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Some Math Problems Seem Impossible. That Can Be a Good Thing

109 points by sonabinu 6y ago

ipsin 6y ago
My problem with the example is that it relies on (and illustrates) a convention that seems arbitrary. Why is a polygon with three consecutive collinear vertices not a polygon? Convention. Does anyone here have insight into why it's a good convention? To me it sounds something like saying "9/12 is not a valid fraction because it can be simplified".
- sandworm101 6y ago
  Because it has no net surface area? It is a 1-dimensional line/curve rather than a 2d surface? I thinks such definitions are a little more substantive than mere language conventions.
  - tomstuart 6y ago
    The parent comment meant three consecutive collinear vertices among other vertices, not a zero-area “flat triangle”.
- redvenom 6y ago
  The reason is because you want to be able to usefully talk about 'true vertices'. Another way to define a polygon is any convex hull of finitely many points in Euclidean 2-space with nonzero area. Then the 'real' vertices are those points on the boundary curve that are nondifferentiable ('pointy' or 'sharp'). Those are the special points you are interested in when you want to distinguish between vertices and other points on the boundary.
- zests 6y ago
  How do you define vertex in a way such that this would make sense? A square is a square, not an octagon with invisible vertices.
  - contravariant 6y ago
    It takes less effort to define it without that restriction, then an octagon is just eight vertices with lines between them. You can add various restrictions such as not allowing it to intersect itself or requiring that it is convex, but not allowing 3 successive vertices to be colinear doesn't really give you any useful properties so I'm not sure why you'd want that.
- QuesnayJr 6y ago
  Conventions are not set in stone. In a math paper, you could say you allow for this case, if it makes sense. You could even allow for two of the vertices to be the same.
- recursive 6y ago
  According to wikipedia, a polygon with three adjacent collinear points can be convex. https://en.wikipedia.org/wiki/Convex_polygon https://en.wikipedia.org/wiki/Convex_polygon
MauranKilom 6y ago
The illustration at the top where sticks are used immediately made me want to provide the "outside the box" solution of arranging the sticks in 3D. But I guess an octagon is pretty clearly defined as a planar object. Something the article didn't touch on but that I think is fundamental when trying to construct proofs: Often, you can get further in a proof by trying to prove the opposite. Assume that what you're trying to prove is impossible and try to construct a counterexample to the original proof. You will reach a point where the construction can't go further - examine closely what invariant prevents you from completing the counterexample and you will happen across something that will advance your actual proof. Note that this need not result in a proof by contradiction - it's usually more about illuminating the way forward. In a sense, the octagon problem is teaching exactly this step. That is, the "real" problem here (and the proof that the students should arrive at) is "prove that you cannot construct a convex octagon with 4 right angles". How do you do that? You assume that it's wrong and try to find a counterexample to the original proposition - a convex octagon with 4 right angles. The ensuing "productive struggle" leads exactly to the insights you need to prove the original proposition, as detailed in the article.
- sonabinu 6y ago
  Proof by contradiction was a thought that went thru my mind as well when reading this article. Not sure if high school teaches that stream of thought, can't recall using such methodology in school.
  - MauranKilom 6y ago
    Proof by contradiction is how we were shown that sqrt(2) cannot be a rational number. I'm not from the US though (and I don't even know how this generalizes within my country).
    - monktastic1 6y ago
      Am in US and can confirm that sqrt(2) is irrational here also.
  - 6y ago
EForEndeavour 6y ago
In the theme of this excellent article: solving proof-based problems takes skill, but designing good problems in the first place is even harder. I wonder to what extent solving such problems automatically makes you better at formulating similar problems. Reminds me of how being able to understand a spoken language doesn't help with speaking as much as I expected it would (for me, anyway).
- redvenom 6y ago
  It does make you much better at formulating problems. In fact, from my experience as a mathematician and watching others do math, the vast majority of good problems comes after someone has solved a problem and then asked, "what if we change this instead?"
Someone 6y ago
One needs only a tiny bit of space curvature to get a space where a convex octagon with four right angles exists. One could draw one on the surface of a sphere, for example, just as one can draw a triangle with three right angles (https://www.quora.com/How-can-I-draw-a-triangle-with-three-90-degree-angles-on-a-sphere?top_ans=2573947 https://www.quora.com/How-can-I-draw-a-triangle-with-three-9...)
- mhandley 6y ago
  Wouldn't you need to draw the convex octagon on the inside surface of the sphere?
  - deleted 6y ago
    [deleted]
  - dmurray 6y ago
    It's the same whether you draw it on the inside or the outside. Imagine the sphere is see-through, draw it on the inside then go out and look on the outside. The definition of "convex", "octagon", "right angle" or "straight line" doesn't change when you move from looking at the interior surface to the exterior one.
    - Someone 6y ago
      Thanks both. I still can’t visualize that octagon, but I now think you need saddle points to draw it, so a sphere wouldn’t do. If that’s true, a torus would do.
      - dmurray 6y ago
        I'm pretty certain it's possible on the sphere, but I'm currently failing to demonstrate that with a pencil and a table tennis ball. Seems it should be possible to draw the octagon that's shown in the article: draw a rectangle and flex the sides out a little bit.
        6y ago
academia_hack 6y ago
The coin toss example given here seems tautological. Why would my unfair coin have to abide by some norm of probability? If I were designing such a coin, it would obviously include a mechanism for tracking state such that if the last toss was heads the next would always be tails. For example, it could literally redraw the upwards-facing side to the desired outcome an instant before striking the table. I've never had a great head for mathematics because it feels like most impossibilities arise from arbitrary constraints and definitions. Reading such proofs feels like: 1) I've created an arbitrary universe with made up rules 2) This condition does/doesn't break one of those rules I made up 3) QED For example, why on earth can't a straight line in a rectangle have a 180 degree angle at its center making that rectangle an octagon too? It's not clear why one system is epistemologically valid and the other is not. The proof of impossibility feels like a "no true scotsman"/"no true octagon" fallacy disguised by a bit of added notation.
- firethief 6y ago
  > The coin toss example given here seems tautological. Why would my unfair coin have to abide by some norm of probability? If I were designing such a coin, it would obviously include a mechanism for tracking state such that if the last toss was heads the next would always be tails. If a stateful coin is permitted, the problem is indeed trivial; but if we start with the assumption that the problem isn't trivial, we can assume a "passive" coin more in line with the intuitions most people would have about coins.
DougN7 6y ago
I’ll share one I’ve been pondering for 30 years since my geometry teacher told me it’s impossible: It’s not possible to trisect an arbitrary angle using only a straightedge and compass.
- redvenom 6y ago
  The reason basically comes down to the fact that every time you perform a compass and straight-edge construction, you are constructing quadratic extensions of number fields, whereas trisection (in general) needs a cubic extension. The 'in general' part is important: of course, you can trisect specific angles like a 90 degree angle. (The construction of number field extensions is happening because let's say you construct a 90 degree angle, well that allows you to construct an isosceles triangle with equal sides length 1, and therefore you construct the square root of 2, so the extension Q(sqrt(2))
  - murkle 6y ago
    You need to be careful how you phrase it though - it's possible with a _marked_ edge https://www.geogebra.org/m/cWfHr7pk https://www.geogebra.org/m/cWfHr7pk
  - dmurray 6y ago
    That's a long way from being a convincing argument that compass and straight-edge can only produce quadratic extensions to number fields, though. (I know it's true, just saying it looks like you've offered an explanation in elementary terms, but I don't think one exists).
    - redvenom 6y ago
      Absolutely true, there is a lot to prove there. It's not an elementary explanation by any means, rather it is an outline that someone who knows Galois theory well could probably reconstruct.
6gvONxR4sf7o 6y ago
> Proving that something is impossible is a powerful act of mathematics. It shifts our perspective from that of rule follower to that of rule enforcer. And to enforce the rules, you must first understand them. You need to know not just how to apply them, but when they don’t apply. And you also need to be on the lookout for situations where rules might conflict with one another. Our octagon exploration exposes the interplay between polygons, convexity, right angles and angle sums. And it highlights how S = (n – 2) × 180º isn’t just a formula: It’s one condition in a world of competing conditions. This paragraph puts to words something I’ve been mulling over for decades. Looking back to math class, or just looking at any discussion of math education, there’s always a discussion about students who just memorize and plug in the formulas, mentioned as the kinds of students who don’t understand what they’re doing. But that’s an incomplete explanation of what those students are missing that I believe the quoted paragraph completes. It’s not about knowing how to apply a formula. It’s about understanding a formula as a property of the system you’re working with. It’s a great insight and one I’m probably going to mull over for a long while.
- carapace 6y ago
  "Young man, in mathematics you don't understand things. You just get used to them." John Von Neumann to Felix Smith
- contravariant 6y ago
  One level of mathematical understanding further would be to understand which properties (or even concepts) are the necessary consequences of other properties. To given an example: the Pythagorean theorem, the parallelogram law, angles, transposes etc. are all properties/concepts a space has because it has an inner product.
  - orf 6y ago
    > To given an example: the Pythagorean theorem, the parallelogram law, angles, transposes etc. are all properties/concepts a space has because it has an inner product. Could you elaborate a bit? This sounds really interesting.
ramshorns 6y ago
The unfair coin that can't be more likely to give two different results than two the same assumes that each flip has the same probabilities (so k doesn't depend on how many times you flipped it or what you got). Which is fair, but maybe a coin could be built that tends to toggle between outcomes. If it was half full of honey, and you flipped it over once in your hand after catching to read it, it might be more likely to give the other result the next time. Anyway, this is a bit outside the box; the point was that within a set of rules there was a problem that was impossible to solve.

109 points by sonabinu 6y ago

My problem with the example is that it relies on (and illustrates) a convention that seems arbitrary. Why is a polygon with three consecutive collinear vertices not a polygon? Convention. Does anyone here have insight into why it's a good convention? To me it sounds something like saying "9/12 is not a valid fraction because it can be simplified".

sandworm101 6y ago
Because it has no net surface area? It is a 1-dimensional line/curve rather than a 2d surface? I thinks such definitions are a little more substantive than mere language conventions.
- tomstuart 6y ago
  The parent comment meant three consecutive collinear vertices among other vertices, not a zero-area “flat triangle”.
redvenom 6y ago
The reason is because you want to be able to usefully talk about 'true vertices'. Another way to define a polygon is any convex hull of finitely many points in Euclidean 2-space with nonzero area. Then the 'real' vertices are those points on the boundary curve that are nondifferentiable ('pointy' or 'sharp'). Those are the special points you are interested in when you want to distinguish between vertices and other points on the boundary.
zests 6y ago
How do you define vertex in a way such that this would make sense? A square is a square, not an octagon with invisible vertices.
- contravariant 6y ago
  It takes less effort to define it without that restriction, then an octagon is just eight vertices with lines between them. You can add various restrictions such as not allowing it to intersect itself or requiring that it is convex, but not allowing 3 successive vertices to be colinear doesn't really give you any useful properties so I'm not sure why you'd want that.
QuesnayJr 6y ago
Conventions are not set in stone. In a math paper, you could say you allow for this case, if it makes sense. You could even allow for two of the vertices to be the same.
recursive 6y ago
According to wikipedia, a polygon with three adjacent collinear points can be convex. https://en.wikipedia.org/wiki/Convex_polygon https://en.wikipedia.org/wiki/Convex_polygon

MauranKilom 6y ago

The illustration at the top where sticks are used immediately made me want to provide the "outside the box" solution of arranging the sticks in 3D. But I guess an octagon is pretty clearly defined as a planar object. Something the article didn't touch on but that I think is fundamental when trying to construct proofs: Often, you can get further in a proof by trying to prove the opposite. Assume that what you're trying to prove is impossible and try to construct a counterexample to the original proof. You will reach a point where the construction can't go further - examine closely what invariant prevents you from completing the counterexample and you will happen across something that will advance your actual proof. Note that this need not result in a proof by contradiction - it's usually more about illuminating the way forward. In a sense, the octagon problem is teaching exactly this step. That is, the "real" problem here (and the proof that the students should arrive at) is "prove that you cannot construct a convex octagon with 4 right angles". How do you do that? You assume that it's wrong and try to find a counterexample to the original proposition - a convex octagon with 4 right angles. The ensuing "productive struggle" leads exactly to the insights you need to prove the original proposition, as detailed in the article.

sonabinu 6y ago
Proof by contradiction was a thought that went thru my mind as well when reading this article. Not sure if high school teaches that stream of thought, can't recall using such methodology in school.
- MauranKilom 6y ago
  Proof by contradiction is how we were shown that sqrt(2) cannot be a rational number. I'm not from the US though (and I don't even know how this generalizes within my country).
  - monktastic1 6y ago
    Am in US and can confirm that sqrt(2) is irrational here also.
- 6y ago

EForEndeavour 6y ago

In the theme of this excellent article: solving proof-based problems takes skill, but designing good problems in the first place is even harder. I wonder to what extent solving such problems automatically makes you better at formulating similar problems. Reminds me of how being able to understand a spoken language doesn't help with speaking as much as I expected it would (for me, anyway).

redvenom 6y ago
It does make you much better at formulating problems. In fact, from my experience as a mathematician and watching others do math, the vast majority of good problems comes after someone has solved a problem and then asked, "what if we change this instead?"

Someone 6y ago

One needs only a tiny bit of space curvature to get a space where a convex octagon with four right angles exists. One could draw one on the surface of a sphere, for example, just as one can draw a triangle with three right angles (https://www.quora.com/How-can-I-draw-a-triangle-with-three-90-degree-angles-on-a-sphere?top_ans=2573947 https://www.quora.com/How-can-I-draw-a-triangle-with-three-9...)

mhandley 6y ago
Wouldn't you need to draw the convex octagon on the inside surface of the sphere?
- deleted 6y ago
  [deleted]
- dmurray 6y ago
  It's the same whether you draw it on the inside or the outside. Imagine the sphere is see-through, draw it on the inside then go out and look on the outside. The definition of "convex", "octagon", "right angle" or "straight line" doesn't change when you move from looking at the interior surface to the exterior one.
  - Someone 6y ago
    Thanks both. I still can’t visualize that octagon, but I now think you need saddle points to draw it, so a sphere wouldn’t do. If that’s true, a torus would do.
    - dmurray 6y ago
      I'm pretty certain it's possible on the sphere, but I'm currently failing to demonstrate that with a pencil and a table tennis ball. Seems it should be possible to draw the octagon that's shown in the article: draw a rectangle and flex the sides out a little bit.
      - 6y ago

academia_hack 6y ago

The coin toss example given here seems tautological. Why would my unfair coin have to abide by some norm of probability? If I were designing such a coin, it would obviously include a mechanism for tracking state such that if the last toss was heads the next would always be tails. For example, it could literally redraw the upwards-facing side to the desired outcome an instant before striking the table. I've never had a great head for mathematics because it feels like most impossibilities arise from arbitrary constraints and definitions. Reading such proofs feels like: 1) I've created an arbitrary universe with made up rules 2) This condition does/doesn't break one of those rules I made up 3) QED For example, why on earth can't a straight line in a rectangle have a 180 degree angle at its center making that rectangle an octagon too? It's not clear why one system is epistemologically valid and the other is not. The proof of impossibility feels like a "no true scotsman"/"no true octagon" fallacy disguised by a bit of added notation.

firethief 6y ago
> The coin toss example given here seems tautological. Why would my unfair coin have to abide by some norm of probability? If I were designing such a coin, it would obviously include a mechanism for tracking state such that if the last toss was heads the next would always be tails. If a stateful coin is permitted, the problem is indeed trivial; but if we start with the assumption that the problem isn't trivial, we can assume a "passive" coin more in line with the intuitions most people would have about coins.

DougN7 6y ago

I’ll share one I’ve been pondering for 30 years since my geometry teacher told me it’s impossible: It’s not possible to trisect an arbitrary angle using only a straightedge and compass.

redvenom 6y ago
The reason basically comes down to the fact that every time you perform a compass and straight-edge construction, you are constructing quadratic extensions of number fields, whereas trisection (in general) needs a cubic extension. The 'in general' part is important: of course, you can trisect specific angles like a 90 degree angle. (The construction of number field extensions is happening because let's say you construct a 90 degree angle, well that allows you to construct an isosceles triangle with equal sides length 1, and therefore you construct the square root of 2, so the extension Q(sqrt(2))
- murkle 6y ago
  You need to be careful how you phrase it though - it's possible with a _marked_ edge https://www.geogebra.org/m/cWfHr7pk https://www.geogebra.org/m/cWfHr7pk
- dmurray 6y ago
  That's a long way from being a convincing argument that compass and straight-edge can only produce quadratic extensions to number fields, though. (I know it's true, just saying it looks like you've offered an explanation in elementary terms, but I don't think one exists).
  - redvenom 6y ago
    Absolutely true, there is a lot to prove there. It's not an elementary explanation by any means, rather it is an outline that someone who knows Galois theory well could probably reconstruct.

6gvONxR4sf7o 6y ago

> Proving that something is impossible is a powerful act of mathematics. It shifts our perspective from that of rule follower to that of rule enforcer. And to enforce the rules, you must first understand them. You need to know not just how to apply them, but when they don’t apply. And you also need to be on the lookout for situations where rules might conflict with one another. Our octagon exploration exposes the interplay between polygons, convexity, right angles and angle sums. And it highlights how S = (n – 2) × 180º isn’t just a formula: It’s one condition in a world of competing conditions. This paragraph puts to words something I’ve been mulling over for decades. Looking back to math class, or just looking at any discussion of math education, there’s always a discussion about students who just memorize and plug in the formulas, mentioned as the kinds of students who don’t understand what they’re doing. But that’s an incomplete explanation of what those students are missing that I believe the quoted paragraph completes. It’s not about knowing how to apply a formula. It’s about understanding a formula as a property of the system you’re working with. It’s a great insight and one I’m probably going to mull over for a long while.

carapace 6y ago
"Young man, in mathematics you don't understand things. You just get used to them." John Von Neumann to Felix Smith
contravariant 6y ago
One level of mathematical understanding further would be to understand which properties (or even concepts) are the necessary consequences of other properties. To given an example: the Pythagorean theorem, the parallelogram law, angles, transposes etc. are all properties/concepts a space has because it has an inner product.
- orf 6y ago
  > To given an example: the Pythagorean theorem, the parallelogram law, angles, transposes etc. are all properties/concepts a space has because it has an inner product. Could you elaborate a bit? This sounds really interesting.

ramshorns 6y ago

The unfair coin that can't be more likely to give two different results than two the same assumes that each flip has the same probabilities (so k doesn't depend on how many times you flipped it or what you got). Which is fair, but maybe a coin could be built that tends to toggle between outcomes. If it was half full of honey, and you flipped it over once in your hand after catching to read it, it might be more likely to give the other result the next time. Anyway, this is a bit outside the box; the point was that within a set of rules there was a problem that was impossible to solve.