Dear all,
I am a mathematics teacher at an Italian middle school. For the past six years, I have been working on a formal system using only a compass and a line. Finally, my work has reached a point where it can be objectively evaluated and perhaps proven. The problem I propose is of a geometric and computational nature with potential significant implications in probability theory.
It involves a conjecture based on a simple and well-defined harmonic curve that raises an interesting question: can this curve be calculated and thus represented by a classical computational system (based on two values)?
As you can see, the question is very direct and unambiguous. I have reasons to believe that this may not be possible, or at least only with a precisely calculable probability of 1 in 16,777,216 (due to 8 points of indecision with 8 different choices each).
I offer it for evaluation because I think it is an interesting problem in many respects. Additionally, my mathematical skills are limited, and this would require a computational proof, which is beyond my current capabilities.
If you are not interested, I will not be offended.
Here follows my explanation of the problem.
Thus, the problem is computational/representational relative to a specific curve that I will call "00Curve." It has two closely related but distinct sides: one geometric and the other computational.
The geometric side is explained by the booklet attached here, which contains all the necessary instructions to draw that curve. Please understand that the geometric structure underlying the curve can be built and perfectly understood by any classical computational system. The crucial and therefore interesting step is exactly the last one, where an unusual but legitimate request is made: "draw the shortest path that connects the +, 0, and - points without intercepting the 8 circles." This might be a critical step for a classical computational system since the curve naturally follows an unusual shape, which can be correctly identified "by eye" by a human. The possibility that a classical computational system can do the same is doubtful. Why? Because of the computational side of the problem.
The computational side of the problem is explained by two simple statistical problems written in the file "Mr. X." Please do not judge my apparent silliness in proposing problems, I have my good reasons for choosing this particular style (firstly because I work with children). However, the problems are easy enough to solve. Even a 13-year-old can do it. Again, without the need to calculate the correct answer. Note that the red and blue segments are naturally of different lengths (but this does not affect the solution of the problem). Even a classical computational system can do it, but how? By imagining/reducing the real shape of the curve to a simple circle (we might call it "the imaginary shape" of the curve). Thus, the real reason why the statistical problems are complementary to the geometric side is that they contain all the information necessary to correctly draw that curve in its unique and real form.
The explanation of this is kindly provided by ChatGPT at this link.
It's true that artificial intelligence does not provide reliable opinions, but given the geometric context already analyzed, the judgment is to be considered correct, as well as an ironic way to propose a computational problem of a statistical nature.
Appreciate that the only measurement act referred to in the conversation is nothing but the first opening of the compass as explained in the booklet. This act of measurement is really the only necessary link to join the two sides of the problem and to indicate the curve as unique.
This is a simple but mathematically vital requirement.
The computational description of the curve is also peculiar if we consider the shape of its derivative (speed) and acceleration which indicate the real geometric and computational issues critical to the curve: the points marked with +, 0, and -. Here are some graphical representations of the problems attached (imagined and drawn by hand since the functions are still unknown, at least to me). The speed curve at those points is determined by periodic semicircles, which cannot be modeled by sinusoidal waves. Therefore, the acceleration peaks as spikes at the + and -, making it difficult to calculate them with due precision.
These considerations, combined with the geometric issues, offer the possibility that this could be a challenging problem for a classical computational system based on two-value logic. Why do I say this? Because of the particular shape of the derivatives of that curve and because of the number and specific placement of the curve's zero points (8 and evenly distributed along it).
If I'm right, a classical computational system will face those points as points of indecision (1 correct choice out of 8 for each point) making it nearly impossible to correctly RENDER the real shape of the curve.
If I am right, this could offer very interesting insights in mathematics, geometry, computational science, and physics.
I remain at your disposal.
Thanks
P.S. Please understand that so far the curve and its derivative can only be drawn by hand since its equation is unknown (at least to me).
I am a mathematics teacher at an Italian middle school. For the past six years, I have been working on a formal system using only a compass and a line. Finally, my work has reached a point where it can be objectively evaluated and perhaps proven. The problem I propose is of a geometric and computational nature with potential significant implications in probability theory.
It involves a conjecture based on a simple and well-defined harmonic curve that raises an interesting question: can this curve be calculated and thus represented by a classical computational system (based on two values)?
As you can see, the question is very direct and unambiguous. I have reasons to believe that this may not be possible, or at least only with a precisely calculable probability of 1 in 16,777,216 (due to 8 points of indecision with 8 different choices each).
I offer it for evaluation because I think it is an interesting problem in many respects. Additionally, my mathematical skills are limited, and this would require a computational proof, which is beyond my current capabilities.
If you are not interested, I will not be offended.
Here follows my explanation of the problem.
Thus, the problem is computational/representational relative to a specific curve that I will call "00Curve." It has two closely related but distinct sides: one geometric and the other computational.
The geometric side is explained by the booklet attached here, which contains all the necessary instructions to draw that curve. Please understand that the geometric structure underlying the curve can be built and perfectly understood by any classical computational system. The crucial and therefore interesting step is exactly the last one, where an unusual but legitimate request is made: "draw the shortest path that connects the +, 0, and - points without intercepting the 8 circles." This might be a critical step for a classical computational system since the curve naturally follows an unusual shape, which can be correctly identified "by eye" by a human. The possibility that a classical computational system can do the same is doubtful. Why? Because of the computational side of the problem.
The computational side of the problem is explained by two simple statistical problems written in the file "Mr. X." Please do not judge my apparent silliness in proposing problems, I have my good reasons for choosing this particular style (firstly because I work with children). However, the problems are easy enough to solve. Even a 13-year-old can do it. Again, without the need to calculate the correct answer. Note that the red and blue segments are naturally of different lengths (but this does not affect the solution of the problem). Even a classical computational system can do it, but how? By imagining/reducing the real shape of the curve to a simple circle (we might call it "the imaginary shape" of the curve). Thus, the real reason why the statistical problems are complementary to the geometric side is that they contain all the information necessary to correctly draw that curve in its unique and real form.
The explanation of this is kindly provided by ChatGPT at this link.
It's true that artificial intelligence does not provide reliable opinions, but given the geometric context already analyzed, the judgment is to be considered correct, as well as an ironic way to propose a computational problem of a statistical nature.
Appreciate that the only measurement act referred to in the conversation is nothing but the first opening of the compass as explained in the booklet. This act of measurement is really the only necessary link to join the two sides of the problem and to indicate the curve as unique.
This is a simple but mathematically vital requirement.
The computational description of the curve is also peculiar if we consider the shape of its derivative (speed) and acceleration which indicate the real geometric and computational issues critical to the curve: the points marked with +, 0, and -. Here are some graphical representations of the problems attached (imagined and drawn by hand since the functions are still unknown, at least to me). The speed curve at those points is determined by periodic semicircles, which cannot be modeled by sinusoidal waves. Therefore, the acceleration peaks as spikes at the + and -, making it difficult to calculate them with due precision.
These considerations, combined with the geometric issues, offer the possibility that this could be a challenging problem for a classical computational system based on two-value logic. Why do I say this? Because of the particular shape of the derivatives of that curve and because of the number and specific placement of the curve's zero points (8 and evenly distributed along it).
If I'm right, a classical computational system will face those points as points of indecision (1 correct choice out of 8 for each point) making it nearly impossible to correctly RENDER the real shape of the curve.
If I am right, this could offer very interesting insights in mathematics, geometry, computational science, and physics.
I remain at your disposal.
Thanks
P.S. Please understand that so far the curve and its derivative can only be drawn by hand since its equation is unknown (at least to me).
