or would be if the system worked.

[Nusselt]

...yeah so commercially sensitive but apparently partly funded by the dod because you've told them they'll get phantom death radars

[Nodrog]

Originally Posted by acropolis

what they proved is that you can't do it in a finite number of steps.

but often i hear the question phrased as 'can you trisect an angle with only an (unmarked) straight edge and compass?'

to which the answer is obviously 'yes'
'

Actually, the answer is 'no, not in the general case' since you cannot physically peform infinitely many steps. Your algorithm wouldnt terminate when given a 60 degree angle.


Here is a proof that there exists a largest integer:

Lemma: If I can count through the whole sequence of natural numbers in ascending order, then there must be a largest integer

Proof: Let the last number I say be x. If x isnt the largest integer, then there is a y st y > x. But since I said x and I counted in ascending order, I must have said y hence y < x.


Theorem: I can count through the whole sequence of natural numbers in ascending order

Proof:

defun count(n): for (i = 1 to n) print n;
defun count_to_infinity: int n = 1, lim(n->inf) count(n);

[acropolis]

Originally Posted by Nodrog

Actually, the answer is 'no, not in the general case' since you cannot peform infinitely many steps. Your algorithm would never terminate.

why can't i perform infinitely many steps? if someone states the trisection problem with a finite step limitation, then fine.

'physically performable' seems an argument against it, but a true bisection is also not physically performable due to imperfections in any given straightedge and compass (as well as spacetime curvature issues). But we simply do our geometry in an idealized euclidean space, with ideal straightedge and compass. And in that space there is no reason I can't imagine each step of my algorithm taking half the time of the previous one, and so terminating in just a few seconds of euclid space time

One valid criticism of the Greeks (amongst acres of criticism-for-the-sake-of-criticism bullshit) is their unwillingness to accept the use of the concept of the infinite (and infintesimal) to solve any number of problems. It's a large part of the reason that we have space travel and supercomputers and they had olive oil and sex with young boys.

[acropolis]

Originally Posted by Nodrog

defun count(n): for (i = 1 to n) print n;
defun count_to_infinity: int n = 1, lim(n->inf) count(n);

this limit DNE.

And what was it about 60 degrees?

[Nodrog]

Originally Posted by acropolis

why can't i perform infinitely many steps? if someone states the trisection problem with a finite step limitation, then fine.

The finite step limit is implicit in the context of greek geometry problems. You cant construct a line of infinite length, but you can construct one which is arbitrarily long. It's similar to asking 'why cant I move my rook diagonally in chess?'.

If you want to introduce non-finite constructions, then (I would guess, although I'm not sure) all real numbers will become constructible, and youve defeated the point of geometry problems since you can now do pretty much whatever you like (eg doubling the cube and squaring the circle, which are equivalent to constructing the third root of 2 and pi respectively - impossible using finitary methods).

'physically performable' seems an argument against it, but a true bisection is also not physically performable

I should have said implementable. A bisection algorithm can be implemented as a Turing machine which is guaranteed to terminate for all inputs. A trisection algorithm cant.

[Nodrog]

Originally Posted by acropolis

this limit DNE

I'm not using the limiting value - its an instruction to perform a function call infinitely many times (like yours was, unless I've misread your notation)

And what was it about 60 degrees?

There is no finitary algorithm for trisecting a 60 degree angle using ruler and compass alone.

[JamMak]

Wow. This thread turned interesting.

I'm impressed GD!