1230: "Polar/Cartesian"

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The Synologist
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1230: "Polar/Cartesian"

Postby The Synologist » Wed Jun 26, 2013 6:05 am UTC

Image

http://xckd.com/1230

Alt-text: "Protip: Any two-axis graph can be re-labeled 'coordinates of the ants crawling across my screen as a function of time"

...yeah, someone's going to have to explain this one to me

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Re: 1230: "Polar/Cartesian"

Postby rhomboidal » Wed Jun 26, 2013 6:09 am UTC

Maybe it's a quantum graph demonstrating polar/Cartesian duality.

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Re: 1230: "Polar/Cartesian"

Postby justusranvier » Wed Jun 26, 2013 6:22 am UTC

Well done.

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Re: 1230: "Polar/Cartesian"

Postby Djehutynakht » Wed Jun 26, 2013 6:27 am UTC

I'm going to chirp in for the "totally lost" camp here.

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Re: 1230: "Polar/Cartesian"

Postby phlip » Wed Jun 26, 2013 6:34 am UTC

Explaining the joke:

A Cartesian graph is the type of graph you're probably used to, where the domain of the function (in this case, time) goes from left to right, and the value of the function goes up and down. So, in this case, the graph would indicate that we started at about 50/50 confidence, but ended at 0% confidence - ie we were sure it was a Cartesian graph.

A polar graph, on the other hand, is one where the domain of the function is an angle, and the value of the function is represented by the distance from the centre. So, starting with the angle "up", we see that the graph indicates we started at about 50/50 confidence, and as it rotates around to the angle "right", the distance now indicates we ended at 100% confidence - ie we were sure it was a polar graph.

Or, in short: the graph can be read in two ways, and because the graph is self-referential, both ways of reading the graph lead you to the conclusion that the one you chose was correct.

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Re: 1230: "Polar/Cartesian"

Postby slinches » Wed Jun 26, 2013 6:40 am UTC

He wouldn't have this problem if he had labeled his axes. This reminds me of the old adage "Those who do not learn from their own comics are doomed to repeat them." Or something like that, I forget.

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Re: 1230: "Polar/Cartesian"

Postby The Synologist » Wed Jun 26, 2013 7:08 am UTC

phlip wrote:Explaining the joke:

A Cartesian graph is the type of graph you're probably used to, where the domain of the function (in this case, time) goes from left to right, and the value of the function goes up and down. So, in this case, the graph would indicate that we started at about 50/50 confidence, but ended at 0% confidence - ie we were sure it was a Cartesian graph.

A polar graph, on the other hand, is one where the domain of the function is an angle, and the value of the function is represented by the distance from the centre. So, starting with the angle "up", we see that the graph indicates we started at about 50/50 confidence, and as it rotates around to the angle "right", the distance now indicates we ended at 100% confidence - ie we were sure it was a polar graph.

Or, in short: the graph can be read in two ways, and because the graph is self-referential, both ways of reading the graph lead you to the conclusion that the one you chose was correct.

Wow that is cool. Is there a reason I've never heard of polar graphs despite having gone through school and gotten a Comp Sci degree?

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Re: 1230: "Polar/Cartesian"

Postby Mjb » Wed Jun 26, 2013 7:30 am UTC

The Synologist wrote:Wow that is cool. Is there a reason I've never heard of polar graphs despite having gone through school and gotten a Comp Sci degree?

The idea usually shows up somewhere between geometry and precalculus. Standard 2nd-semester calculus includes function integrals for the polar graph's 3D cousins, spherical and cylindrical.

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Re: 1230: "Polar/Cartesian"

Postby DBPZ » Wed Jun 26, 2013 9:21 am UTC

There are two probabilities: Pr( polar | curve ) and Pr ( Cartesian | curve ). The sum of the two probabilities must be less than one.

Which Pr is the Y-axis?

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Re: 1230: "Polar/Cartesian"

Postby Klear » Wed Jun 26, 2013 9:30 am UTC

DBPZ wrote:There are two probabilities: Pr( polar | curve ) and Pr ( Cartesian | curve ). The sum of the two probabilities must be less than one.

Which Pr is the Y-axis?


Flip a coin =P

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Re: 1230: "Polar/Cartesian"

Postby brandbarth » Wed Jun 26, 2013 9:40 am UTC

DBPZ wrote:The sum of the two probabilities must be less than one.


No, If it's polar or cartesian and there is no third option, P(A) + P(nonA) = 1

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Re: 1230: "Polar/Cartesian"

Postby Mjb » Wed Jun 26, 2013 9:54 am UTC

DBPZ wrote:There are two probabilities: Pr( polar | curve ) and Pr ( Cartesian | curve ). The sum of the two probabilities must be less than one.

Which Pr is the Y-axis?

It's a "certainty" graph, so it can reach 1 for a given definition of "sure". For both systems, the vertical axis is "Cert(Polar|Curve)". The horizontal one is more interesting.

brandbarth wrote:No, If it's polar or cartesian and there is no third option, P(A) + P(nonA) = 1

Mathematically true, but Polar and non-Cartesian are not identical. And on the internet, P("this is not a graph") = ?

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Re: 1230: "Polar/Cartesian"

Postby Sneftel » Wed Jun 26, 2013 10:16 am UTC

Comics like this are hyperbolae.

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Re: 1230: "Polar/Cartesian"

Postby jc » Wed Jun 26, 2013 11:33 am UTC

Mjb wrote:
DBPZ wrote:There are two probabilities: Pr( polar | curve ) and Pr ( Cartesian | curve ). The sum of the two probabilities must be less than one.

Which Pr is the Y-axis?

It's a "certainty" graph, so it can reach 1 for a given definition of "sure". For both systems, the vertical axis is "Cert(Polar|Curve)". The horizontal one is more interesting.

brandbarth wrote:No, If it's polar or cartesian and there is no third option, P(A) + P(nonA) = 1

Mathematically true, but Polar and non-Cartesian are not identical. And on the internet, P("this is not a graph") = ?


Indeed. On the internet (and in most of the mass media), there's always a nonzero probability that it's a pseudo-graph, intended to persuade rather than inform. The usual tipoff is that one or both axes are unlabelled, or labelled with numbers without units.

Note that the "graph" in the cartoon has a semi-of unit ("%", but we don't know what it's a percent of) on the vertical axis, but none on the horizontal axis. This may have been intentional, to hint that the graph actually contains no information in the same way that media graphs often do.

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Re: 1230: "Polar/Cartesian"

Postby PM 2Ring » Wed Jun 26, 2013 11:42 am UTC

phlip wrote:Or, in short: the graph can be read in two ways, and because the graph is self-referential, both ways of reading the graph lead you to the conclusion that the one you chose was correct.

With the minor quibble that in a conventional polar graph the polar axis is horizontal, with the angle increasing in the anticlockwise direction. My brain would rather see this comic as a conventional cartesian graph than as an unconventional polar one.

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Re: 1230: "Polar/Cartesian"

Postby gmalivuk » Wed Jun 26, 2013 11:53 am UTC

jc wrote:"%", but we don't know what it's a percent of

Percent certainty, as clearly stated above the graph.
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Re: 1230: "Polar/Cartesian"

Postby Barstro » Wed Jun 26, 2013 12:14 pm UTC

I'm a little surprised the alt-text went with ants instead of red spiders.

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Re: 1230: "Polar/Cartesian"

Postby cellocgw » Wed Jun 26, 2013 12:45 pm UTC

Barstro wrote:I'm a little surprised the alt-text went with ants instead of red spiders.


I think it's an homage to some fella's explanation of statistics failure in economics. He wrote, more or less, that if you let loose a horde of ants on a page of stock reports, and pick the ant which landed on a winning stock as the "smart investor," you're pretty much doing what people do when adulating some (currently successful) hedge fund manager.
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Re: 1230: "Polar/Cartesian"

Postby drachefly » Wed Jun 26, 2013 1:19 pm UTC

The alt text is false. A circle does not describe such a motion unless the axes do not extend past the circle (what, did two ants suddenly materialize on your screen and then disappear later?). You can however say 'plot of the X and Y positions occupied by the ants on my screen'.

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Re: 1230: "Polar/Cartesian"

Postby yellow103 » Wed Jun 26, 2013 1:35 pm UTC

Alt-text: "Protip: Any two-axis graph can be re-labeled 'coordinates of the ants crawling across my screen as a function of time"

Any THREE axis graph can be re-labeled; coordinates of the flys buzzing in front of my screen as a function of time.

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Re: 1230: "Polar/Cartesian"

Postby eran_rathan » Wed Jun 26, 2013 1:37 pm UTC

oh look, another graph joke. And not a very good one either.

1/5 stars. would not recommend.
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Re: 1230: "Polar/Cartesian"

Postby Locoluis » Wed Jun 26, 2013 1:56 pm UTC

"Joke"-spoiling graph fix.

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Re: 1230: "Polar/Cartesian"

Postby Barstro » Wed Jun 26, 2013 3:00 pm UTC

cellocgw wrote:
Barstro wrote:I'm a little surprised the alt-text went with ants instead of red spiders.


I think it's an homage to some fella's explanation of statistics failure in economics. He wrote, more or less, that if you let loose a horde of ants on a page of stock reports, and pick the ant which landed on a winning stock as the "smart investor," you're pretty much doing what people do when adulating some (currently successful) hedge fund manager.


Praising random "success" and Malcolm Gladwell's presentations showing that early success leads to greater opportunities for training, practice, and eventual skill than those without said early success causes me to wonder about some people's smugness about how they deserve their elevated positions.

Anyway; glad to see that there is a particular reason for the use of ants.

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Re: 1230: "Polar/Cartesian"

Postby San Fran Sam » Wed Jun 26, 2013 3:10 pm UTC

phlip wrote:Explaining the joke:

A Cartesian graph is the type of graph you're probably used to, where the domain of the function (in this case, time) goes from left to right, and the value of the function goes up and down. So, in this case, the graph would indicate that we started at about 50/50 confidence, but ended at 0% confidence - ie we were sure it was a Cartesian graph.

A polar graph, on the other hand, is one where the domain of the function is an angle, and the value of the function is represented by the distance from the centre. So, starting with the angle "up", we see that the graph indicates we started at about 50/50 confidence, and as it rotates around to the angle "right", the distance now indicates we ended at 100% confidence - ie we were sure it was a polar graph.

Or, in short: the graph can be read in two ways, and because the graph is self-referential, both ways of reading the graph lead you to the conclusion that the one you chose was correct.


Thanks for the explanation and links. i had forgotten what a polar plot was. but...

doesn't the existence of the axes themselves define away the possibility that this picture/graph is a polar plot?

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Re: 1230: "Polar/Cartesian"

Postby eran_rathan » Wed Jun 26, 2013 3:28 pm UTC

San Fran Sam wrote:
phlip wrote:Explaining the joke:

A Cartesian graph is the type of graph you're probably used to, where the domain of the function (in this case, time) goes from left to right, and the value of the function goes up and down. So, in this case, the graph would indicate that we started at about 50/50 confidence, but ended at 0% confidence - ie we were sure it was a Cartesian graph.

A polar graph, on the other hand, is one where the domain of the function is an angle, and the value of the function is represented by the distance from the centre. So, starting with the angle "up", we see that the graph indicates we started at about 50/50 confidence, and as it rotates around to the angle "right", the distance now indicates we ended at 100% confidence - ie we were sure it was a polar graph.

Or, in short: the graph can be read in two ways, and because the graph is self-referential, both ways of reading the graph lead you to the conclusion that the one you chose was correct.


Thanks for the explanation and links. i had forgotten what a polar plot was. but...

doesn't the existence of the axes themselves define away the possibility that this picture/graph is a polar plot?


Nope. the conversion between polar and cartesian coordinates is expressed by

r =sqrt(x2+y2)
theta = tan-1(y/x)

edit: or you can do r*cos(theta) = x and r*sin(theta) = y to go the other way.
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Re: 1230: "Polar/Cartesian"

Postby zogwarg » Wed Jun 26, 2013 3:31 pm UTC

The math for this one seems iffy:

randall seems to have graphed in cartesian mode f(t)=(1 - t²)/2

This is valid on the assumption that (x,y)_cartesian(t)=(x,y)_polar(t)

if p(t) is the probability expressed in the opening sentence:

x_cart(t)=t
y_cart(t)=1- p(t)
n_cart(t)²= t² + 1 - 2*p(t) + p(t)²

x_pol(t)=sin(t*pi/2)*p(t)
y_pol(t)=cos(t*pi/2)*p(t)
n_pol(t)²=p(t)²

(n_c²==n_p² )
- -> p(t)= (1 + t²)/2
- ->f(t)=1-p(t)= (1-t²)/2

unfortunately (x,y)_cartesian(t)=/=(x,y)_polar(t) as :
if (x,y)_cartesian(t)=/=(x,y)_polar(t)
then n_c²==n_p² and x_c==x_p
then p(t)= (1 + t²)/2 and t= sin(t*pi/2) * ((1 + t²)/2)
(the last statement being obviously false for example with t=0.5)

Now obviously the graph isn't exactly correct,
my questions are:
-how big a mistake is made (if x_c - x_p is small enough it might not be significant)
-Is this problem even solvable ? (meaning can you overlay self consistent cartesian and polar graphs for this question)

(I would have included an overlay of the comic with a plot of f(t) [which matches perfectly] but I haven't got enough posts yet)

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Re: 1230: "Polar/Cartesian"

Postby rmsgrey » Wed Jun 26, 2013 4:13 pm UTC

San Fran Sam wrote:doesn't the existence of the axes themselves define away the possibility that this picture/graph is a polar plot?


A polar plot requires some form of scale too - and having the scale shown at both ends of the curve is not unreasonable...

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Re: 1230: "Polar/Cartesian"

Postby Fantomius » Wed Jun 26, 2013 5:32 pm UTC

slinches wrote:He wouldn't have this problem if he had labeled his axes.

You are absolutely right: he wouldn't have this problem if he had labeled his axes -- but then, he wouldn't have a joke, either.

The joke comes from the fact that you can't tell if it's a common Cartesian graph, or a less-common Polar graph.

At first I was a bit confused at the comic, because I misread the text as "Certainly this is a clockwise polar plot, ..." instead of "Certainty that this is a clockwise polar plot, ...". Once I re-read (and properly understood) the text, I realized what the joke was:

If you think that the plot is a Cartesian graph, then you're less likely to think it's a Polar graph. (This makes sense.) But the opposite is also true: If you think that the plot is a Polar graph, then you're more likely to think it's a Polar graph. (This also makes sense.)

So the plot is sort of like a self-fulfilling prophecy: The more you believe it's either a Cartesian or Polar plot, the more certain you are of it, as time goes by.

I think it's pretty clever! Of course, you need to be familiar with both Cartesian and Polar graphs for this cartoon to make sense.

(First time poster.) :)

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Re: 1230: "Polar/Cartesian"

Postby orthogon » Wed Jun 26, 2013 6:08 pm UTC

As any fule no, on a Cartesian plot, the two orthogonal axes are "mind" and "body".
xtifr wrote:... and orthogon merely sounds undecided.

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Re: 1230: "Polar/Cartesian"

Postby janhe » Wed Jun 26, 2013 8:07 pm UTC

The title-text got to me.

If a graph represents a function from time to coordinates, that impies that the output of the function is a tuple of at least 2 numbers ("coordinates").
e.g.: f(1 second) = (2,3)
Since you are plotting a function of time, convention suggests the x-axis represents time. How would the "coordinates" be represented on the y-axis? Randall did specify that we're dealing with a 2-axis graph.

I'm assuming a carthesian coordinate system to keep things less complicated.

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Re: 1230: "Polar/Cartesian"

Postby Copper Bezel » Wed Jun 26, 2013 10:10 pm UTC

Yeah, that bothered me, too. X, Y, and time make three axes. I suppose the ant is (or ants are) moving uniformly rightward, which seems like a very odd thing for ants to do. But if you let them roam free, you'll get a non-monotonic X and it won't be a function at all. Humph.
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Re: 1230: "Polar/Cartesian"

Postby Turing Machine » Thu Jun 27, 2013 1:13 am UTC

Barstro wrote:
cellocgw wrote:
Barstro wrote:I'm a little surprised the alt-text went with ants instead of red spiders.


I think it's an homage to some fella's explanation of statistics failure in economics. He wrote, more or less, that if you let loose a horde of ants on a page of stock reports, and pick the ant which landed on a winning stock as the "smart investor," you're pretty much doing what people do when adulating some (currently successful) hedge fund manager.


Praising random "success" and Malcolm Gladwell's presentations showing that early success leads to greater opportunities for training, practice, and eventual skill than those without said early success causes me to wonder about some people's smugness about how they deserve their elevated positions.

Anyway; glad to see that there is a particular reason for the use of ants.


Malcolm Gladwell hasn't shown anything except that he doesn't know what an eigenvalue is. Seriously, dude is wrong.

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Re: 1230: "Polar/Cartesian"

Postby candybrie4zo » Thu Jun 27, 2013 3:21 am UTC

janhe wrote:The title-text got to me.

If a graph represents a function from time to coordinates, that impies that the output of the function is a tuple of at least 2 numbers ("coordinates").
e.g.: f(1 second) = (2,3)
Since you are plotting a function of time, convention suggests the x-axis represents time. How would the "coordinates" be represented on the y-axis? Randall did specify that we're dealing with a 2-axis graph.

I'm assuming a carthesian coordinate system to keep things less complicated.


Perhaps it's a parametric graph with time as the parameter. Of course you'd need to indicate direction, but it wasn't a very well labeled graph to begin with.

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Re: 1230: "Polar/Cartesian"

Postby ptityeti » Thu Jun 27, 2013 2:34 pm UTC

My take on the graph.
- It shows the probability that the graph is cartesian. You would plot this in a cartesian graph, right? So P(cartesian)=y because it is shown on the y-axis (and the x-axis is time).
- It shows the probability that the graph is polar. You would plot this in a polar plot, wouldn't you? So P(polar)=r=sqrt(x^2+y^2) (and theta would be time).
- Now, using axiom 1230, this graph is either cartesian or polar, hence P(cartesian)+P(polar)=1.

It is left as an exercise for the reader to show that y+sqrt(x^2+y^2)=1 is a parabola which not so coincidentally is shown in the graph. Well, the part of the parabola in the first quadrant at least.

So, depending on what you would like to see (cartesian or polar probability) the graph is cartesian and polar at the same time. Which is why the x-axis is not labeled: it is either time or P(polar), depending on what you are reading from the graph.

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Re: 1230: "Polar/Cartesian"

Postby neremanth » Thu Jun 27, 2013 3:34 pm UTC

candybrie4zo wrote:
janhe wrote:The title-text got to me.

If a graph represents a function from time to coordinates, that impies that the output of the function is a tuple of at least 2 numbers ("coordinates").
e.g.: f(1 second) = (2,3)
Since you are plotting a function of time, convention suggests the x-axis represents time. How would the "coordinates" be represented on the y-axis? Randall did specify that we're dealing with a 2-axis graph.

I'm assuming a carthesian coordinate system to keep things less complicated.


Perhaps it's a parametric graph with time as the parameter. Of course you'd need to indicate direction, but it wasn't a very well labeled graph to begin with.


Yeah, that's what I took it to mean: each point on the graph shows the coordinates of an ant by the x coordinate of the point being the x coordinate of the ant, and the y coordinate of the point being the y coordinate of the ant, and then you get time by joining the dots (or by taking measurements so frequently over time that you have so many points they seem to form a line). You then have a line which you can follow: start at one end and you're at t=0, and if the measurements were at regular intervals then when you get to the next point you're at t=<gap between measurements>, at the next point you're at t=2*<gap between measurements> and so on. Of course, as you point out, without an indication of direction you don't know which end of the line is t=0. Also if the ant stops at any point longer than the gap between measurement occasions you don't have the correct time information any more (e.g. if the measurements are every second and the ant stops 2.5 seconds in for 3 seconds before setting off again, then you'll be interpreting the graph correctly for the first 3 points, corresponding to t=0s, t=1s, and t=2s, but the third point you will think is just t=3s whereas it's actually t=3s, t=4s and t=5s, the fourth point you will think is t=4s whereas it's actually t=6s and so on).

Also contrary to the title text's claim this would only work for graphs that consist of lines of joined up points - you'd have to say the labelling was really bad if you tried doing that to a scatterplot, as there wouldn't be any indication of time at all.

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Re: 1230: "Polar/Cartesian"

Postby oliphaunt » Fri Jun 28, 2013 10:50 am UTC

Barstro wrote:
cellocgw wrote:
Barstro wrote:I'm a little surprised the alt-text went with ants instead of red spiders.


I think it's an homage to some fella's explanation of statistics failure in economics.

[...]

[...]

Anyway, glad to see that there is a particular reason for the use of ants.


I suppose the reason you asked was because of the old "red spiders" comics?

http://xkcd.com/8
http://xkcd.com/43
http://xkcd.com/47
http://xkcd.com/126

Well spotted, it would have fitted well with the red data line in the comic.
I'd guess he simply didn't think of the possibility.
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Re: 1230: "Polar/Cartesian"

Postby Flumble » Fri Jun 28, 2013 2:23 pm UTC

janhe wrote:The title-text got to me.

If a graph represents a function from time to coordinates, that impies that the output of the function is a tuple of at least 2 numbers ("coordinates").
e.g.: f(1 second) = (2,3)
Since you are plotting a function of time, convention suggests the x-axis represents time. How would the "coordinates" be represented on the y-axis? Randall did specify that we're dealing with a 2-axis graph.


One is perfectly able to map n real (or complex or another superset) dimensions to one dimension using a space-filling curve. :D

Though the ants might need to follow a very difficult path to get from 50% to 0%.


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