M4TH3M4T1C5 - let's prove .99999.... != 1 - The Something Awful Forums

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ol qwerty bastard: Dec 13, 2005; If you want something done, do it yourself!

i think this is my absolute favourite bit of applied math: http://en.wikipedia.org/wiki/Planimeter

because at first you're like "huh? you can't compute the area just by measuring the perimeter" and then it hits you that it's actually doing integrals

mechanically it is so simple that it could have been built in ancient greece, but nobody invented it until we had the proper math for it. makes you wonder what sort of stuff might be theoretically possible with current technology that we still haven't figured out due to lack of maths

# ? Oct 30, 2014 15:37

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# ? Apr 26, 2024 08:37

motedek: Oct 9, 2012

Snapchat A Titty posted:

augh i still dont get this loving algorithm and ive emailed the authors weeks ago but they dont write me back

ive done the first part where i segment an image into vertical zones, and segment the zones into regions. then comes a HMM part where im supposed to refine the zones, but the way its described is impenetrable for me.

you compute the matrix at your data points, not per zone. it's the transition matrix for the process starting at the last data point and ending at the current data point. the zones shouldn't enter into it, if i understand it.

# ? Oct 30, 2014 17:12

Mr. Apollo: Nov 8, 2000

# ? Oct 30, 2014 17:49

LP0 ON FIRE: Jan 25, 2006; _{beep boop}

i like your avatar and your registration year, Mr. Apollo. it is very math

# ? Oct 30, 2014 17:54

tony police: Sep 22, 2006

rotor posted:

if there's a single goddamn math course they should teach in high school beyond basic algebra it's loving statistics

i hated stats in uni because the prof was a giant goon who always wore disgusting worn-out super hero emblem t-shirts on rotation and used the shittiest most confusing examples for basic concepts and it drove me up the loving wall

# ? Oct 30, 2014 18:17

tony police: Sep 22, 2006

ol qwerty bastard posted:

i think this is my absolute favourite bit of applied math: http://en.wikipedia.org/wiki/Planimeter

because at first you're like "huh? you can't compute the area just by measuring the perimeter" and then it hits you that it's actually doing integrals

mechanically it is so simple that it could have been built in ancient greece, but nobody invented it until we had the proper math for it. makes you wonder what sort of stuff might be theoretically possible with current technology that we still haven't figured out due to lack of maths

this is cool

# ? Oct 30, 2014 18:21

quiggy: Aug 7, 2010; [in Russian] Oof.

Dash O Pepper posted:

i hated stats in uni because the prof was a giant goon who always wore disgusting worn-out super hero emblem t-shirts on rotation and used the shittiest most confusing examples for basic concepts and it drove me up the loving wall

i hated my stats prof because he was a far-right wingnut who openly hated women and queers in class but the class itself was interesting and important

# ? Oct 30, 2014 18:23

Fuzzy Mammal: Aug 15, 2001; Lipstick Apathy

https://www.youtube.com/watch?v=s1i-dnAH9Y4

owns bones

Also here's a post on one of my favourite math sites:

http://acko.net/blog/how-to-fold-a-julia-fractal/

# ? Oct 30, 2014 18:29

quiggy: Aug 7, 2010; [in Russian] Oof.

Fuzzy Mammal posted:

Also here's a post on one of my favourite math sites:

http://acko.net/blog/how-to-fold-a-julia-fractal/

this is really cool, thanks

# ? Oct 30, 2014 18:38

Fuzzy Mammal: Aug 15, 2001; Lipstick Apathy

it looks like it stopped rendering properly in chrome, unfortunately. anyways he's got lots of other cool visualizations on the site.

# ? Oct 30, 2014 18:52

Dixie Cretin Seaman: Jan 22, 2008; all hat and one catte; Hot Rope Guy

Fuzzy Mammal posted:

Also here's a post on one of my favourite math sites:

http://acko.net/blog/how-to-fold-a-julia-fractal/

if you like the geometric approach to complex numbers, the book "Visual Complex Analysis" (whose full PDF seems to be the 2nd hit for the title on google, just sayin') goes through most of classical complex analysis this way. it's impressive how far the author gets with mostly geometric arguments.

i would flip what the link above says, though. instead of saying all numbers are really imaginary i would say complex numbers aren't really numbers, in the usual sense. they're a set of rotation and scaling operators in 2 dimensions that happen to share enough important algebraic properties with traditional numbers that you can treat them like numbers for the sake of many computations.

if you're willing to drop multiplicative commutativity there's a 4-dimensional extension of complex numbers called quaternions, and in 8 dimensions there's a non-associative extension called octonions. it's been proven that those are all the possibilities for defining sets like these (extensions of real numbers that let you add, multiply, and divide by non-zero elements).

# ? Oct 30, 2014 19:15

Broken Machine: Oct 22, 2010

Dixie Cretin Seaman posted:

if you're willing to drop multiplicative commutativity there's a 4-dimensional extension of complex numbers called quaternions, and in 8 dimensions there's a non-associative extension called octonions. it's been proven that those are all the possibilities for defining sets like these (extensions of real numbers that let you add, multiply, and divide by non-zero elements).

Quaternions have an interesting history. Sir William Rowan Hamilton was walking along with his wife in Dublin, near the River Liffey. As he passed a bridge called Broombridge, he had a flash of insight and sketched the fundamental equation defining quaternions on the stones of the bridge with a rock. There's a plaque there to this day; neat place to visit if you find yourself in Dublin. Also the first non-commutative operation (wrt multiplication).

Broken Machine fucked around with this message at 19:43 on Oct 30, 2014

# ? Oct 30, 2014 19:24

bump_fn: Apr 12, 2004; _{two of them}

I want to make a joke about algebraic rings and goatce but I should really pay attention to this seminar

# ? Oct 30, 2014 19:29

Chorrax: May 11, 2007

Notorious QIG posted:

solution:

with no information, there are four options
1) girl/girl
2) girl/boy
3) boy/girl
4) boy/boy

since at least one child is a boy, option 1 is eliminated. there are now three remaining options, of which only one has both children being boys. the intuitive answer, 1/2, is true only if the couple specifies which child is a boy

If girl/boy and boy/girl are distinct options than why aren't boy1/boy2 and boy2/boy1 not options if you're going to lack any specifications about which sex which child is.

I still just can't quite wrap my head around it but that's probably because i'm a dumb baby that stopped math after the 201 level.

# ? Oct 30, 2014 20:07

quiggy: Aug 7, 2010; [in Russian] Oof.

Chorrax posted:

If girl/boy and boy/girl are distinct options than why aren't boy1/boy2 and boy2/boy1 not options if you're going to lack any specifications about which sex which child is.

I still just can't quite wrap my head around it but that's probably because i'm a dumb baby that stopped math after the 201 level.

flip a penny and a quarter. you have four options:

1) both are tails
2) penny is tails, quarter is heads
3) penny is heads, quarter is tails
4) both are heads

theres only one possibility in which you get both heads. if you dont believe me grab some coins and flip them a lot, you'll get h/t twice as often as h/h

# ? Oct 30, 2014 20:10

Chorrax: May 11, 2007

Notorious QIG posted:

flip a penny and a quarter. you have four options:

1) both are tails
2) penny is tails, quarter is heads
3) penny is heads, quarter is tails
4) both are heads

theres only one possibility in which you get both heads. if you dont believe me grab some coins and flip them a lot, you'll get h/t twice as often as h/h

That is a hell of excellent explanation. tyvm

# ? Oct 30, 2014 20:15

Fuzzy Mammal: Aug 15, 2001; Lipstick Apathy

Dixie Cretin Seaman posted:

if you like the geometric approach to complex numbers, the book "Visual Complex Analysis" (whose full PDF seems to be the 2nd hit for the title on google, just sayin') goes through most of classical complex analysis this way. it's impressive how far the author gets with mostly geometric arguments.

i would flip what the link above says, though. instead of saying all numbers are really imaginary i would say complex numbers aren't really numbers, in the usual sense. they're a set of rotation and scaling operators in 2 dimensions that happen to share enough important algebraic properties with traditional numbers that you can treat them like numbers for the sake of many computations.

if you're willing to drop multiplicative commutativity there's a 4-dimensional extension of complex numbers called quaternions, and in 8 dimensions there's a non-associative extension called octonions. it's been proven that those are all the possibilities for defining sets like these (extensions of real numbers that let you add, multiply, and divide by non-zero elements).

yeah that's v cool.

this site actually has an entry on quaternions! it's in the context of animation and it's not as well explained as his complex numbers one. i remember having to use them in 4th year graphics class but i clearly didn't understand them fully enough because i don't 'get' them when i see them come up nowadays.

# ? Oct 30, 2014 20:18

quiggy: Aug 7, 2010; [in Russian] Oof.

Chorrax posted:

That is a hell of excellent explanation. tyvm

# ? Oct 30, 2014 20:19

Chorrax: May 11, 2007

i looked into that problem on wikipedia: http://en.wikipedia.org/wiki/Boy_or_Girl_paradox

"Many people argued strongly for both sides with a great deal of confidence, sometimes showing disdain for those who took the opposing view."

lol ohh the autism

# ? Oct 30, 2014 20:31

Carthag Tuek: Oct 15, 2005; Tider skal komme,
tider skal henrulle,
sl�gt skal f�lge sl�gters gang

motedek posted:

you compute the matrix at your data points, not per zone. it's the transition matrix for the process starting at the last data point and ending at the current data point. the zones shouldn't enter into it, if i understand it.

thx 4 reply

it still doesnt make sense, though. the papers all say the hmm parameter matrices are calculated based on properties of regions (each region i will result in different parameters), but they are applied to observations of entire zones containing several regions, and its not mentioned how to select the single region to use as parameter for the zone

# ? Oct 30, 2014 21:15

Carthag Tuek: Oct 15, 2005; Tider skal komme,
tider skal henrulle,
sl�gt skal f�lge sl�gters gang

like in pseudecode:

code:

m[0] = median height for zones of state text
m[1] = median height for zones of state gap

for r in regions:
  // observations for hmm:
  observations = r.zones.state
  
  // parameters for hmm:
  states = [0, 1]
  initial_prob = [0.5, 0.5]
  i = ?????????
  a00 = exp(-r.zones[i].height/m[0])
  a11 = exp(-r.zones[i].height/m[1])
  transition_prob = [a00, 1-a00, 1-a11, a11]
  
  // cant continue without i...

# ? Oct 30, 2014 21:34

Dixie Cretin Seaman: Jan 22, 2008; all hat and one catte; Hot Rope Guy

Fuzzy Mammal posted:

yeah that's v cool.

this site actually has an entry on quaternions! it's in the context of animation and it's not as well explained as his complex numbers one. i remember having to use them in 4th year graphics class but i clearly didn't understand them fully enough because i don't 'get' them when i see them come up nowadays.

if you're using quaternions for 3-D rotations there's a nice geometric way to think about it. if i remember correctly, you represent the desired axis of rotation by identifying (i,j,k) with the basis vectors of R^3, form a quaternion whose imaginary part points along that direction and makes angle (theta/2) with the real part, then conjugate a vector to effect the rotation of theta around the chosen axis. there was an old book i read on them a long time ago and forgot the name of that had a nice geometric explanation of everything..

# ? Oct 30, 2014 22:13

fritz: Jul 26, 2003

Dixie Cretin Seaman posted:

if you're using quaternions for 3-D rotations there's a nice geometric way to think about it. if i remember correctly, you represent the desired axis of rotation by identifying (i,j,k) with the basis vectors of R^3, form a quaternion whose imaginary part points along that direction and makes angle (theta/2) with the real part, then conjugate a vector to effect the rotation of theta around the chosen axis. there was an old book i read on them a long time ago and forgot the name of that had a nice geometric explanation of everything..

the way i think about quaternions and rotation is you have your axis of rotation (the xyz parts of the quaternion) being a point on the boundary of the sphere, and the magnitude of the rotation (the w part) determines how far out to the boundary you go

so all the (0,x,y,z) quats represent no actual rotation and are at the origin, then as you go out along an axis you increase w, the magnitude of the rotation, until you get to the boundary at (1,0,0,0), and all the boundary points are the same

negative w is a mirror copy of the sphere with the same origin and boundary but different interior points

i dont think i explained that rigorously or maybe even "correctly" but go read this: http://www.amazon.com/Visualizing-Quaternions-Kaufmann-Interactive-Technology/dp/0120884003

# ? Oct 30, 2014 23:05

Deacon of Delicious: Aug 20, 2007; I bet the twist ending is Dracula's dick-babies

Fuzzy Mammal posted:

Also here's a post on one of my favourite math sites:

http://acko.net/blog/how-to-fold-a-julia-fractal/

heh, you have favorite math sites. nerd

...

...

*adds to reading list*

# ? Oct 31, 2014 01:01

bump_fn: Apr 12, 2004; _{two of them}

quarternions are easy if u understand su2 fyi

# ? Oct 31, 2014 01:22

motedek: Oct 9, 2012

Snapchat A Titty posted:

thx 4 reply

it still doesnt make sense, though. the papers all say the hmm parameter matrices are calculated based on properties of regions (each region i will result in different parameters), but they are applied to observations of entire zones containing several regions, and its not mentioned how to select the single region to use as parameter for the zone

i looked at one of the papers and it's pretty incoherent but i think what they do is, use the initial segmentation to break up a vertical zone into regions, and use it to get the estimated heights, but then forget the assignment of each region as text/gap. then, do the HMM with regions as the observations, computing a different transition matrix at each region based on its height, and reassign text/gap using viterbi.

# ? Oct 31, 2014 01:22

Carthag Tuek: Oct 15, 2005; Tider skal komme,
tider skal henrulle,
sl�gt skal f�lge sl�gters gang

motedek posted:

i looked at one of the papers and it's pretty incoherent but i think what they do is, use the initial segmentation to break up a vertical zone into regions, and use it to get the estimated heights, but then forget the assignment of each region as text/gap. then, do the HMM with regions as the observations, computing a different transition matrix at each region based on its height, and reassign text/gap using viterbi.

that makes a bit of sense, but it seems like it would be easier and give pretty much the same result to just use a threshold and drop the hmm altogether

ill try out both and see what results are best

# ? Oct 31, 2014 01:31

Dixie Cretin Seaman: Jan 22, 2008; all hat and one catte; Hot Rope Guy

bump_fn posted:

quarternions are easy if u understand su2 fyi

that depends on how explicitly geometric ur double covers are, m8

# ? Oct 31, 2014 01:47

bump_fn: Apr 12, 2004; _{two of them}

Dixie Cretin Seaman posted:

that depends on how explicitly geometric ur double covers are, m8

idgi all lie groups are group manifolds whats not geometric

# ? Oct 31, 2014 01:56

Dixie Cretin Seaman: Jan 22, 2008; all hat and one catte; Hot Rope Guy

bump_fn posted:

idgi all lie groups are group manifolds whats not geometric

yeah, but the lie group geometric viewpoint is treating each quaternion as a point. im thinking about the geometry of quaternions as maps in SO(3) (via the conjugation action, and identified mod -1 as per the standard double cover). this is the way quaternions are used in the common applications we were discussing; as a convenient way to compute 3D rotations specified by axis and angle of rotation.

my point is that you can do algebra to show all the SU(2) --> SO(3) stuff but that doesn't necessarily describe the very nice straightforward geometric way to go from (axis, angle) to q=a+bi+cj+dk such that conjugation by q effects the desired rotation. again, if i remember correctly it works out to something like a=cos(theta/2), |im(q)|=sin(theta/2), and im(q) points in the direction of the axis you want to rotate around.

# ? Oct 31, 2014 02:47

Symbolic Butt: Mar 22, 2009; (_!_); Buglord

Bloody posted:

lol i skipped 2 pages of this loving terrible thread

I'm unironically proud of this thread so suck my dilz bloody

# ? Oct 31, 2014 04:32

Deacon of Delicious: Aug 20, 2007; I bet the twist ending is Dracula's dick-babies

Symbolic Butt posted:

suck my dilz bloody

ok not that this is great in context, but uh

# ? Oct 31, 2014 04:45

Symbolic Butt: Mar 22, 2009; (_!_); Buglord

I tried finding some article about how quaternions are bad for programming and how they don't add anything. I wanted to read it again and see if it had any merit but I couldn't find it.

in my experience complex numbers really do make a lot of computations easier to express when you're working on the plane. so I kinda expect that quaternions have the same effect but I idk because I never did any programming with quaternions

# ? Oct 31, 2014 04:48

bump_fn: Apr 12, 2004; _{two of them}

Symbolic Butt posted:

I tried finding some article about how quaternions are bad for programming and how they don't add anything. I wanted to read it again and see if it had any merit but I couldn't find it.

in my experience complex numbers really do make a lot of computations easier to express when you're working on the plane. so I kinda expect that quaternions have the same effect but I idk because I never did any programming with quaternions

cx numbers are important fir quantum emchanhics idk if quaternions are (grassmanians def are hth) qwelp thats my story

# ? Oct 31, 2014 04:59

Broken Machine: Oct 22, 2010

One reason why quaternions are popular for applications, especially 3d graphics is they avoid Gimbal lock. Helps prevent your camera from going all over the place.

# ? Oct 31, 2014 05:01

El Wombato: Mar 19, 2008; Mexican Marsupial

Yea and interpolation in quaternion space actually works pretty well unlike a lot of other representations.

# ? Oct 31, 2014 05:06

Brain Candy: May 18, 2006

Symbolic Butt posted:

I tried finding some article about how quaternions are bad for programming and how they don't add anything. I wanted to read it again and see if it had any merit but I couldn't find it.

in my experience complex numbers really do make a lot of computations easier to express when you're working on the plane. so I kinda expect that quaternions have the same effect but I idk because I never did any programming with quaternions

http://number-none.com/product/Understanding%20Slerp,%20Then%20Not%20Using%20It/ ?

# ? Oct 31, 2014 05:19

LP0 ON FIRE: Jan 25, 2006; _{beep boop}

i use this iphone app called Nabla that's a little rough around the edges, but i like seeing how many problems i can get done before the timer runs out. my record is 95, but today for some reason the math part of my mind is not working well. what's up with that? i'm averaging about 26-30 lol

# ? Oct 31, 2014 15:32

Malcolm XML: Aug 8, 2009; I always knew it would end like ｔｈｉｓ．

Dixie Cretin Seaman posted:

if you like the geometric approach to complex numbers, the book "Visual Complex Analysis" (whose full PDF seems to be the 2nd hit for the title on google, just sayin') goes through most of classical complex analysis this way. it's impressive how far the author gets with mostly geometric arguments.

i would flip what the link above says, though. instead of saying all numbers are really imaginary i would say complex numbers aren't really numbers, in the usual sense. they're a set of rotation and scaling operators in 2 dimensions that happen to share enough important algebraic properties with traditional numbers that you can treat them like numbers for the sake of many computations.

if you're willing to drop multiplicative commutativity there's a 4-dimensional extension of complex numbers called quaternions, and in 8 dimensions there's a non-associative extension called octonions. it's been proven that those are all the possibilities for defining sets like these (extensions of real numbers that let you add, multiply, and divide by non-zero elements).

Visual Complex Analysis is the greatest math book ever written

Complex Analysis is super nice and far more interesting than real analysis and ends up explaining tons of things in applied math. Fourier series, function approximation and poo poo are just trivially done in C

# ? Oct 31, 2014 16:26

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# ? Apr 26, 2024 08:37

Malcolm XML: Aug 8, 2009; I always knew it would end like ｔｈｉｓ．

El Wombato posted:

Yea and interpolation in quaternion space actually works pretty well unlike a lot of other representations.

yeah there are other charts of SO(3) that are just as good, like mobius rotations on the riemann sphere but quaternions are nicer i guess?

# ? Oct 31, 2014 16:31

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