Spiral of Theodorus
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- čas přidán 9. 06. 2024
- This is a short, animated visual proof demonstrating how to construct square roots of any positive integer using the Spiral of Theodorus
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#manim #math #mathshorts #mathvideo
#construction #geometry #mtbos #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #iteachmath #spiral #theodorus #squareroot
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no, but this will be a great tool for drawing seashells in the future.
Seashells look more like the Fibonacci Spiral which is also easier to construct
@CatOnACell My thoughts, exactly 🎯!
@@mentallyderanged888 I’m pretty sure that seashells (at least nautilus shells) are nowhere near golden, in terms of their featured spiral. Only that they’re approximately logarithmic. See the Mathologer-video: ”Visual Infinite Descent”, and follow the link, mentioned therein, for more. 🤔
@@mentallyderanged888 i was thinking the same thing.
@@mentallyderanged888 seashell are logarithmic spirals not Fibonacci
"Spiral of Theodorus" sounds like some maguffin from a new Indiana Jones movie
lol true
Funny thing. The maguffins from those movies are all real things, too.
😂
I don't think I'd be able to construct sqrt(200), except as 10sqrt(2).
Much easier way for sure :)
I guess you'd get less error by using a 2-14 right triangle
(or a 5-15 right triangle and a little bit of Thales)
I was thinking about the same 🫣
@@hallfiry 5-15? That would produce 5sqrt(10) instead!
@@wyattstevens8574 Nope, you use 15 as the hypotenuse and construct yourself a right triangle over that with 5 as one of the short sides. 15²-5²=200, so the other short side will be sqrt(200)
Nice! finally something new to put on every image besides the golden ratio
😂😂
The lore behind that first triangle is quite... "irrational"
Ba dum tss 🥁🥁
Fun fact: hippasus the guy who discovered irrational numbers was thrown of a ship and drowned by Pythagorean because he made a religious based on math and irrational numbers messed with his beliefs
That's very interesting @@DarkoStevanovic-wr5xu thanks for sharing
@@DarkoStevanovic-wr5xu no way! thats so cool
Is that a mother fudging Reverse 1999 reference!? (Sorry. Someone had to say it)
my teacher made us draw an entire page of this thing, thanks for reminding me of this traumatic experience
Kinda feel bad but it's funny haha😂😂
same
L my 8th grade teacher made us make one and make it into whatever we wanted so i drew a snail
@@LordFishTheSecond yo I think we could also make it into something, pretty sure I also made a snail (or one of my friends at least)
I made a crab that was very ugly
"Do you think you could construct this by hand?"
Ammonites: "I don't even need hands"
Genius
These follow the Fibonacci sequence... similar but not the same
This is the kind of comments I like
damn he really wanted to know if I think I could construct this by hand
You gonna tell him or leave him hanging?
you look like Aliensrock
@@Rev_Erser no he looks like me
Yeah you can construct it by hand, And a ruler (for straight lines)
The compass could even be locked to draw circles of 1 unit radius it would take time but it could be done by hand. I wanna know how he constructed it.
Once it gets bigger it kinda looks like a fancy spiral seashell. It's really pretty.
I like the pacing of this short. Very contrary to the seemingly rushed speech and lack of breaks of other shorts
I heard spiral out. The TOOL fan in me has been awoken.
Yup
Keep. Going.
Was looking for this comment
I remember learning this 9th class but couldn't fully understand it back then
...
Cbse board?
@@abhidababy6746yep
I smell CBSE
In case anyone here is in the same boat:
This happens because of the Pythagorean theorem - because the relationship between the length of the hypotenuse ( _c_ ) and those of the other two sides ( _a_ and _b_ ) is given by the formula _a²+b²=c²_ , we can express the length of the hypotenuse directly by taking the square root of both sides of the equation: _c=√(a²+b²)_ .
Now, if we take _a_ to be the square root of some positive integer _n_ , and _b_ to be 1 (as is the case in the video); we can fill in the expression for _c_ we got earlier. This results in the equation: _c=√([√n]²+1²)_ - notice that we are squaring both a square root (its inverse operation which cancels it out) and the number 1 (1 raised to any power is 1), so we can simplify it to receive the expression: _c=√(n+1)_ - which is exactly the relationship described in the video.
I SMELL CBSE
My geometry teacher in high school would have us do constructions every week where we’d make a little piece of “art” using whatever formulas we were learning about at that time. This would be right up her alley 😂
Lol I actually found this by myself just doodling some triangles. Super cool that you can get measurements for basically any square root’s values this way!
Sounds like a cool way to compute the square roots. Actually, I wonder how computers do that in the first... New rabbit hole, here I go!
Back then computation wasn’t as straightforward in geometry-greeks
Most computers use Newtons method I assume. If you know calculus, you probably know that a derivative gives you a rate of change. This rate of change corresponds to the slope of the tangent line at a point (Think of it like deconstructing a curvy line into many tiny straight lines.)
You can use this fact that a derivative is a tangent line to solve equations of the form f(x) = 0 by starting at an arbitrary point on the graph and repeatedly drawing tangent lines and finding the point they intersect the horizontal axis to approximate the solution x of the equation.
In our case, if we want e.g. sqrt(2) we are really trying to solve the equation
x = sqrt(2)
x^2 = 2
x^2 - 2 = 0
and we can apply newtons method starting at X = 1 to find the values
step 1 -> 1.5
step 2 -> 1.416
step 3 -> 1.41421
and we already found the first 5 digits after the decimal point with 3 steps.
@@mrocto329Good explanation. As computers have limited precision they'll just stop when they hit that limit. Some applications that require super fast computation, e.g. games, will sacrifice precision for speed.
Newton's method requires lots of division and that's an expensive operation.
I remember a long time ago trying to write a really fast circle drawing program in 6502 assembler. It required square roots. Division was really hard so I instead went for a process that took advantage of the fact that n^2 is the sum of the odd numbers from 1 to 2n-1. I had a loop that repeatedly subtracted odd numbers until the result would be less than zero, then rounded up or down as appropriate. It was accurate enough and fast enough. A binary lookup table would have been even faster.
I believe square roots have an infinite sum series, and stop after the series stops affecting the last-most digit displayed.
This is how most computer systems calculate Linear Transformations in different cores, each core with a different transformation, then all summed up after they're all done.
Each core can calculate the square root series for different values (indeces) of n , then add up the result and repeat!
there's a really good book on euclidean geometry that's just all constructions beginning with a line is that which subtends the distance between two points or some shit like that.
But it does the Pythagorean theorem and it does square roots and it does all sorts of crazy stuff with just constructions. basically validates math
There is a much simpler and non-recursive way to construct sqrt(n) using the fact that sqrt(n)=sqrt(n*1) which is the geometric mean of n,1. The geometric mean of two numbers a,b can be seen as a perpendicular to a diameter of a circle with length n+1 when the perpendicular stops when it touches the circle. In other words, you can first construct n+1, which is a pretty simple task, then bisect the segment to get the center of the circle. Then you can draw the circle, draw a perpendicular line 1 units from the end of the segment and voila your sqrt(n) is just the length of that perpendicular segment.
correct me if i'm wrong but i think this also lets you take the square root of any rational number (or constructive number in general)! so yeah, this is the much preferred method :)
@@niuniujunwashere Yes indeed this method can construct not only the square root of any rational, but the square root of any CONSTRUCTIBLE number. Every number that you know how to construct, you can construct it's square root with this geometric mean method.
sorry I didn't get what you mean by "draw a perpendicular 1 units from the end of the segment" like the length would be always 1?? so how does it represent sqrt(n)?
@@chinmay1958 the horizontal line segments have length 1 and n respectively. the vertical line segment (i.e. the perpendicular) is what represents sqrt(n), not any of the horizontal ones.
I would just blow up (1, 1, √2) to be (n/2, n/2, √n)
Clearest explanation ever
I can't figure out the point of using the compass, since you don't show using it to find the perpendicular of your √ line. You can make this construction with just a right-angle triangle ruler for your straight edge.
All the lines must be length 1.
@@MathVisualProofs If my straight edge doesn't have markings on it, yeah, a compass would help. But any point on the circumference of that circle will be length 1 from the center, so how does having it help you make sure your new line is perpendicular to the √ line?
@@KalliJ13 Ah, still have to also construct the perpendicular line. Didn't want to show that full construction here :)
Straightedge and Compass means you can only draw straight lines and radians. Right-angles can be drawn using these two tools. There is no right-angle tool allowed.
No right angle tool allowed. Thats a luxury. Straight lines and circles only. The seeds of all angles.
then visual representation of the spiral motivates the conjecture, that the difference of the radius between the loops remain constant. Then one could draw the spiral with a pencil limited by a thread winded up around a cylinder with radius=1 in the center which is rolling off by drawing. The difference between loops therefore is constantly 2*pi.
Well, this is the best thing I've seen the whole day.
Thank you for this amazing performance.❤
I love that it says sqrt(4) instead of two
Spiral out, keep going 🤘
I actually learned something...Bravo!
We were given a question on this in a maths - we were told that all the outside lengths were 1, and told to find the 9th hypotenuse (which, after working it out, was sqrt(9) which is 3!)
Didnt realise it was an actual mathematical thing tho lol
Constructing this by hand would be difficult. Even a tiny early error would compound with every iteration. Very very cool though.
It could be manually kept _near_ correct by measuring lines with a ruler (and making adjustments if needed) when the root is also an integer, e.g. 3/9, 6/36, 11/121...
This one is pretty beautiful
Is he ever gonna stop surprising us?
Beautiful
Didn't know this. Very neat. Might try to draw
The urge to extend this spiral into a smooth curve and find a polar equation for it ↗️↗️↗️
haha I was JUUUUUUUUST typing this
The fact you explained this better than my teacher is scary
This is spiraling out of control
Wow. That was a movie. This is some of the best content on youtube Ive ever seen. It was so interesting and exciting to follow you along on this adventure, and it really makes me want to do something like this myself. And I probably will. I have actually been in that exact mine once, so it was cool to see it again and also see more of it. Absolutely amazing video, I will definitely subscribe!
I remember reading somewhere that using visuals is absolutely necessary to teaching math, that if you just tell people how to do the calculations and expect them to understand just through repetition it's going to be much harder for them than using this kind of image/animation...
i HAVE constructed this by hand
and so has jason padgett
This song feels so soft and calming. Can't wait to see more music!
This is beautiful
And its not even a logarithmic spiral, that is what is so beautiful and strange about this.
So you are saying that anything that is not a logarithmic spiral is beautiful and strange?
@@Tommy_007 your mom's not a logarithmic spiral
Me: Not understanding a thing
Also me: Ohhh interesting
Wait yes: 1 could have just drawn 2 lines with an 90° angle with lenght 10*1 each. The connecting line would be 10*sqrt(2) = sqrt(100)*sqrt(2) = sqrt(100*2) = sqrt(200)
Or am I missing something?
Very interesting stuff.
what a beauty.
This is my favorite comedy channel.
Great way to train and check the precision of your drawing skills with pencil, compass and ruler.
(when square number pops up you measure length of the constructed line)
Thats actually neat
Source mappers will use this for elaborate stairwells
This was part of my maths syllabus last year
Just imagine how much free time and BOREDOM there was in ancient times for Theodoras to doodle SO MUCH that he came to those realizations 😮
This could probably also be useful for drawing spiral staircases
"Spiral out... KEEP, GOING..."
Spiral Out. Keep Going.
its also great for drawing top-down arial shots from the center of a spiral staircase
Maths and geometry are soo cool! And ammonitoidea to.
I did it jn 7th or 8th standard to locate √2 or any other irrational number on number line.
You learned this in (USA) junior high? Neat way to find them!
Well this is not true. Not every real number can be constructed this way, only numbers of the form sqrt(n). Although all constructible numbers (en.wikipedia.org/wiki/Constructible_number) can be drawn by a ruler and compass.
But it is not true that you can locate any irrational number since some numbers are non-constructible, like pi or e or all transcendental numbers or ever third roots of rational numbers. In fact it is known that there are more non-constructible numbers than constructable numbers. So what you have probably did is locate constructable numbers, which are dense in the reals so they may have given you an illusion that they cover all of the real number line.
*PRAISE THE HELIX FOSSIL*
*FOREVER SHALL ANARCHY REIGN*
A mathematically perfect spiral. My ocd thanks you as I now have an acceptable spiral formula I can use in designs without feeling nasty.
I absolutely can construct this by hand. In my mind.
the minecraft nautilus shell in my inventory:
That's dope shit.
Hey this is a really clean art style - what software did you use to make this animation?
EDIT: Manimgl - found it in your bio, including it in case others want to know the answer!
😀👍
"The perfect golden rotation"
I used to do this by hand in secondary school
It looks like a fossil seashell, it's like the math is embedded in nature 😮❤
That's exactly how it works, actually.
Yes. It's easy to construct square roots, not just of integers, but of anything in the field you're working over. Start at the origin. Draw a line segment of length 1+x. Draw a circle of radius (1+x)/2 at the midpoint of the segment. The line drawn perpendicularly between 1 along the segment and its intersection with the circle is of length sqrt(x)
I love your music where can I listen to it by itself
Architects gotta learn from this
We got this as the maths holiday homework till √50 , it was fun
“Do you think you could construct this by hand?”
Nautiluses: 🎉
Oh look 👉 a cool transparent spiral staircase!
Cool!
my 7th grade math book's cover has this, when i found out about it, i found it very amazing
Btw this is also an amazing visualization of the concept behind Pythagorean theorem ✌️
Damn, what a house of leaves place where you basically have no mouth and you must scream. Just the way the jaunts and how IT keeps pace make me want to quote the raven tale. Amazing video, as well as the multi-layered narative, and the camera making it feel like that house had people in it
The way he asks with such an intensity if I'm able to construct this by hand is somehow intimidating
Wait so… if you do this to infinity, you would have a hypotenuse of length root(infinity) = infinity, as n -> infinity, and a side of length root( n - 1 ) which is still infinity as n -> infinity, but have a line segment of 1 because of the circle drawn with radius 1
Un triangolo isoscele di base 1 e altezza infinita ha due lati di lunghezza infinita e DUE angoli retti.
Neat!
Theodorus means "GODS GIFT"
I thought it meant "the smelly one", as in "The odoures" 😅
Oh!I did that all the time when younger lol
That last bit sounded pained, like "do you really think you can do what I did?? do you really think it's that easy?!?" XD
As the pattern evolves, I’d like to see the relationships between the perfect squares, 4,9, 16, 25, 36 etc. Or how the primes relate.
I’ve drawn this before. Very satisfying.
I don't think that primes would give you an interesting pattern here, but perfect squares might.
I just got an idea for the most needlessly complex spiral staircase in history
Great Mosque of Samarra?
Fun fact: If you make each full turn rise a constant distance, as Samarra does, you have to grade the rise of each step, from smallest rise at the bottom, where the radius is largest. I.e. to have a constant slope to the eye, it has to grow steeper underfoot as you ascend. Or to have a constant slope underfoot, it would have to have a dome shape visually.
fuggin nautilus lore thats sick
Could I draw that? Probably, if I was really bored.
The staircase that the characters have to go up (it's for comedic relief)
U givin me flashbacks to a problem we had to solve In School a while back. We had to figure out the perimeter of any given shape, N.
“Do you think you could construct this by hand?”
Do I have the dedication: no
Do I have the patience: no
Do I have the time: yes
- In short no.
that looks like its a spiral to heaven
Math is so cool
I'd rather turn this guy into my math teacher
My mind is blown
Spiral out, keep going
Spiral out, keep going
Spiral out, keep going
Now slap a z-coordinate for extra funkiness
Given that you have a unit line and can presumably directly draw lines of integer lengths, the easiest way would be to draw a triangle with hypotenuse length 15 and one side length 5, and the other side would be length sqrt(225-25) = sqrt(200). Otherwise, if you really wanted to use this spiral somewhere, you could start with a 14-1-sqrt(197) triangle
this is the shape we draw when we are board and dont know what to draw
My guess from seeing the spiral was that it would be based on angles, like 90, 45, 22.5... but it wasn't and now I wonder if there is/what is the relationship between the angles.
THIS IS MY SPIN TECHNIQUE
Easy to draw by hand, you showed how. Just need a compass and a ruler and a large enough piece of paper.
Man I was watching the whole video like I knew what they were talking about
"Is that an Archimedean spiral?" -Peter Parker
Before I was suspicious but now I know for sure: snails know math.