This continues from our proof of Ptolemy's Theorem --- kzread.info/dash/bejne/lH6D19yqpZazhbQ.html

@jasethesmiff5683

4 жыл бұрын

Why do they not teach this in preschool!

@MS_950

4 жыл бұрын

Ptolemy sounds like it's the name of a disease.We shall use the real name: Ptolemaios.

@DRMath

4 жыл бұрын

Numberphile ✌️

@leif1075

4 жыл бұрын

Bradyhow did you guess that so quickly at the beginning that the two short lines equal the long line?

@bogdanelbogdan2600

3 жыл бұрын

So... does a = radius?

*any hard math problem* Prof. Zvezda: "I remember struggling to solve this problem at kinder garden"

@metallsnubben

4 жыл бұрын

Jeremias Figueiredo ”Back when I was a mortal”

@user-ot2uf8ne9s

2 жыл бұрын

hahahaha

@jassenjj

2 жыл бұрын

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.

@anoxie1301

Жыл бұрын

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.

@Triantalex

8 ай бұрын

false.

I laughed so hard when Brady reconfirmed that was not the factorial. He must have been so traumatised by numberphile

@PaulPaulPaulson

4 жыл бұрын

Now d is a unary higher order function with the factorial operator as the argument 😉

@AHBelt

4 жыл бұрын

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'.

@gabrielkellar1935

4 жыл бұрын

Its a parker exclamation

@MechaStorm7

4 жыл бұрын

@@gabrielkellar1935 underrated comment

@Triantalex

8 ай бұрын

false.

All Numberphile guests are wonderful, but Zvezda is my personal favorite. :)

@Triantalex

8 ай бұрын

false.

Everyone deserves a math teacher who is as passionate as Zvezda.

@Triantalex

8 ай бұрын

false.

1+1=2! Checkmate factorials

@justinjustin7224

4 жыл бұрын

1+0=0!

@DiegoMathemagician

4 жыл бұрын

1!+2!=3!!

@Peterwhy

4 жыл бұрын

1!=1 Checkmate programmers

@alexismiller2349

4 жыл бұрын

@@DiegoMathemagician The fact that you used the notation of double factorial without meaning twice factorial means that your brain is very big

@justinjustin7224

4 жыл бұрын

@@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.

fun fact: zvezda «звезда» means “star” in many slavic languages

@DerekRoss1958

Жыл бұрын

So, "Stella" in English.

She's a gifted teacher.

The sequel is even better than the first! Blown away.

@DRMath

4 жыл бұрын

Linda Ristevski ✌️

@lemonenjoyer6410

4 жыл бұрын

Is is first factorial or......

@Triantalex

8 ай бұрын

??

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 ? !

@marios1861

4 жыл бұрын

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.

@additionaddict5524

4 жыл бұрын

Well it was proved long (ancient Greece) before inversion transform / geometry came along in 1831.

@theshuman100

4 жыл бұрын

its fun inverting random shapes until you have to justify your algorithm to other people

@nahidhkurdi6740

4 жыл бұрын

For sure, inversion was not conceived for proving Ptolemy's Theorem.

@abdullahal-ahmati5030

4 жыл бұрын

Inversions are really just 1/x but in 2 dimensions.

“ oh look an homogeneous equation I can do miracles with that” I’ve laughed so hard😂😂😂😂

@Triantalex

8 ай бұрын

??

What an ingenious proof, the stark elegance of it all is just mind blowing

@DRMath

4 жыл бұрын

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)

@washizukanorico

4 жыл бұрын

Smart, I solve X^2-X-1 every time ...

@sergey1519

4 жыл бұрын

i remember it as (1+sqrt(5))/2...

@SocksWithSandals

4 жыл бұрын

I like 1.62, as it is almost exactly the ratio of kilometres to miles. And miles to kilometres is about 0.62.

@susmitamohapatra9293

4 жыл бұрын

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

@jaroslavsevcik3421

4 жыл бұрын

@@susmitamohapatra9293 So the Golden ratio is an irrational number that can be computed?!

Somehow, "!" Is unintendedly one of the highlights.

@steffen5121

4 жыл бұрын

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.

Theorem: every Bulgarian math teacher has this accent in any language they speak.

@00bean00

4 жыл бұрын

I had a Bulgarian math professor/instructor (Kumchev) and he had an interesting accent.

@TheSimplesAreFree

4 жыл бұрын

Why did I read your comment in an accent?

@kuretaxyz

4 жыл бұрын

@@TheSimplesAreFree Maybe you are a Bulgarian math teacher?

@daddymuggle

4 жыл бұрын

It's only a theorem if it can be proven.

@jaroslavsevcik3421

4 жыл бұрын

@@daddymuggle It is only theorem until it is proven.

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?

@Craznar

4 жыл бұрын

Not really a famous ratio if it isn't even called the ratio :)

@alephnull4044

4 жыл бұрын

lol fair point

@vitalspark6288

4 жыл бұрын

@@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.

@linusandersen5608

4 жыл бұрын

@@vitalspark6288 lol fair point

@newkid9807

4 жыл бұрын

Vital Spark 911, it’s Christopher Burke he’s been killed

The way she draws b really makes you see how it evolved from B

@miorioff

4 жыл бұрын

It's actually because of the russian "B" written small as "в". Old habits of hers :)

@LakeReeder

4 жыл бұрын

Yes, it's в in cursive. Also it would be actually pronounced like v.

@justincronkright5025

4 жыл бұрын

Wrong word... progressed, not evolved. I know this is maths here, but still. We're all speaking English - more or less, here anyway.

@justincronkright5025

4 жыл бұрын

@@miorioff That's Cyrillic not Russian.

@miorioff

4 жыл бұрын

@@justincronkright5025 sure, whatever it is

"The Bermuda triangle?" lol :-)

@GeorgePlaten

4 жыл бұрын

Brady's brain is fried at this stage!

@tylisirn

4 жыл бұрын

He wasn't far wrong though. The Bermuda Triangle has sides 1669 km, 1663 km and 1545 km. It is almost an equilateral triangle.

@silkwesir1444

4 жыл бұрын

@@tylisirn there are many different definitions of the Bermuda Triangle

@gcewing

4 жыл бұрын

So it's a Parker equilateral triangle?

@tylisirn

4 жыл бұрын

@@silkwesir1444 I used the one they showed on the video which is one of the most common. Vertices at Bermuda - Miami - San Juan

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!

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.

And one can use the Pentagon relationship to show that cos(36 degrees) = golden ratio /2, and cos(72 degrees) = (golden ratio - 1)/2

@polettix

4 жыл бұрын

Trigonometry... ewwww! ;-)

@tramquangpho

4 жыл бұрын

.

@patriknovosad3113

4 жыл бұрын

And you can continue with this to show that sin(666°) = - golden ratio/2 ;)

I love Zvezda's smile.

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 is insane! We have to inform the Pentagon! Oh wait, they already know.

@alveolate

4 жыл бұрын

they about to use phi to justify another 700bn raise.

@voorth

4 жыл бұрын

@@alveolate Semper phi ?

@survivordave

2 жыл бұрын

This is the type of joke I live for

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!

This might be my favourite Numberphile video ever

Yeah Zvezda, I also remember struggling with this problem in kindergarten.

These last two videos have been brilliant.

Wow, I could listen to Zvezda explain math things all day.

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...

@letao12

4 жыл бұрын

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...

Since we've worked so hard to prove it 😂. My favourite teacher on Numberphile!!!

Greetings from Bulgaria! 🇧🇬

That is beautiful; it makes me love math even more. Live == learn. Thank you for the video: you made my day!

I've seen this numerous times. I can't believe I've just liked it now! It's a laugh but a great lesson.

Incredibly elegant

thank you for answering my question AND showing the math!!

I could genuinely listen to Professor Stankova’s accent all day.

The beauty of mathematics properly explained

Love it! More, more!

This is pretty cool, time to add this to my problem-solving arsenal

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

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

Zvezda is such a Star!

Thank you Prof. ♦️

In Euclid's Elements there is a compass and ruler construction of the Golden Ratio and thence a regular pentagon.

Ptolemy is my new favorite mathematician.

Amazing, I miss the circle making part.

Love your voice. My French teacher 2.0.

Finally you have talked about phi

Very interesting!

Numberphile is now at Pi million subscribers in the world of Indianna.

Mum: do you want some pizza? Me: 1:20 Great video, as always

Beautiful

Was nice with a meaty video (pt 1+2) :-)

She's a very lovely and fantastic teacher. Can we get more videos?

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.

It's beautiful

fantastic!

When you do too much math you start seeing exclamation point as factorial d!

I literally solved a family of problems without knowing Ptolemy's theorem. 💀

Thank you.

Awesome Leibniz portrait in the background 6:28

Another reason to love the golden ratio.

@fogsmash6914

4 жыл бұрын

may i introduce you to jojo part 7

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.

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.

Sh'es actually a gorgeous mathematician

Make a video on the burning ship fractal? Would be interested to learn how it works.

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 like the videos with cliff

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.

Bermuda pentagon

A regular distribution curve is Pentagon. E equal mc2 follow m central and line 🕸.

Nice!

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

Hahaha :D really enjoyed the proof. Reminds me of my young myself doing geometry at school :D

It looks like NUMBER OF THE BEAST at diagonals on pentagon!!! ... and the pentagon is mystical figure )

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)

Now I really want to know what that negative root means. Feels at a glance like a link between Ptolemy, inversion and complex numbers.

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

2:00 wow, that's a lovely letter b.

OK, what shapes would one have to do this with to get the other famous ratios. like the silver ratio?

Quite plainly, if Zvezdelina ever offers a course in Bulgarian middle school geometry, I'll be first in line.

I didn't know Yennefer was so well-versed in math too. Respect.

@flowerwithamachinegun2692

4 жыл бұрын

Wind's howling

You know, a simplified version of this is in my book about typography and in particular talking about the 8 1/2 x 11 sheet of paper Or does it? I need to have another look at that chapter.

Oooow... memories to over 25 years ago...

I have a Bulgarian colleague at work, he is just as mad as Professor Zvezda, he cracks me up xD

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.

@ca-ke9493

4 жыл бұрын

Looking at phi this way is so elegant somehow

The negative solution corresponds to a pentagram (whose diagonals form a pentagon), I think

Nice

2 videos in one day?!?!

I think that a non-euclidian triangle with three 90 degree angles is more spectacular to consider.

How can you tell if a curve is logarithmic/exponential, or a golden ratio curve (like the golden spiral), or an algebraic expression mapped onto a 3D shape (like a sphere or something like it) that creates curve, or if a curve is just a product of an infinite series (that may even have no discernable pattern to it)? For example y = e^x scaled so that the x and y axis are invtan of their values (e.g. 1 becomes 45 degrees) approximates y = x/2 + pi/4 (rads), which itself mapped onto a sphere forms almost a golden spiral. For example, which curve could be repeated in a neat dodecahedron pattern say where all the curves either touched at tangents or crossed at near right-angles or intersected at regular points? - I once found something like this but had no idea what the expression for the curves were? And the pattern appeared to form neatly even if smaller versions (like a russian doll) were placed inside of it. Was way too complex for me to figure out.

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

Extend to regular polygons with more sides?

My internal dialogue... I know I'm getting smarter watching these videos, so why do I feel more stupider?

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.

Boom!! 🤯

First thing I thought about was Disney's Donald in Mathmagic Land. I watched it this exact thing demonstrated in a cartoon!

This video is not visible through the numberphile videos list, I'm having troubles looking for old extra videos in some of your channels, I think they are lost, like this one. cuz I'm pretty sure I saw them before, but now I can't find them anywhere I love Zvezda's explanations and personality.

@ideallyyours

4 жыл бұрын

Unlisted. Some of these videos are linked to in descriptions of other Numberphile videos.

@b3z3jm3nny

4 жыл бұрын

Since this is a part two, it’s unlisted for a bit so people see the part one first.

@Barzx

4 жыл бұрын

@@ideallyyours its unfair for other platforms that doesn't offer views for comments and so

@Barzx

4 жыл бұрын

btw, It is listed now, so thanks @Numberphile