Hearted Youtube comments on Mathologer (@Mathologer) channel.
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Nice work! Gives me some flashbacks to Paul Lockhart's lovely book Measurement, which is like the one textbook that I, as a soon-to-be math teacher, actually really like; of course, I can't base my teaching on it because of these soul-crushing standardized requirements and tests, though (I guess I can at least use it for inspiration). One of my favorite related proofs when it comes to understanding the logarithm based on the graph of the function defined by f(x)=1/x that could have been mentioned, though, is a lovely visual argument that the alternating harmonic series sums to the natural logarithm of two. You restrict yourself to the little region from 1 to 2 and ponder the area under the graph – it's the natural logarithm of two, of course, but you can also approximate it with rectangles. If you use a 1x1 square, you're overshooting a lot, but you can remedy that by taking a rectangle out horizontally to remove the second half, where the overcounting is most egregious. You then add the smallest rectangle you can that still covers the desired area in that region, which has a height of 1/(3/2)=2/3 and a length of 1/2, so its area is a third. I think most of you will know where this is going, right? Our estimation, which still overcounts but is much more accurate, is 1 - 1/2 + 1/3. Now, we remove the second half of the first rectangle (area 1/4) and add a better fit back in that has an area of 1/5, and we also remove the second half of the second big rectangle we have (area 1/6) to get a better approximation with an area of 1/7. And so on and so forth – it's not too hard to convince yourself that this approximation strategy does indeed amount amount to evaluating partial sums of the alternating harmonic series, and the argument shows that they approach the logarithm of 2. Isn't that beautiful, even if it's tricky to describe everything just with words? This is exactly the kind of thing I want to teach eventually.
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In fact, I've seen these diagrams as divisibility test. Make the multiplier equal to 10 (or whichever base you're working in), and the modulus equal to the divisor you want to test divisibility/remainder by. Then, with the arrows as you suggest in your third wish, start by moving clockwise from 0 as many steps as the first (most significant) digit, then follow exactly one arrow in the diagram, move clockwise as many steps as the second digit, follow one arrow, so on. Do not follow an arrow after your last digit. If you follow these instructions correctly, you'll end up at the remainder (and if and only if you end up at zero, the number is divisible by the modulus).
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The primitive pythagorean triples are given by the general formula (a,b,c)=(u^2-v^2, 2uv, u^2+v^2) where u and v are positive integers, u>v, coprime, and of different parity (i.e. one is odd the other even). This shows that the 4k+1 primes are always also pythagorean triples' hypotenuses. The converse is not true though, for example (7, 24, 25) is a primitive Pythagorean triple but 25 is not prime.
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Taking a shot at the "Multiplication Partition" problem at the end...
Empirically, the numbers seem to follow the pattern F(2n), where F(n) is the Fibonacci function. So the pattern is every-other Fibonacci number, henceforth called the "Skiponacci sequence". I will prove the hypothesis that the sum of products generated from partitions of the number n follows the Skiponacci sequence.
Here, S(n) is the Skiponacci function, and P(n) is the partition-product-sum function.
Firstly, analizing the stacks of equations, we can utilize the "recursion" mentioned early in the video. Looking at the example given for n=4, we see that the products are:
4
3 * 1
1 * 3
2 * 2
2 * 1 * 1
1 * 2 * 1
1 * 1 * 2
1 * 1 * 1 * 1
We focus on the products ending with "1", and removing the "* 1" we see:
3
2 * 1
1 * 2
1 * 1 * 1
Oh look, its the products for n=3 ! Looking at the products ending with "2" and removing it:
2
1 * 1
Its the products for n=2. The pattern is becoming clearer. Looking at the remaining products:
4
1 * 3
The "1 * 3" clearly follows the pattern, being 3 times the n=1 product. The 4 sticks out, but for now its easy to write it off as just "n". The final formula for this pattern is:
P(n) = P(n-1) + 2P(n-2) + ... + (n-1)P(1) + n
The reason for this formula makes sense. The "recursion" is because the partition products that are multiplied by 2 are made from partitions that are 2 less than n. Hence the "+2" in the partition list becoming a "*2", giving us the 2*P(n-2) part of the equation.
Now how does this fit into the Skiponacci sequence? It becomes clearer if we write the terms out into a pyramid. For instance, for n=5, the answer is the sum of these numbers. Each row has n copies of P(n-1), except the last row, which is written as n "1"s, for the "+n" term.
21
8 8
3 3 3
1 1 1 1
1 1 1 1 1
To aid in making sense of this, here is the pyramid for n=4:
8
3 3
1 1 1
1 1 1 1
Notice the recursion? The n=5 pyramid contains the n=4, just with the extra diagonal. This makes sense, since every time n increases by 1, each P(n-k) factor's coefficient increases by 1 (and the "+n" term increases by 1, naturally). All this means that this equation holds:
P(n) - P(n-1) = P(n-1) + P(n-2) + ... + P(1) + 1.
This gives us a neater equation for P(n) if you add P(n-1) to both sides, but for now lets test our hypothesis and replace P(n) with S(n).
S(n) - S(n-1) = S(n-1) + S(n-2) + ... + S(1) + 1
The left side is easy to simplify, because S(n) = F(2n)
S(n) - S(n-1)
F(2n) - F(2n-2)
F(2n-1)
For the right side, we can recursively replace the two right-most elements with another fibonacci number, until we are left with F(2n-1)
S(n-1) + S(n-2) + ... + S(1) + 1
F(2n-2) + F(2n-4) + ... + F(4) + F(2) + F(1)
F(2n-2) + F(2n-4) + ... + F(6) + F(4) + F(3)
F(2n-2) + F(2n-4) + ... + F(8) + F(6) + F(5)
...
F(2n-2) + F(2n-4) + F(2n-5)
F(2n-2) + F(2n-3)
F(2n-1)
This leaves us with this equation, which is obviously true:
F(2n-1) = F(2n-1)
Therefore, because we were able to replace P(n) with S(n) in our equation, we showed that P(n) = S(n). QED.
Also I did the math and found that the general equation for S(n) and P(n):
S(n) = 2/sqrt(5) * sinh(2 * ln((1+sqrt(5))/2) * n)
This is a long way of saying I think the next number is 55 :)
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Congrats on 100 videos, mate. You really made it as a maths educator and content creator on YouTube, and I'm looking forward to seeing you do even better in the future. Hope you blow our minds again with each video you make
That said...
CHALLENGES!
11:18
I have no army of middle school minions but I am still ready to attack
Same reasoning as before. This time we start with the second paraboloid, the one carved from the cylinder. Makes the maths a little easier.
The paraboloid has radius R and height H. Cutting it at height h will leave a ring with outer radius R and inner radius r.
The paraboloid is modeled after a parabola y=ax^2, and so we should have H = aR^2 and h = ar^2. So it's possible to solve for r and get r = Rsqrt(h/H).
The ring thus has area pi * R^2 - pi * R^2 * h/H, or piR^2(1-h/H).
The first paraboloid should also have that area. Thus its radius should be Rsqrt(1-h/H).
Now the inverted paraboloid can be modeled by another quadratic, but the important takeaways are H = bR^2 and H - h = br^2. Solving for r this time gives Rsqrt(1-h/H), exactly the same as what was predicted by the circles area argument.
Or you can use integrals. Whatever floats your boat
17:42
The layers of the onion look almost like surface areas stuck together. That can be written as: V(R) = the integral from 0 to R of SA(r)dr
By FTC1, we can also write this as V'(R) = SA(R)
And so the derivative of the volume is the surface area. Even in 420 dimensions.
18:04
The base of the cylinder has area piR^2. The height is 2R, and the circumference is 2piR. In total, the surface area of the cylinder is 6piR.
With the surface area of the sphere being 4piR, the ratio of the surface areas of the two shapes really is 3:2.
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I am absolutely trash at math but when I stumbled across a numericology video on 369 and they explained the theory I immediately commented that it was clearly because we use a base ten system. For instance base 14 was used in the Middle East for a long time. Based on counting the inner pads of the fingers, I am a behavioral scientist and Buddhist philosopher so I am well studied but I really am horrible at math. I can’t even do basic division if the numbers go into the hundreds without taking a long time. It was actually obvious because from a behavioral psychological point of view when you have something people can’t explain the culprit is most often a blind spot in your paradigm. Anytime we attempt to build a thought model of what we believe is true we create a type of tunnel vision because our imagination becomes constrained by your base assumptions. If you think a certain action is always bad you will be blind to anything good that comes from that action and when it’s pointed out you won’t be readily able to accept the evidence, it literally requires an individual to rewrite the most fundamental assumptions about life because our thoughts are like cards stacked upon each-other our conclusions come as a result of previous conclusions, as this happens our ability to recognize anything new or aberrant narrows and our ability to explain concepts becomes reliant upon what we have “figured out” thus creating a blindness in our personal philosophy. Most people assume that math is always a base ten system. But there are as many systems as there are numbers and historically societies that weren’t connected to the western tradition came up with novel calculation systems. Most notably the Indians who were the first to recognize the need for the number zero. Incendently this comes from Hindu philosophy which posited emptiness or void and singularity or oneness. Both of these concepts are represented by the number zero which like consciousness can be posited and observed in its function but cannot be qualified or countrified as it is immaterial like space. Anyways from a purely psychological perspective the answer was obvious I didn’t need to comprehend the math to see that this was merely a blind spot made by a false axiom.
Rant about Ether vs Space:
I will add that Tesla was correct about ether. When people talk about bending space they are talking about bending ether. Space is not a medium and cannot be acted upon it is a potentiality for expansion and position. If something bends in a given position that which is bent must rest within something. Without a medium the distance between all things would be zero, the fact that we can imagine folding space means that something which has qualities is being manipulated, however actual void ness, emptiness, the expanse of nothing allows for all laws of physics to be unobstructed and to function within their own constraints. If space was not completely void there would be no possibility for anything to bend, 0 the absence of all is what gives freedom to bend the ether. When you bend ether that fold is still a location in space and relies upon the void as a platform for existence. Nothing has freedom or capability to be anything without a complete void to appear within. This void exists outside of time and was present as the externality of the singularity but void or emptiness is not a thing and so it is completely logically coherent to say nothing existed outside the singularity because emptiness has no qualities and cannot be proven to exist. It can only be pointed out as being the function of unobstructed freedom of material phenomena. If space was void you could not bend it, it has qualities and so it cannot be said to be the expanse which allowed for the Big Bang. Without this freedom of nothingness and infinite indescribable void the singularity would never have been able to exspand as it’s size would be fixed.
Even the diagrams of bending space require the illustration of grid planes which automatically describes a medium which has properties. True space ie the infinite void has no properties it is the absence of properties and can only be proven to exist because the bending of space happens within a free unobstructing openness. It a real shame because I suspect this distinction is likely an incredible resource for computation. Again if there was no ether than all of existence would not have any measurable distance, distance in space is a fixed property which is measurable as space/time, but something has to exist that undergirds and existed before space time in order to allow it to come into being and to exist in an exspansive way. Nothingness is that thing. Without the absence of all nothing can expand out into infinity. Without void everything would be permanently constrained. Space itself is a constraint as it is governed by the speed of light.
Space what I would call ether acts on objects because it is a medium with properties subject to the qualities of other objects like planets whose gravity actually warps it. True void is inert and has no properties it cannot be affected or measured it is merely the absence of all qualities which allows for qualities to come into being without being constrained.
Can you comprehend emptiness or is your paradigm constrained by preconceived notions of what your science books have told you. I have made a coherent argument that we are missing a fundamental piece of reality in our current scientific paradigm. Exists in math as zero it is already proven to exist in math and many higher calculations cannot even be done without zero proving that as a model of reality zero is acknowledged as indispensable to model our reality. ... yet we ignore emptiness. We already know zero cannot be affected by any other number zero cannot be divided subtracted or multiplied because it is an absence of qualities. It can be added to because it is inert and unobstructed and so it is a vessel which is truly and absolutely infinite in its allowance for qualities to exist within it. We already know all this yet our current models of physics refuse to acknowledge that what we call space has qualities that occupy the infinite void as a basis for all. Without void we cannot posit existence we understand that two things cannot occupy the same place which means for anything to occupy any space there must be a void which does not obstruct position, characteristics or behavior. As I said if what we call space can bend it is bending within emptiness. There must be a basis for reality which is without qualities or existence would be constrained. If what we call space was truly the basis for reality it wouldn’t have qualities. If space was not a medium there would be no time or distance to travel between objects, you could move from earth to Mars in an instant because there would be nothing to travel through. Traveling through something means it is a medium especially if it has coherent properties, it’s qualities are consistent not chaotic meaning like atoms it has a particular measurable property which can be counted on to act and behave in a uniform manner. Having inherent measurable qualities means it exists as an object.
Please don’t reply if you’re just going to quote text book physics as my entire point is to point out something missing in the current assumed model of reality. We know the number zero is real functional and vital yet we deny it’s actual existence in the world despite our model of reality requiring zero to be calculated. The absence of everything came before anything this should be obvious as only an absence of things allows for an unobstructed exspansion. Without an absence of qualities qualities could not come into being without being affected by constraints. If there was no external void the singularity would have nothing to expand to.
This is important for positing the big crush as there may be a limit to this ether or space time, it may have an elastic property where it can only stretch so far, space time may be capable of thinning metaphorically like a gas perhaps creating a gravitational vacuum. Just like as gas thins it creates a vacuum which can literally implode or crush objects. Once we see space as a medium this new paradigm means measuring the qualities of this ether may actually allow for incredible breakthroughs. I believe humanity is constraining it’s progress by resting on a preconceived notion that space/time is the undergirding foundation of reality when In my mind you always must have nothing before you have something. If
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Very interesting! This reminds me of a result I came across recently. Take a regular polygon positioned anywhere in a circle. Extend the sides to form two sets of segments with lengths x_1,x_2,...,x_n and y_1,y_2,...,y_n, so that each y_i is counterclockwise from x_i. Then applying power of point to each vertex, we get
x_k(L+y_(k+1))=y_k(L+x_(k-1)),
where L is the side length of the polygon. Adding all these equations and cancelling the x_ky_(k+1) terms, we get
x_1+x_2+...+x_n=y_1+y_2+...+y_n.
So the two groups of segments have equal sums.
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