What's the chance of an acute triangle?

[superbonzo]

I'll try being more specific. This is what nuzzle did: he defined an angle preserving transformation f from the set of triangles T to the set of triangles; he parametrized the image of that transformation as a subset of the plane (what he describes as an "American football"); then he uniformly picked points from the football taking the probability that such points represent acute triangles. Recapitulating, he argues that : (*) being the transformation angle preserving and being triangle acuteness an angle dependent property then such probability is "the chance that a triangle is acute":

now, if A is the set of acute triangles, F is the "American football" set in R^2, you have

A -j-> T -f-> T -p-> F

( where j is the natural inclusion and p a function sending each transformed triangle to the corresponding point in the american football set )

the probability he computed is equal to L{ p(f(j(A))) } / L{ F } , where L{} is the Lebesgue measure in the plane ( aka the "area" ).

now, the statement (*) tantamounts saying that:

if f and f' preserve acuteness and F and F' are bounded then L{ p(f(j(A))) } / L{ F } = L{ p'(f'(j(A))) } / L{ F' }

this statement is false ( I provided a counter example in post #10, but it should be evident from the very definition of a measure ).

Would it make any difference if the points could somehow... "pop-up" on their own? If there wasn't any sentient-being-introduced picking involved? Or this is just the same?

what do you mean ?

[nuzzle]

So, essentially: what is the percentage of acute triangles in the set of all possible triangles? Is there an inherent flaw in such a problem proposition, too?

That would remove the seemingly problematic issue of whether one can pick triangles at random in the plane.

It would give the chance of an acute triangle an a-priory probability just like there's an a-priori probability attached to each side of a coin (1/2) and each side of a dice (1/6) for example.

[nuzzle]

[b]
I'd appreciate it too , but there are so many foundational issues involved here, ranging from the foundation of mathematics to physics, probability and plain natural philosophy... it would take pages only for introducing the general problem rigorously...

I know and that's what makes if fun. -

for example, given a triangle translate and scale it in such a way that the circumscribed circle is centered in the origin and has radius one. Then rotate it in such a way one of its side is vertical. Obviously our assumptions are satisfied; also, the image is representable as a simple pair of angles, thus it have the same "dimensionality" of the image of your transformation. Let's take such pair of angles uniformly on the circle. Is the result the same ?

a quick calculation (non tested, but I suppose is correct) with wolfram mathematica gives me 0.360+-0.001 for your transformation and 0.248+-0.001 for the transformation above. Both angle preseving, different "chances".

I have argued that limiting the plane to a region skews the acute triangle fraction because not all triangles will fit within the region. Transforming all triangles to a region will also skew the triangle fraction in a different way. In this case the region won't be evenly populated so the acute triangle fraction will vary within the region.

My transformation is not to a region. It preserves both the internal angles and the percentage of acute triangles in relation to all triangles. Fixing the triangles to a region takes place after the transformation.

My transformation makes the longest side of every triangle equal to 1 and all triangles are aligned along this side with the opposite corner in the same direction. This means all opposite corners will fall within a half American football shaped region. If the opposite corners are not evenly distributed my solution falls apart of course.

BTW, you could think that there exist a "physically" intuitive notion of "uniformly random"; you'd be wrong; I'll try to give you an example ( that does not nearly exhaust the many foundational issues involved here ):

There doesn't even exist a physically intutive notion of the infinite plane. All physical planes are regions. But I argue that if one accepts the notion of an infinite plane as an abstraction, nothing stands in the way of accepting the notion of random points in it.

[superbonzo]

My transformation is not to a region. It preserves both the internal angles and the percentage of acute triangles in relation to all triangles. Fixing the triangles to a region takes place after the transformation.

...If the opposite corners are not evenly distributed my solution falls apart of course.

so your making the assumption that your "transformation preserves ... the percentage of acute triangles in relation to all triangles" to prove that you have found "the percentage of acute triangles in relation to all triangles" without ever having defined what "the percentage of acute triangles in relation to all triangles" is ?

in order to compute a "percentage" you need to count two sets A and B and take the ratio between them, right ? well, how do you count sets ?

there are essentially two ways:

1) if you want your "counting" to produce a unique object representing the "number of its elements" you'll came up with the concept of cardinality: two sets have the same cardinality if there exist a bijection between them. For example, every finite set is in bijective corrispondence with a set {1,2,..,n} for some natural number n, so we simply say its cardinality is 'n'. The next cardinal greater then every finite set is the cardinality of the set of natural numbers. Note that the sets of integers, pair of integers, tuple of intergers, rational numbers, odd numbers, prime numbers, algebraic numbers,... have all the cardinality of the set of natural number. So things like the ratio of (the cardinality of) odd numbers to the ratio of (the cardinality of) even numbers is not defined.

2) the other way is to literally count them, that is attach labels 1,2,3,.. to elements of the two sets. This tantamounts defining a sequence of strictly increasing finite subsets of A x B. Each such subset has finite cardinality, thus, you can take their ratios. Finally, you take the limit of that ratio. For example, you could count odd to even numbers as 1,2,3,4,5,6,... and concluding that their "inifinte" ratio is 1/2. The problem is that if you count them in a different way you'll get different limits (including non converging ones): for example, 1,3,2,5,7,4,.. is a well defined such sequence but the ratio converges to 1/3.

Now you could add an ad hoc rule to impose that, say, the "right" counting sequences are those that preserve ordering, in this way you could demonstrate that for such sequences the ratio always converge to 1/2. Fine. But what happens if a set has no canonical definition of order (the set of triangles is such a set) ?

Of course, you can always add ad hoc rules more and more restricting the way of counting elements; but at the end you'll end up with tautological results simply restating your starting assumptions.

Anyway, can you describe what do you mean exactly by "Fixing the triangles to a region takes place after the transformation." ?
that is, try writing down what properties should a transformation satisfy in order to be considered "acute triangle percentage preserving".

There doesn't even exist a physically intutive notion of the infinite plane. All physical planes are regions. But I argue that if one accepts the notion of an infinite plane as an abstraction, nothing stands in the way of accepting the notion of random points in it.

There's no need to "accept" the idea of an infinite plane which is a perfectly defined concept in every set theory that includes the natural number system and the axiom of choice ( all this is related but totally indipendent from any tenet coming from other philosophical fields, including existentiality conditions of actual infinities ).

Regarding the concept of "physical plane" you're again taking things too naively. In physics and mathematics the geomertical structure of the plane (and more often its higher dimensional analogues) happens in many different contexts; in some of them they indeed have an intuitive physical interpretation: for example, if you assume the realism of special relativity then spacetime is a Minkowsky space whose elements have a physically well defined interpretation. if you assume the realism of general relativity then the Einstein field is a 2-2 tensor field defined over tangent spaces that in turn have a well defined physical interpretation.

BTW, you could think that objects described in physical theories can never corresponds exactly to the "real" things they represents, they are always approximated. Sometimes this is the case ( for example, a rigid body in classical mechanics is never assumed really rigid, or a electric AC current is never assumed really sinusoidal ), sometimes not. Notable examples are the solutions of Einstein equations, that are not equations in its common sense, because its solutions are geometrical structures. So, if you assume the realism of general relativity then, say, the Kerr black holes are in a strict sense isomorphic to their geometrical abstract representations ( that is, if you take a region of space time surrounding it and whose density satisfy certain conditions, then the Einstein field has exactly that form, no matter how energy is microscopically distributed there ).

[TheGreatCthulhu]

what do you mean ?

I mean, since the problem is related to and since the result depends on the process of picking (or the procedure used to pick) the points, would it make a difference, would the problem still exist if no one would actually do the picking, but if the points that would form the triangle emerged on their own, in some natural, random way. Like... I don’t know... raindrops hitting the ground, thus defining random points on a plane, or random impacts on the surface of the Moon.
Maybe my understanding of the notion of randomness lacks some important aspect of it.
I’ll quote the Wolfram Math World article again:
"Unfortunately, the solution of the problem depends on the procedure used to pick the "random" points (Portnoy 1994)." (http://mathworld.wolfram.com/ObtuseTriangle.html; Portnoy 1994 paper PDF: http://projecteuclid.org/DPubS?servi....ss/1177010497).
Note that the word 'random' is between quotation marks, indicating that there’s no true randomness involved once the pick procedure is defined.

But, back to the idea of the points emerging on their own. As I’ve asked before - is this jut the same as picking the points? If there’s no person, or rather, if there’s no method defined that is used to pick them, but if the points emerged in some natural (truly random?) process - would anything change, would the question still be ill-posed? Or there would be some underlying rules/laws that would govern this emergence, and thus these rules could be considered equivalent to the pick method? Are these just two ways of expressing the same thing, or there’s a fundamental difference.

In nature, processes that would be considered random are actually the result of existing starting conditions and forces at work, they are the result of the laws of physics. Even a simple act of throwing a coin is the result various physical interactions governed by these laws. So, one could argue that everything that happens is just a result of a long cause & effect chain, and there’s nothing truly random. But I refuse to believe that. There must be some truly random process in nature, at the very fundamental level of it (maybe at quantum level?). I won’t go further into it, since with this we enter the realm of philosophy.

As you’ve said: “you could think that there exist a "physically" intuitive notion of "uniformly random"; you'd be wrong”.

[nuzzle]

I'll try being more specific.

I guess I had it coming but this kind of sophisticated math reasoning is above my head. Still your argumentation is beginning to sink in.

You've showed that two kinds of angle preserving transformations from infinite plane to finite plane leads to different results. One was to a unit circle and one was to a unit line segment (as I suggested).

There are two ways to continue here for me.

1. You say there's no difference between the two but I claim there is.

2. I have to admit I cannot come up with why one should be considered more natural or logical than the other.

How much I would like it to be 1 it seems I will have to go for 2. I'll have to regroup and see if I can come up with something. If not, at least now I have a much better understanding of the fundamental issues at play here.

To use my urn analogy. An urn can have an infinite number of balls and the fraction of white balls exists and can be measured. The problem is that there are many urns and the white ball fraction varies between them.

[nuzzle]

I realized this while I used Paint to visualize what you said, but I just saw that the idea is not new - in fact, there's a text on this topic on the page linked in the thread where all this started (Wolfram Math World: http://mathworld.wolfram.com/ObtuseTriangle.html).

I definately recognize my "half American football" in that link. And when I simulate it I get the analytically expected result.

So although "my" solution came 100 years too late there is at least some consolation in that I arrived at it independently.

I'll just have to accept that when one goes from infinite to finite, strange things happens to probability.

[nuzzle]

Here are the simulations I promised.

They're modifications of the code I used in the other thread.

There are two. The first is the situation where all triangles in the infinite plane are transformed to the unit circle. In the second all triangles are transformed with their longest side on the unit line segment. The latter is "my" solution in this thread.

double rnd() { // random double between 0.0 and 1.0 (inclusive)
	return double(rand()) / double(RAND_MAX);
}

void test1() {
	srand(unsigned(time(0)));

	int N=1000000; // number of tries
	double PI = 3.1415926535897932384626;
	int n = 0; // hit counter
	   // unit circle
	for (int i=0; i<N; i++) {
		double fi = 2.0*PI*rnd();
		double ax = sin(fi); // 3 random points on unit circle
		double ay = cos(fi);
		fi = 2.0*PI*rnd();
		double bx = sin(fi);
		double by = cos(fi);
		fi = 2.0*PI*rnd();
		double cx = sin(fi);
		double cy = cos(fi);
		
		double a2 = (ax-bx)*(ax-bx) + (ay-by)*(ay-by); // side lengths squared
		double b2 = (bx-cx)*(bx-cx) + (by-cy)*(by-cy);
		double c2 = (cx-ax)*(cx-ax) + (cy-ay)*(cy-ay);
		
		if (a2+b2>c2 && b2+c2>a2 && c2+a2>b2) n++; // count acute triangles
	}	
	std::cout << double(n) / double(N) << "\n"; // print frequency (probability)

	n = 0; // hit counter
	   // unit line segment
	for (int i=0; i<N; i++) {
		double ax = 0.0;
		double ay = 0.0;
		double bx = 1.0;
		double by = 0.0;
		double cx;
		double cy;
		do { // repeat until point is within half American football
			cx = rnd();
			cy = rnd();
		} while ((cx*cx + cy*cy > 1.0) || ((cx-1.0)*(cx-1.0) + cy*cy > 1.0));

		double a2 = (ax-bx)*(ax-bx) + (ay-by)*(ay-by); // side lengths squared
		double b2 = (bx-cx)*(bx-cx) + (by-cy)*(by-cy);
		double c2 = (cx-ax)*(cx-ax) + (cy-ay)*(cy-ay);

		if (a2+b2>c2 && b2+c2>a2 && c2+a2>b2) n++; // count acute triangles
	}	
	std::cout << double(n) / double(N) << "\n"; // print frequency (probability)
}

[TheGreatCthulhu]

So, regarding the "percentage of all acute triangles" issue: if I understood this well - because both the set of all triangles and the set of acute triangles have the same cardinality, since there exits a corresponding bijection, it doesn't make sense to compare this sets in terms of percentage/ratio? (Same goes for the set of obtuse triangles, or any subset of the set of all triangles on which a similar bijection may be defined.)

Let us step back from the triangle problem a bit, and take a look at this simple example.
Consider the segment [0,1], and the following question: What is the percentage of all elements x e [0,1] that satisfy x <= 0.5, in relation to all the elements of [0, 1]?

Following the above logic, asking this question would also make no sense. Clearly, these elements are all the elements of [0, 0.5] (there are infinitely many), and one could intuitively say that the answer is 50&#37;, but, since there's a bijection from [0,1] to [0, 0.5], the cardinality is the same, so the question is nonsensical.
Am I right?

If so, then this provided me with a new insight in the properties of infinite sets, and I'm rather glad about it.

"The more you understand what is wrong with a figure, the more valuable that figure becomes."
— Lord Kelvin

But, is there some property of these sets that would enable as to quantify the relation between a set of all triangles and it's subset of acute triangles? Does mathematics define such a property? Something that would enable as to speak of these sets in a fashion similar to "percentage"? A numerical value that can tell us in more detail how one set relates to the other, enabling us to know more than just the fact that one is a subset of the other?

If nothing like that is defined, is there any reason that such a property doesn't exist? Maybe our (human) knowledge of sets can be further expanded.

"When you can measure what you are speaking about, and express it in numbers, you know something about it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts advanced to the stage of science."
— Lord Kelvin (again)

[superbonzo]

... because both the set of all triangles and the set of acute triangles have the same cardinality, since there exits a corresponding bijection, it doesn't make sense to compare this sets in terms of percentage/ratio? (Same goes for the set of obtuse triangles, or any subset of the set of all triangles on which a similar bijection may be defined.)

Not exactly. As I sead in post #19, the concept of "numbers that count things" is modeled as cardinal numbers. Cardinal numbers are "usual" integers only for finite sets, thus the usual notion of "ratio" can be defined only for them.

Now, you can do two things:

1) try defining what is a ratio of general cardinal numbers; you can do that excatly in the same way you construct the set of rational numbers from the set of natural numbers ( with some added tecnicalities, it all depends on the axiomatic system you choose... ).
But as you noted, cardinality is a very coarse grained way of looking at sets: the interval [0,1] is the same as the interval [0,0.5], but it's also the same as any subset of R^n, like the interior of a square, a cube or a 100-dimensional sphere...

2) conclude that probability has to do with ratios and countings only when finite sets are involved. With infinite sets you need different concepts. The most common way of representing a probabilistic model is through a "probability space" ( that in turn is a special case of a measure space ) and of "random variables" ( that are simply measurable functions between probability spaces ).

In this way you can see the interval [0,1] as a probability space and you'll be able to conclude that the measure (ie the probability) of the interval [0,0.5] is 0.5.

But, is there some property of these sets that would enable as to quantify the relation between a set of all triangles and it's subset of acute triangles?

yes, you can define a probability structure over the set of triangles ( as an exotic example, you can send bijctively each triangle to a point in the interval [0,1] and use its natural probability measure directly ).

What you cannot do is defining a translation invariant probability measure over that set, because such a thing does not exist ( eg. there's no such probability space in the class of probability spaces ).

[nuzzle]

But, is there some property of these sets that would enable as to quantify the relation between a set of all triangles and it's subset of acute triangles? Does mathematics define such a property? Something that would enable as to speak of these sets in a fashion similar to "percentage"? A numerical value that can tell us in more detail how one set relates to the other, enabling us to know more than just the fact that one is a subset of the other?

I was pretty convinced there existed a percentage of acute triangles among all triangles in the infinite plane, but arguments in this thread has lead me to believe that may not be the case. At least to be able to calculate the percentage you need to transform all triangles to the finite plane and by doing that you influence the percentage. I though I had found a transform which would keep the percentage intact but appearantly that's not even possible.

This is disturbing and my belief is that if you "instead of taking triangles to finity took probability to infinity" you could find the one true percentage. Maybe some matematician can figure that out but I can't.

[TheGreatCthulhu]

This is disturbing [...]

Funny how, when the foundations of our own beliefs are shaken, or when we need to readjust our way of seeing the world, we find it disturbing...

[D_Drmmr]

D_Drmmr:
As for the urn example, I think that it is implied that nuzzle was not talking about an urn with any sort of special arrangement of the balls ("top 100 balls are white"). Also, the urn is just a more "reader-friendly" term for a set, so you shouldn't think of it in terms of it's physical properties, like only being able to pick the topmost balls - imagine that, somehow, you are able to pick any of the balls.

My intention was not to obfuscate the example with practical issues, but to show the importance of implicit assumptions that were not mentioned. In the case of the urn drawing uniformly random is the implicit assumption. In case of the original problem, there are a lot of implicit assumptions along the way. I think superbonzo has done a great job to bring them to light.

Let us step back from the triangle problem a bit, and take a look at this simple example.
Consider the segment [0,1], and the following question: What is the percentage of all elements x e [0,1] that satisfy x <= 0.5, in relation to all the elements of [0, 1]?

Following the above logic, asking this question would also make no sense. Clearly, these elements are all the elements of [0, 0.5] (there are infinitely many), and one could intuitively say that the answer is 50%, but, since there's a bijection from [0,1] to [0, 0.5], the cardinality is the same, so the question is nonsensical.
Am I right?

If I can change your example a bit to make it precise. The probability that a uniformly random point on the interval [0, 1) lies in the interval [0, 0.5) is 0.5. I agree that you cannot count the number of points in either of these intervals, but you can show that for any point in the interval [0, 0.5) there is exactly one uniquely corresponding point in the interval [0.5, 1). The function f(x) = 0.5 + x defines this relation. This property, combined with the definition of a uniform random probability measure (that every possible outcome has the same probability) yields the probability of 0.5.

So, even though we can't count the number of points in either of the intervals above, we can deduce properties for these points that lead to the conclusion that the probability is 0.5. That's the cool thing about mathematics.

[nuzzle]

Funny how, when the foundations of our own beliefs are shaken, or when we need to readjust our way of seeing the world, we find it disturbing...

Yes I find it disturbing. I want an infinite-sided dice and I'm sure I will have one some day.

But for the time being I have to accept that such a dice would be sphere shaped and wouldn't have a side to stop on when thrown. It would keep rooling forever. In short, a round dice has no outcome. This is at the heart of this problem isn't it.

I've noticed that there's a more probabilistically sound approach to this problem called Gaussian triangles. The Gaussian distribution is defined in the plane so it's possible to generate triangles according to this distribution. When I have some time to spare I'll check out so called heavy-tail distributions which I believe also are defined in the plane.

[TheGreatCthulhu]

I’ve done some reading and I think I’ve got it.
But, first a little intro to some concepts, since it might of help to someone who is interested, but nor all too familiar with the concepts that were discussed here.
Others may skip next few passages (unless you want to check if I didn’t misinterpret something).

Probability space is a mathematical tool (or one of available tools) used to create a model of a real-world probabilistic process, or experiment (or a mind experiment – why not?). If we shift our focus form all the mathematical details and definitions, the concept is really quite simple.

Probability space, also called probability triple (O, E, p) is simply a following construction:

O: A set of all possible outcomes - called the sample space.

E: A set of events, where each event is a set of zero or more outcomes (a set of other sets containing elements o, such that o e O). It is with these that we work with - I’ll expand on this in a moment.

p: Is the probability measure. A moment to explain this to anyone who might be reading this, and is not familiar with the topic; this is a special kind of a measure, which is simply a function that assigns numbers to sets, with the limitations that these numbers are >= 0, that p(emptySet) = 0, and that p(unionOfDisjointSets) = p(set1) + p(set2) + ... The numbers that a probability measure assigns are in the range [0, 1], or 0&#37; - 100% if you prefer, and represent the probabilities of events.

The concept of outcomes is straightforward; anything that can happen and is relevant to the problem is an outcome - if you throw a die, an outcome would be getting a specific number. You can model this with {1, 2, 3, 4, 5, 6}. An outcome is the result of a single execution of the model.
But, since individual outcomes are rarely the point of interest, the events are used to model a wider range of possibilities. For example, what if, instead of being interested in the probability of getting a specific number when you roll a die, you wanted to know the possibility of getting an even number. An event {2, 4, 6} e E models this – and it is considered that it has happened if any of its outcomes occurred. Only those events that are interesting or relevant in relation to the problem are considered. There are related rules, but I’ll left them out.

The probability measure p, being a function, returns the probabilities of events.
p: E --> [0, 1]

Now we enter the core of our discussion. The probabilities of the events, returned by the probability measure, are related to the probability mass function (in a discrete case - like with dice) or to a probability distribution function (in a continuum case - like with points of a plane). These ascribe probabilities to the outcomes.

First, some of us have (or had?) this notion that there is some natural probability related to each outcome (that a result of a die roll, or of a point selection has some inherent probability), but this was wrong. The probabilities are given (ascribed!) so that they fit a specific problem. You might say - but a chance for a specific outcome of a die roll is 1/6 and that’s it, this cannot be changed! But, these chances are "ascribed" by the physical properties of a die. Let us abstract this object. It becomes {1, 2, 3, 4, 5, 6}. With this, you can do whatever you want. You can use the same object to model a rigged die, say a one that is weighted so that a specific number has a better chance of appearing, by ascribing different probabilities.

Now, this becomes even more apparent in the point-picking problem. Why would any outcome have any inherent chance in this case?

The second important aspect of the problem which most of us failed to see is that, in order to solve the triangle problem, we approached it by transforming our infinite plane into something more manageable. Different approaches yielded different results, all seemingly valid. But, as superbonzo said, there's no way to define this transformation in such a way that our probability measure p still gives the same results for the same events (such an event specifically being "an acute triangle is formed").
Someone somewhere proved it. Period.
By applying the transformation, we change the output of p.

If I'm right, this is what it's all about.
There might be some technicalities here that I overlooked, but I'm sure others will point them out if it's necessary.


A side thought:
About nuzzle’s infinite sided die - yes, it would be a sphere, but why would it roll forever? Introduce friction, and it would eventually stop. The topmost point is the result of the roll. But, the surface of a sphere is a continuum, so what would be the right way to model such a die? Would the sample space be the set of, say, non-negative reals [0, +inf. ), in respect to its physical surface, or the set of natural numbers {1, 2, 3, ...}, in relation to the concept of "infinite-sided" object?

Some replies:

My intention was not to obfuscate the example with practical issues, [...] there are a lot of implicit assumptions along the way. I think superbonzo has done a great job to bring them to light.

Of course, I agree. A bit of irony: I assumed that you are trying to obfuscate it with practical issues, which lead to this conversation - another proof that there's a good reason to be explicit

Now, you can do two things:

1) try defining what is a ratio of general cardinal numbers [...] But as you noted, cardinality is a very coarse grained way of looking at sets [...]

2) conclude that probability has to do with ratios and countings only when finite sets are involved. With infinite sets you need different concepts. The most common way of representing a probabilistic model is through a "probability space" [...]
In this way you can see the interval [0,1] as a probability space and you'll be able to conclude that the measure (ie the probability) of the interval [0,0.5] is 0.5.

So, if I tied (1), I’d get some ratio-like expressions that wouldn’t really tell me much more that what I had without them?

If I tried (2), someone might argue that this is only one of possible results, since there's really no "correct" probability distribution function.

Not exactly. As I sead in post #19, the concept of "numbers that count things" is modeled as cardinal numbers. Cardinal numbers are "usual" integers only for finite sets, thus the usual notion of "ratio" can be defined only for them.

Yes. But, I wasn't talking about cardinality in general, I was referring to the cardinal number of the specific sets used in my example. (Being the same as the cardinal number of the set of reals, unless I'm mistaken.)