Spiral of Theodorus
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- Опубліковано 2 жов 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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#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.
😂
SCP foundation
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)
"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
this is very relateable lol
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
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.
I heard spiral out. The TOOL fan in me has been awoken.
Yup
Keep. Going.
Was looking for this comment
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 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
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)
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
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!
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.
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 😂
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.
Is he ever gonna stop surprising us?
*PRAISE THE HELIX FOSSIL*
*FOREVER SHALL ANARCHY REIGN*
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
Could i produce this by hand? Obviously, yes. Will i? Obviously, no.
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...
“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.
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.
It actually starts from sq rt of 1 ;)
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?
Clearest explanation ever
Explanation of what? For what it is, yes. For why it is, no.
@@xinpingdonohoe3978to me the why is also clear
Even when you are not a math person, this is all the way fascinating…
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!
😀👍
Haha this is at my school i had to draw this and the roots
This could also be seen as a proof that the roots of all integers are constructable numbers
For sure! In fact the longer form version of this mentions that :)
i HAVE constructed this by hand
and so has jason padgett
Who?
I can't even construct it with compass and edge, forget about "ByHaNd" 😅😅😅
Well, this is the best thing I've seen the whole day.
Thank you for this amazing performance.❤
Yeah because sqrt(200) is just 10*sqrt(2) which is the diagonal of a square of side length 10
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
Is this what they mean when they say:
"You're spiraling out of control" ? 😂😂😂
I did like 300+ for a math project. I did the most and got a free 10. I don't know exactly the number.
See, this is why math can't represent real life, because if I keep spiraling down I'm gonna have a Michael Douglas "Falling Down" moment
As N goes to infinity does the length of the inner spiral equal to the latest external spiral ..what is the radius of that external spiral/circle ??
سؤال جيد
اعتقد يساوي طول منحنى الدالة جذر n
@@user-awrssadk yeah but after 360 degrees of rotation is the inner and latest spiral circle close enough to minimize the gap ? or it still increases geometrical? to infinity...and last after a full rotation 360 degrees in which N we end up? is it steady a fixed number or it increases to with every full rotation? it takes more steps to fill to 360??
يزيد مع كل دورة جديدة إلى ما لا نهاية على ما اعتقد ، ولا ينتهي في الدورة الداخلية ابداً
I love that it says sqrt(4) instead of two
I'm annoyed it doesn't start with √1. We're on the positive branch, might as well make it complete. But would there be a √0 length if we tried to make a circle in the other direction?
İs that JoJo reference !!!
'Do you think you can construct this by hand'
Are you really going to tell me you made this video with your feet
Could I do it by hand. Maybe. But I’d rather draw it with python, or LaTex.
Very cool! Although to make root of 200, I would make a 10, a root 2 and multiply them XD
This could probably also be useful for drawing spiral staircases
Colorful, cool and a little trippy
👌👍😎 ( Sweet )
in my school in 9th grade , in our maths chapter Real Numbers we had an exercise in which we were asked to represent irrational numbers like root 2 , root 5 using this same method.
Now make a triangle of squareroot of 1 or 0 and I'll be happy
Could I draw that? Probably, if I was really bored.
I can construct this with my complex number library
How do you do this by computer? I would like to program this ! :D
I'd just draw a 10x10 square and get the diagonal EZ
Well, I can construct it with hand. That part is not my concern. But consistently, no.
The reason this works is because pythagoras theorem, we add 1² in (sqrt n)² and take root of it as a whole
Maths is kool
Huh, I actually did this for fun once when I was a kid. stopped after it made a shell shape.
Isn’t this basically just the golden spiral?
Huh. I was just thinking the same… can I construct this by hand
this actually sounds really fun to do by hand, maybe i will if i ever have free time (student rizz)
Yeah, I could construct that by hand, piece of cake. Oh, you want it to be accurate? In that case, no, not a chance ;)
No, the inaccuracy will compound drastically
I accidentally read it spiral of thermodynamics💀
The urge to extend this spiral into a smooth curve and find a polar equation for it ↗️↗️↗️
haha I was JUUUUUUUUST typing this
Probably not easy. Wouldn't this one satisfy something like θ=sum k=0 to r²-1 of arccot(√k)? Unless we can get a partial sum formula, it won't be easy to smooth.
brothe my holiday homework of class 9 was to create this spiral till 200 and we did it and all other clildren
by hand if any one think its li is if i lie i know you dont need to say in my comments
It looks like a fossil seashell, it's like the math is embedded in nature 😮❤
That's exactly how it works, actually.
If you spiral in, instead of out, you will find the secret to infinity.
Now slap a z-coordinate for extra funkiness
I actually learned something...Bravo!
Like the Golden Ratio? but different math?
Spiral out
Keep going
Spiral out
Keep going
Spiral out!
what the hell? nah mathematics is the most beautiful and amazing thing i will ever see
I absolutely can construct this by hand. In my mind.
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
As an Indian, we learn this is class 9 maths as "Spiral Root"😅
Source mappers will use this for elaborate stairwells
Cool but they never intersect. Thats anooying
Well, I made one before I watched this video.
Pov: guy with OCD wants to draw a spiral
Didn't know this. Very neat. Might try to draw
But what is the angle of the tangent lines drawn from the circle? [Edit: oh I see, you are making right triangles the whole way
I think it would be easier done without those circles
We actually had to do this at school only like a month ago till root17
Reminds me of the cover of House of Leaves
Spiral of the odorous smells fishy... 🤔
the minecraft nautilus shell in my inventory:
babe wake up, new golden ratio just dropped
How does the perpendicular line extend in length if it's only length 1? The circle is radius 1 so the line can't ever actually get bigger, can it?
It's clearly not radius 1. However, the outer line (the "opposite", relative to the angle touching that centre) is always 1, for each triangle.
"Spiral out... KEEP, GOING..."
I can construct sqrt(200) easily yes
A mathematically perfect spiral. My ocd thanks you as I now have an acceptable spiral formula I can use in designs without feeling nasty.
Jagged lines don't really constitute a "perfect spiral" in any sense.
Spiral out, keep going 🤘
Yeah thats a root spiral we did in 6th grade 😂
and how can I draw √2 exactly ?
Construct a square of side length 1 and draw the diagonal
Am I stupid or is this the golden spiral?
It is called saligram diagram according to sanatan
I learned this in algebra! Didnt know it had a name
I've never seen such an unnecessary use for circles lol
Easy for me, I am really good in spiraling