I hope this is OK, I may have written this story before as a reply to a Numberphile video, but I once got a copy of ;The Penguin Dictionary of Curious and Interesting Numbers' by David Wells, and somewhere (I couldn't find where) he jokes about people might misunderstand a number followed by a '!', and at the end, writing about Graham's number, he end with "...who suspect that the answer is 6 !!", seeming to be very careful to put in at least one space after the '6'.
Funny, because these proofs are actually from the high school mathematics that was thought in Eastern Europe up to 20 years ago... Ptolemy's theorem was in the 8th or 9th grade.
Sure, she's a genius, but we forget how hard were math books in the 80's in Eastern Europe, and how poor the math level is in high school in most Western countries in 2022.
@@Peterwhy checkmate how? If that's code (or at least code where "!=" is the inequality operation), it just returns false. Nothing wrong with that (other than how horrible your code is). And if your just expressing it purely mathematically, that's still perfectly valid.
It's one thing to see this elegance in the explanation, but how mind bending must it have been to originally conceive the inversion principle and apply it ? !
I could feel it when you said "I'll be the one editing this video." This was probably a lot of work, but I am super happy you stuck around to take us through the entire lecture. That was very interesting. Thanks for making this, both of you. You reveal the world to be full of interesting mysteries and complexities, many far beyond my scope of understanding, but I feel like I get a glimpse of those complexities through your videos, which to me is incredible. Thanks for making the future what it was promised to be.
I always felt, since we use exclamation mark naturally for the continuous product of things, we should equally use the question mark naturally, as the continuous sum of things.
What I find interesting about this is the connexion between the five-sidedness of a pentagon and the root five that happens to show up as the major feature of the Golden Ratio.
This was in my recommendation and it’s been years since I did this type of math, but surprisingly I didn’t hate it. You explained it beautifully and simply!
2:50 okay that's a pretty crummy regular pentagon. Get me Professor Eisenbud, a ruler and compass and a gratuitously complicated list of instructions...
The instructions for constructing a regular pentagon using ruler and compass involves constructing a segment whose length is golden ratio * side length, so that kind of gives it away...
Whaaat! How did the golden ratio end up here? What a legendary number! The video with the proof that it's the most irrational number is one of my favorites of the channel
Pentagon's are cool! I discovered that if you draw out a Pentagon, remove one of the triangle segments and close up the gap the arrangement lifts into a 3d shape very similar to the pyramids in Egypt. I suspected that the angles were similar too.
Well that fact a little bit surprised me, somehow I was still unaware of that, but several minutes and I understood how naturally golden ratio pop up here. My math teacher once showed me how do u derive sin(36°) using a very unique shape isosceles triangle with angles 36° and 72°. If you haven't heard of this triangle before try it yourself, draw it and draw several angle bisectors. So with this triangle you will be able to write sin(36°) as a radical and it contains sqrt(5). And ye, pentagon's angles are 36°*3 = 108°. Cosine law and you get the result! But still a fancy fact
Looking at the comment section feels like sneaking your way into a prestigious university with lots of smart people. I can't comprehend what the people here are even talking about. I also feel that if I subscribe on this channel it'll charge me for a thousand dollars without me knowing.
What is an equilateral triangle. The triangle that can be fit inside a circle and that CG can be at the centre. Usually most pyramids are different base. Base is like the number base. Ten base and two base usually give seven at centre. That's why π. Ratio is a binary system to other base numbers. What is prime. The joint between different base. Duality of systems are because of lowest base.
Hi, Wonderful prouf! I didn't know. And I discovered the Ptolemeys's theorem as well. Just one point: the so cold golden ratio ... is not one (ratio), since it is a non rational number. However, I know this phrase is very often used for that number.
Hey Y'all I just ran (1 minus (the square root of 5)) divided by 2, and the answer is −0.61803398875 if you're curious. This happens to be equal to -1 over 1.61803398875. 1.61803398875 being the golden ratio. I just found that out by messing around. pretty cool my dudes
Yeah! You've essentially put the algebra into words. What number x is such that -1+x is 1/x? Well, it solves -1+x=1/x or x^2-x-1=0. Exactly that quadratic equation.
You’ve also stumbled onto one of the definitions of the golden ratio. A golden ratio rectangle is one where if you cut out a square of length equal to the shorter side, you are left with a rectangle of the same proportion as the original. So if one side is 1, and the other side is x (making the ratio x/1 or just x), and you cut out a 1x1 square, you’ll be left with a rectangle that has sides 1 and x-1 (making the ratio 1/x-1). The golden ratio is the value of x where those two are the same. ie, x = 1/x-1. If you rearrange, you get the equation in the video: x^2 - x - 1 = 0. So coming to what you found, that’s why the two roots (if you ignore the minus sign) are the same number with a difference of 1. If you have a golden rectangle and look at the ratio of longer side to shorter side, you get 1.618... If you look at the ratio of the shorter side to the longer side, you get the same number minus 1. The golden ratio is the only number for which this is true.
A circle means constraints. Spiral means for pentagon. To reduce to get golden ratio. How to get other ratios. Increase the number of sides. You get Pi. So Pi is a constraint of line. Or splitting factor of constraint waves generating frequency. A circular dish gives frequency spectrum transmission of three. One five etc. Elliptical ones always give triplets of frequency. Like the orbital. Eggs shape for one equals other. Or two identical frequency for one mid way. That's why force transfer is stable. A circle inside egg touch at five points. Somewhat like u cords. How to prove.
Could you redraw using the 8/4 sided pyramids in the sphere? Then calculate the angle of refraction along the radius? Then add squares to that in their correct place. I would like to see the breakdown of the solids fitting in the sphere. How many will fit is the quiz.
I don't remember how I did it, but I remember that one day in middle school i found "by accident" that there were many golden ratios in a 5 pointed star
This reminds me of a video by 3blue1brown on the stability of phi and phi's little brother/-0.618 the other solution for this quadratic formula. He basically used one of the formulas of phi (I think its 1/(1+1/(1+1/(1+1/...))) to transform the number line so that we could see how the points converged to phi and "inverse phi" or -0.618... Interestingly, the transformation looks like a eclipse, not sure if its related to inversion or not...
Just watched the Conway Checkers proof and now this. Think it’s just a coincidence that both hold great value to the golden ratio as well as the number 5?
Fantastic and elegant! Is there an equation of ratios for each shape? it looks like it goes to infinity, maybe can be represented by some sort of ln(x)
This continues from our proof of Ptolemy's Theorem --- ua-cam.com/video/bJOuzqu3MUQ/v-deo.html
Why do they not teach this in preschool!
Ptolemy sounds like it's the name of a disease.We shall use the real name: Ptolemaios.
Numberphile ✌️
Bradyhow did you guess that so quickly at the beginning that the two short lines equal the long line?
So... does a = radius?
All Numberphile guests are wonderful, but Zvezda is my personal favorite. :)
false.
I laughed so hard when Brady reconfirmed that was not the factorial. He must have been so traumatised by numberphile
Now d is a unary higher order function with the factorial operator as the argument 😉
I hope this is OK, I may have written this story before as a reply to a Numberphile video, but I once got a copy of ;The Penguin Dictionary of Curious and Interesting Numbers' by David Wells, and somewhere (I couldn't find where) he jokes about people might misunderstand a number followed by a '!', and at the end, writing about
Graham's number, he end with "...who suspect that the answer is 6 !!", seeming to be very careful to put in at least one space after the '6'.
Its a parker exclamation
@@gabrielkellar1935 underrated comment
false.
Everyone deserves a math teacher who is as passionate as Zvezda.
false.
*any hard math problem*
Prof. Zvezda: "I remember struggling to solve this problem at kinder garden"
Jeremias Figueiredo ”Back when I was a mortal”
hahahaha
Funny, because these proofs are actually from the high school mathematics that was thought in Eastern Europe up to 20 years ago... Ptolemy's theorem was in the 8th or 9th grade.
Sure, she's a genius, but we forget how hard were math books in the 80's in Eastern Europe, and how poor the math level is in high school in most Western countries in 2022.
false.
1+1=2! Checkmate factorials
1+0=0!
1!+2!=3!!
1!=1 Checkmate programmers
@@DiegoMathemagician
The fact that you used the notation of double factorial without meaning twice factorial means that your brain is very big
@@Peterwhy checkmate how? If that's code (or at least code where "!=" is the inequality operation), it just returns false. Nothing wrong with that (other than how horrible your code is).
And if your just expressing it purely mathematically, that's still perfectly valid.
She's a gifted teacher.
It's one thing to see this elegance in the explanation, but how mind bending must it have been to originally conceive the inversion principle and apply it ? !
math is just playing with an idea you came up with while holding all other worthwhile ideas of humankind on the back of your mind.
Well it was proved long (ancient Greece) before inversion transform / geometry came along in 1831.
its fun inverting random shapes until you have to justify your algorithm to other people
For sure, inversion was not conceived for proving Ptolemy's Theorem.
Inversions are really just 1/x but in 2 dimensions.
The sequel is even better than the first! Blown away.
Linda Ristevski ✌️
Is is first factorial or......
??
fun fact: zvezda «звезда» means “star” in many slavic languages
So, "Stella" in English.
What an ingenious proof, the stark elegance of it all is just mind blowing
steven wonder ✌️
The way I remember the golden ratio is: taking 5 and 1/2 and applying the operations in descending order: 5 ^(1/2) *(1/2) +(1/2)
Smart, I solve X^2-X-1 every time ...
i remember it as (1+sqrt(5))/2...
I like 1.62, as it is almost exactly the ratio of kilometres to miles.
And miles to kilometres is about 0.62.
I remember golden ratio as the continued fraction of (1,1,1,1...) i.e. 1+1/(1+1/(1+...)) then I just solve x=1+1/x
@@susmitamohapatra9293 So the Golden ratio is an irrational number that can be computed?!
I could feel it when you said "I'll be the one editing this video."
This was probably a lot of work, but I am super happy you stuck around to take us through the entire lecture.
That was very interesting.
Thanks for making this, both of you.
You reveal the world to be full of interesting mysteries and complexities, many far beyond my scope of understanding, but I feel like I get a glimpse of those complexities through your videos, which to me is incredible.
Thanks for making the future what it was promised to be.
Somehow, "!" Is unintendedly one of the highlights.
I always felt, since we use exclamation mark naturally for the continuous product of things, we should equally use the question mark naturally, as the continuous sum of things.
I love Zvezda's smile.
The way she draws b really makes you see how it evolved from B
It's actually because of the russian "B" written small as "в".
Old habits of hers :)
Yes, it's в in cursive. Also it would be actually pronounced like v.
Wrong word... progressed, not evolved. I know this is maths here, but still. We're all speaking English - more or less, here anyway.
@@miorioff That's Cyrillic not Russian.
@@justincronkright5025 sure, whatever it is
4:00 "could be the most famous ratio?" Zvezda, may I introduce you to the ratio of a circle's circumference divided by its diameter?
Not really a famous ratio if it isn't even called the ratio :)
lol fair point
@@Craznar That's like saying Einstein wasn't a famous doctor of physics because nobody refers to him as Doctor Einstein in every day speech.
@@vitalspark6288 lol fair point
Vital Spark 911, it’s Christopher Burke he’s been killed
And one can use the Pentagon relationship to show that cos(36 degrees) = golden ratio /2, and cos(72 degrees) = (golden ratio - 1)/2
Trigonometry... ewwww! ;-)
.
And you can continue with this to show that sin(666°) = - golden ratio/2 ;)
What I find interesting about this is the connexion between the five-sidedness of a pentagon and the root five that happens to show up as the major feature of the Golden Ratio.
“ oh look an homogeneous equation I can do miracles with that”
I’ve laughed so hard😂😂😂😂
??
This was in my recommendation and it’s been years since I did this type of math, but surprisingly I didn’t hate it. You explained it beautifully and simply!
Wow, I could listen to Zvezda explain math things all day.
These last two videos have been brilliant.
This might be my favourite Numberphile video ever
2:50 okay that's a pretty crummy regular pentagon. Get me Professor Eisenbud, a ruler and compass and a gratuitously complicated list of instructions...
The instructions for constructing a regular pentagon using ruler and compass involves constructing a segment whose length is golden ratio * side length, so that kind of gives it away...
Theorem: every Bulgarian math teacher has this accent in any language they speak.
I had a Bulgarian math professor/instructor (Kumchev) and he had an interesting accent.
Why did I read your comment in an accent?
@@TheSimplesAreFree Maybe you are a Bulgarian math teacher?
It's only a theorem if it can be proven.
@@daddymuggle It is only theorem until it is proven.
The ending to this video is like a phone call with my grandmother.
'I think our journey was worth it'. CLICK.
No goodbyes, no I love you, just click!
This is insane! We have to inform the Pentagon!
Oh wait, they already know.
they about to use phi to justify another 700bn raise.
@@alveolate Semper phi ?
This is the type of joke I live for
"The Bermuda triangle?" lol :-)
Brady's brain is fried at this stage!
He wasn't far wrong though. The Bermuda Triangle has sides 1669 km, 1663 km and 1545 km. It is almost an equilateral triangle.
@@tylisirn there are many different definitions of the Bermuda Triangle
So it's a Parker equilateral triangle?
@@silkwesir1444 I used the one they showed on the video which is one of the most common. Vertices at Bermuda - Miami - San Juan
Yeah Zvezda, I also remember struggling with this problem in kindergarten.
Greetings from Bulgaria! 🇧🇬
Since we've worked so hard to prove it 😂. My favourite teacher on Numberphile!!!
I could genuinely listen to Professor Stankova’s accent all day.
Whaaat! How did the golden ratio end up here? What a legendary number! The video with the proof that it's the most irrational number is one of my favorites of the channel
The beauty of mathematics properly explained
Numberphile is now at Pi million subscribers in the world of Indianna.
I've seen this numerous times. I can't believe I've just liked it now! It's a laugh but a great lesson.
I literally solved a family of problems without knowing Ptolemy's theorem. 💀
That is beautiful; it makes me love math even more. Live == learn. Thank you for the video: you made my day!
Incredibly elegant
thank you for answering my question AND showing the math!!
Zvezda is such a Star!
Pentagon's are cool! I discovered that if you draw out a Pentagon, remove one of the triangle segments and close up the gap the arrangement lifts into a 3d shape very similar to the pyramids in Egypt. I suspected that the angles were similar too.
When you do too much math you start seeing exclamation point as factorial
d!
Thank the gods for numberphile. For a while it started looking like I was gonna have to work for a couple of marks in my high school assignment xD
Love your voice. My French teacher 2.0.
Sh'es actually a gorgeous mathematician
Well that fact a little bit surprised me, somehow I was still unaware of that, but several minutes and I understood how naturally golden ratio pop up here. My math teacher once showed me how do u derive sin(36°) using a very unique shape isosceles triangle with angles 36° and 72°. If you haven't heard of this triangle before try it yourself, draw it and draw several angle bisectors. So with this triangle you will be able to write sin(36°) as a radical and it contains sqrt(5). And ye, pentagon's angles are 36°*3 = 108°. Cosine law and you get the result! But still a fancy fact
Ptolemy is my new favorite mathematician.
Finally you have talked about phi
This is pretty cool, time to add this to my problem-solving arsenal
She's a very lovely and fantastic teacher. Can we get more videos?
Amazing, I miss the circle making part.
Quite plainly, if Zvezdelina ever offers a course in Bulgarian middle school geometry, I'll be first in line.
In Euclid's Elements there is a compass and ruler construction of the Golden Ratio and thence a regular pentagon.
Awesome Leibniz portrait in the background 6:28
Bermuda pentagon
Looking at the comment section feels like sneaking your way into a prestigious university with lots of smart people. I can't comprehend what the people here are even talking about. I also feel that if I subscribe on this channel it'll charge me for a thousand dollars without me knowing.
What is an equilateral triangle. The triangle that can be fit inside a circle and that CG can be at the centre. Usually most pyramids are different base. Base is like the number base. Ten base and two base usually give seven at centre. That's why π. Ratio is a binary system to other base numbers. What is prime. The joint between different base. Duality of systems are because of lowest base.
Hi,
Wonderful prouf! I didn't know. And I discovered the Ptolemeys's theorem as well.
Just one point: the so cold golden ratio ... is not one (ratio), since it is a non rational number. However, I know this phrase is very often used for that number.
Thank you Prof. ♦️
Hey Y'all
I just ran (1 minus (the square root of 5)) divided by 2, and the answer is −0.61803398875 if you're curious.
This happens to be equal to -1 over 1.61803398875. 1.61803398875 being the golden ratio. I just found that out by messing around. pretty cool my dudes
Yeah! You've essentially put the algebra into words. What number x is such that -1+x is 1/x? Well, it solves -1+x=1/x or x^2-x-1=0. Exactly that quadratic equation.
You’ve also stumbled onto one of the definitions of the golden ratio. A golden ratio rectangle is one where if you cut out a square of length equal to the shorter side, you are left with a rectangle of the same proportion as the original. So if one side is 1, and the other side is x (making the ratio x/1 or just x), and you cut out a 1x1 square, you’ll be left with a rectangle that has sides 1 and x-1 (making the ratio 1/x-1). The golden ratio is the value of x where those two are the same. ie, x = 1/x-1. If you rearrange, you get the equation in the video: x^2 - x - 1 = 0.
So coming to what you found, that’s why the two roots (if you ignore the minus sign) are the same number with a difference of 1. If you have a golden rectangle and look at the ratio of longer side to shorter side, you get 1.618... If you look at the ratio of the shorter side to the longer side, you get the same number minus 1. The golden ratio is the only number for which this is true.
Tyson Hayter JUST STFU
@@NoWayHaze a number such that adding one to it is equal to it squared seems nicer
@@newkid9807 JUST STFU
A circle means constraints. Spiral means for pentagon. To reduce to get golden ratio. How to get other ratios. Increase the number of sides. You get Pi. So Pi is a constraint of line. Or splitting factor of constraint waves generating frequency. A circular dish gives frequency spectrum transmission of three. One five etc. Elliptical ones always give triplets of frequency. Like the orbital. Eggs shape for one equals other. Or two identical frequency for one mid way. That's why force transfer is stable. A circle inside egg touch at five points. Somewhat like u cords. How to prove.
Mum: do you want some pizza?
Me: 1:20
Great video, as always
Was nice with a meaty video (pt 1+2) :-)
I think that a non-euclidian triangle with three 90 degree angles is more spectacular to consider.
I didn't know Yennefer was so well-versed in math too. Respect.
Wind's howling
Another reason to love the golden ratio.
may i introduce you to jojo part 7
Could you redraw using the 8/4 sided pyramids in the sphere? Then calculate the angle of refraction along the radius? Then add squares to that in their correct place.
I would like to see the breakdown of the solids fitting in the sphere. How many will fit is the quiz.
I don't remember how I did it, but I remember that one day in middle school i found "by accident" that there were many golden ratios in a 5 pointed star
Love it! More, more!
A regular distribution curve is Pentagon. E equal mc2 follow m central and line 🕸.
I have a Bulgarian colleague at work, he is just as mad as Professor Zvezda, he cracks me up xD
It looks like NUMBER OF THE BEAST at diagonals on pentagon!!! ... and the pentagon is mystical figure )
Make a video on the burning ship fractal? Would be interested to learn how it works.
I HATE TO NOT BE TEACHED PTOLOMEUS THEOREM!!! I loved pythagoras theorem... but I am a cheater now!
YOU ACCIDENTALLY THE CAPS LOCK AND ENGLISH! MY BRAIN ASPL0DE, MAKING MEME BUTTER!!!1!!111!!!
It’s ok alright Pythagorean theorem is a special case of Ptolemy
First thing I thought about was Disney's Donald in Mathmagic Land. I watched it this exact thing demonstrated in a cartoon!
Now I really want to know what that negative root means. Feels at a glance like a link between Ptolemy, inversion and complex numbers.
Beautiful
I remember figuring that out on my own and not understanding how that was the case.
Yes, I found that out by playing with shapes in Geogebra
I'm sure this series is going to end up with an arcane pentagram and summon the devil through the power of Ptolemy's ancient theorem
It's beautiful
2:00 wow, that's a lovely letter b.
The negative solution corresponds to a pentagram (whose diagonals form a pentagon), I think
I like the videos with cliff
hmm, looks like we can prove that φ (Golden ratio) and π are related, thus π can be truly expressed using φ and no longer be just approximated!
This reminds me of a video by 3blue1brown on the stability of phi and phi's little brother/-0.618 the other solution for this quadratic formula. He basically used one of the formulas of phi (I think its 1/(1+1/(1+1/(1+1/...))) to transform the number line so that we could see how the points converged to phi and "inverse phi" or -0.618... Interestingly, the transformation looks like a eclipse, not sure if its related to inversion or not...
I always wondered where that square root of 5 came from.
Looking at phi this way is so elegant somehow
Just watched the Conway Checkers proof and now this. Think it’s just a coincidence that both hold great value to the golden ratio as well as the number 5?
Thank you.
We get the \sqrt{2} for an inscribed square too? I.e. actually finally using Ptolemy's Pythagorian limit. Maybe obvious, but also a great number.
Fantastic and elegant!
Is there an equation of ratios for each shape? it looks like it goes to infinity, maybe can be represented by some sort of ln(x)
Very interesting!
neat. these numbers really exist out there somewhere
fantastic!
I'm sure the 5 in the golden ratio is related to the fact that there are 5 sides to the pentagon, but I'm not sure why
2 videos in one day?!?!
still think 72 36 72 is the best triangle, and it does play well with a pentagon
I love math
I can't help but feel like the (1-rt[5])/2 could have an interesting story that we're ignoring here.
Every number has interesting properties if we look hard enough, my friend.
as the equation is symmetric it's a matter of perspective; solving for a/b or for b/a (but in an inverted sense, thus the negative sign)
woooosh