Thank you for sharing this video. As someone in fifty who stopped studying math some 30 years ago, it was hard to understand mathematical terms and concept. I'm tryting to gather some information about space filling curves, which puts me into an Alice in Wonderland situation. Besides I'm translating the concept into Korean. Your video helped me a lot.
Why does the fact that the space filling curve doesn't have area mean that it's not quite two dimensional? I understand it's more unconventional compared to a shape with area, but if it hits points that require two dimensions, even if the shape formed by the line isn't closed off, shouldn't it still be considered two dimensional? And why does the "curve" have to cover all of the points in a space to be considered two dimensional? Wouldn't the line only have to hit enough points to make its first twist in order to be considered two dimensional? Well, I guess I understand this part if the goal is to imitate area in order to make the line two dimensional, but I still don't understand why the line needs area in order to be two dimensional.
A space-filling curve like the Peano Curve is actually cheating. it really doesn't cover all the geometrical points in the space (surface in this case). The reason for this is that only 2ω (where 2ω is the Cantor set) number of points are represented in the x and y direction, corresponding to the set of the real numbers. The real numbers cover the line by the use of the "completeness property", but this works by transforming any infinitesimally small value to zero (zero is the only real infinitesimal), so all points on the geometrical line infinitesimally close to zero is represented by the same real number. This means that the real numbers, and also the points in the space-filling curves have neighboring points that aren't covered by the curve, and the "completeness property" can't be used in this purely geometrical method, using only lines with zero thickness. To really cover the larger space a different method has to be used, using the notion of moving points or similar method, to actually be able to represent the whole geometrical continuum.
I've been thinking about something when it comes to space filling curves and the nature of the real plane. As we know there are different types of numbers, one specific kind has caught my attention the last years, the so called "transcendental numbers" (non algebraic). I've heard people such as Vihart here on youtube explain that a transcendental number (such as pi, or e) basically has a structure similar to a number where you defined each new decimal with a throw of a 10 sided dice (0-9). The chance to get exactly 1/3 = 0.33333... to infinity, is literally 0 cause eventually you'll throw something else with the die. It also turns out that the vast vast majority of numbers are these kind of "structure-less" numbers. My question is then: how can a simple rule, such as the one that gives rise to the Piano curve, go through all the transcendental points? Since it's a geometrical rule it seems that each point that it could possibly pass through is describable in some form of schema, some form of equation, depending on the exact nature of how you twist and turn the iterations. But by definition, the transcendental numbers _can't_ be defined by algebraic equations (with rational coefficients should be added). How do we know that the curve passes through these points? That it can get infinitely close, in the limit sense? Is that really good enough?
You should read Munkre's proof given in his topology book. You can find it for free online if you just search James Munkres Topology PDF in Google. You can find the construction on page 144 of the pdf I am referencing. It's not the same exact construction as this, but this curve does in fact go through every real number. You have to know a decent amount of topology to grasp it fully, but you can still skim it and gain a sufficient understanding as to why it covers more than just rationals.
Mr Eddie Woo, damn, did not expect you teaching this type of mathematics. Inspiration.
oh, u r the space filling curve of my heart! ❤️🤸🤸🤸
Thank you for sharing this video. As someone in fifty who stopped studying math some 30 years ago, it was hard to understand mathematical terms and concept. I'm tryting to gather some information about space filling curves, which puts me into an Alice in Wonderland situation. Besides I'm translating the concept into Korean. Your video helped me a lot.
Why does the fact that the space filling curve doesn't have area mean that it's not quite two dimensional? I understand it's more unconventional compared to a shape with area, but if it hits points that require two dimensions, even if the shape formed by the line isn't closed off, shouldn't it still be considered two dimensional? And why does the "curve" have to cover all of the points in a space to be considered two dimensional? Wouldn't the line only have to hit enough points to make its first twist in order to be considered two dimensional? Well, I guess I understand this part if the goal is to imitate area in order to make the line two dimensional, but I still don't understand why the line needs area in order to be two dimensional.
Enjoyed every part! Learnt something new.
nice teacher. Congrats!
A space-filling curve like the Peano Curve is actually cheating. it really doesn't cover all the geometrical points in the space (surface in this case). The reason for this is that only 2ω (where 2ω is the Cantor set) number of points are represented in the x and y direction, corresponding to the set of the real numbers. The real numbers cover the line by the use of the "completeness property", but this works by transforming any infinitesimally small value to zero (zero is the only real infinitesimal), so all points on the geometrical line infinitesimally close to zero is represented by the same real number. This means that the real numbers, and also the points in the space-filling curves have neighboring points that aren't covered by the curve, and the "completeness property" can't be used in this purely geometrical method, using only lines with zero thickness.
To really cover the larger space a different method has to be used, using the notion of moving points or similar method, to actually be able to represent the whole geometrical continuum.
helps make it clear. thank you. followed. XD
I've been thinking about something when it comes to space filling curves and the nature of the real plane. As we know there are different types of numbers, one specific kind has caught my attention the last years, the so called "transcendental numbers" (non algebraic). I've heard people such as Vihart here on youtube explain that a transcendental number (such as pi, or e) basically has a structure similar to a number where you defined each new decimal with a throw of a 10 sided dice (0-9). The chance to get exactly 1/3 = 0.33333... to infinity, is literally 0 cause eventually you'll throw something else with the die. It also turns out that the vast vast majority of numbers are these kind of "structure-less" numbers.
My question is then: how can a simple rule, such as the one that gives rise to the Piano curve, go through all the transcendental points? Since it's a geometrical rule it seems that each point that it could possibly pass through is describable in some form of schema, some form of equation, depending on the exact nature of how you twist and turn the iterations. But by definition, the transcendental numbers _can't_ be defined by algebraic equations (with rational coefficients should be added). How do we know that the curve passes through these points? That it can get infinitely close, in the limit sense? Is that really good enough?
You should read Munkre's proof given in his topology book. You can find it for free online if you just search James Munkres Topology PDF in Google. You can find the construction on page 144 of the pdf I am referencing. It's not the same exact construction as this, but this curve does in fact go through every real number. You have to know a decent amount of topology to grasp it fully, but you can still skim it and gain a sufficient understanding as to why it covers more than just rationals.
how can line have 1 dimension??
Find one vector that can be a basis!