Straight Lines in Curved Space explained and visualized. Useful for the four dimensional space-time of Einstein’s General Relativity. My Patreon page is at / eugenek
To see subtitles in other languages: Click on the gear symbol under the video, then click on "subtitles." Then select the language (You may need to scroll up and down to see all the languages available). --To change subtitle appearance: Scroll to the top of the language selection window and click "options." In the options window you can, for example, choose a different font color and background color, and set the "background opacity" to 100% to help make the subtitles more readable. --To turn the subtitles "on" or "off" altogether: Click the "CC" button under the video. --If you believe that the translation in the subtitles can be improved, please send me an email.
I hate the music with explanations in all your videos It is distrubing me making me nervous The musuc is loud and pass over the voice of speaker Excuse me but it is realy sh*t your videos are for sciences not videogames Do you want to send any subliminal's message with this music?!
I'm a physics student and I think you're doing an amazing job. This channel encourages me to learn animation and use them to explain physics just like you. Thank you!
You can help translate this video by adding subtitles in other languages. To add a translation, click on the following link: ua-cam.com/users/timedtext_video?ref=share&v=NfqrCdAjiks You will then be able to add translations for all the subtitles. You will also be able to provide a translation for the title of the video. Please remember to hit the submit buttons for both the title and for the subtitles, as they are submitted separately. Details about adding translations is available at support.google.com/youtube/answer/6054623?hl=en Thanks.
Amazing presentation. The accuracy of the graphics and narration kinda split at the end, which is understandable 😇 it’s also easier to learn when our minds start to fill in the blanks.
@@canyadigit6274 Motion is relative... From the perspective of the fallen object, the entire Earth is rushing up and hitting it and from the perspective of the Earth, the falling object is moving towards the ground and hitting the Earth... No reference frame is more true; than the other. Let me ask you this: Say you are inside a rotating room (an empty room with no windows or doors that just rotates around its central axes) Now inside that room you find a tennis ball. You pick it up and you throw it to the opposite wall, what would happen? Will the tennis ball go straight and hit the wall in front of you (since you threw it straight) or will it take a 90 degrees curved path to the right and hit the wall to your right? (or left depending on how the room is turning...) The answer is the latter. The tennis ball will take a curved trajectory and it will not go straight. And you might ask: "why does the tennis ball go to the right when I'm throwing it straight?" That's of course a bad question, because the tennis ball IS NOT actually taking a curved path, its YOU inside the room that is, that see the tennis ball take that curved path! The tennis ball was always going straight, it's only your own motion that made it look like it was curving. Like here: ua-cam.com/video/bJ_seXo-Enc/v-deo.html Now imagine the same idea but with "gravity" and in 4 intrinsically curved dimensions of spacetime and that motion that you have (the one that is equivalent with the motion of the room spinning) is in TIME and not in space! Imagine all of that and you have Einstein's picture of "gravity" and you also have the answer to why Einstein considered this his greatest thought: "A man in freefall will not feel his own weight" No, objects in free fall are not falling, they are not moving in space because there are no forces acting on them at all. The reason why we *see* them fall, is because we exist inside a curved spacetime but we "don't know it", our senses "don't know it" and they in fact interpret our curved minkowski spacetime as a non curved euclidean spacetime. The only way for a human to see reality as it really is, is to go somewhere, where spacetime isn't curved. And we have such a place, it's called "outer space" away from any Star or other Planet sized massive object. There spacetime has a very low curvature, and there, when you let something fall you can clearly see that is not moving in space. Here on Earth is still not moving in space, but our senses by interpreting this world as an euclidean one, (and not as a mikowsky curved one as they should) see the object fall when in fact it's not falling. ua-cam.com/video/QyeF-_QPSbk/v-deo.html :)
The best geodesic explanation I've seen is with paper strips. Orient the strip so that neither side folds or leaves the surface - then you have a geodesic. Putting a straight line around a cone won't work, the side of the strip towards the narrower cone end will not make contact with its surface. It's effectively the same idea as the car wheels, but easier to simulate hands-on!
The car analogy attached on the surface gave it away for me! The wheels rotating with the same speed, cover different distances due to the curvature of the surface and thus the car turns accordingly but always moving fwd thus covering the least distance!
@@wilcovdberg You are right however that if you model gravity as a 'pit' in a surface, less velocity in a direction that escapes it will cause an object to fall earlier
"Straight" is an _ad hoc_ qualification that you have to specify when doing differential geometry; e.g., if you're doing the ordinary Euclidian geometry, two "straight" lines are "parallel" if they conform to Euclid's fifth postulate, and vice-versa...
"Special considerations must be taken when applying this concept to the higher fourth dimensional space-time of general relativity and this will discussed later in the video" *is only a 10 minute video* It's gonna be a good one.
6:50 "The geodesics that are swallowed continue to spiral downwards" that's only if there is no bottom to that hole. If there's a bottom at any point, the geodesics will return. So, if we ever see light emerge from a black hole, could we use that to calculate the spacetime structure?
Something similar to this could be a pulsar which is an object that collapses but doesn't have enough mass to create a black hole, spitting it's matter out in polarized jets and spinning hundreds of times a second, but are 20km across and almost the density of a black hole. Check out the NASA mission IXPE cause we'll get some insane data we've never gotten to explore before when it launches in about a year.
But scientist says the information that gets into blackhole will be destroyed, so if you fire a beam light in to a blackhole maybe when it exits the blackhole it will be in another form or something...Idk
Your videos are fantastic. The visuals, the narration, the music. True works of educational art by masters of their craft. Well worth waiting for. Thank you.
This channel is really great! The explanation is really dynamic with the use of animation and calm explanation....Thank you for the good work and effort!
The Dark Knight when the potential is zero: the only force present is the force normal to the surface: this doesn’t affect the Kinetic Energy hence the Kinetic energy remains constant: hence the speed remains constant
How do you create these animations? They are simply stunning. Imagine if we had these showing in every school, and if all physics was taught in this way. You are creating something powerful, hats off to your efforts!
Last time I was this early people thought time was absolute! I love the title of this video, it shows you don't only appreciate math as a tool but also as something that can be aesthetic and fascinating
I've tried for months to compute the geodesics on a general ellipsoide (x/a)^2 + (y/b)^2 + (z/c)^2 =1 with a b c all different between 2 arbitrary generic points. Your animation shows the geodesic to wrap multiple times around. I had not considered that.
Gravitational time dilation tells you that distant frames in a gravitational field will begin to appear to move relative to you (they will accelerate relative to you) because rather than moving entirely in the time direction through spacetime, they will lose some of their 4-velocity in the time direction, relative to you, and transfer that 4-velocity to a spatial direction, this all appearing to you as a gravitational force pulling these object through space, and slowing them down in time.
Since both happen simultaneously, it is not entirely accurate to say that one causes the other. It might be more accurate to say that gravitational attraction CAN be explained using time dilation. It may be possible to do the reverse, but we do not know of any such way. So in the models that we have built, it appears that time dilation is more fundamental. But, there might be other viable theories (which we don't know of) that could explain time dilation as a consequence of gravity. Or maybe, both are emergent from something else entirely.
0:37 Oh yes it is only deflected by a small amount I totally don’t see two angles opposite of each other Ps 5:54 The reason why they end up converging and diverging because one is positively curve and one is negatively curved.
Your videos are so calming. The music fits perfectly with the amazing animations. And your voice is also calming. You'd think that would make the videos less interesting, but it really doesn't. It makes them a good way to get in a good mood while learning about math and physics.
6:16 I think that sheet can imagined as a distortion in Space created by black hole and that lines as a light And I think this explain why light cannot escape from black hole but some of them can escape after spinning around it ?? *IDK I am right or wrong ?*
Imagine if you had to observe physics from a two dimensional plane. I feel like that's what's going on with black holes from our dimensional perspective. There's a direction we just don't understand 🤷♂️
Another FANTASTIC video! What blows my mind is the animation starting at 3:45 - by the end, all the geodesics looks remarkably like the valence electron shell of an atom! Co-incidence?......
Eugene Khutoryansky, we want to know more about you. Please give some information about where you seek your knowledge! Are you a professor? Do you own a degree? What do you do for a living? Very curious fan here just wondering
Gustovo, I would prefer not to go into personal details about myself on my UA-cam channel, at least not yet. Though, it is not much of a secret, if you Google my name...
![Visualizing 4D Geometry - A Journey Into the 4th Dimension [Part 2]](/img/n.gif)
To see subtitles in other languages: Click on the gear symbol under the video, then click on "subtitles." Then select the language (You may need to scroll up and down to see all the languages available).
--To change subtitle appearance: Scroll to the top of the language selection window and click "options." In the options window you can, for example, choose a different font color and background color, and set the "background opacity" to 100% to help make the subtitles more readable.
--To turn the subtitles "on" or "off" altogether: Click the "CC" button under the video.
--If you believe that the translation in the subtitles can be improved, please send me an email.
I hate the music with explanations in all your videos
It is distrubing me making me nervous
The musuc is loud and pass over the voice of speaker
Excuse me but it is realy sh*t
your videos are for sciences not videogames
Do you want to send any subliminal's message with this music?!
But does the little car have to be a dymaxion?
Αεί Θεός Γεωμετρεί - Πλάτων
6:20 cheerful music plays as innocent geodesics spiral towards their doom
Never to be heard from again!!
I really got goosebumps
That reminded me to the End of Evangelion 👏
A Geodeasid
Me: Don’t do it
Geodesic: I will do it it’s out there in the center?
Die: because this is a 2d black hole
Hmm
1:04 me when i see Eugene posted a new video
Same happens to me! LOL
lmaooo
uhhh that umm ok...next comment
Lol, true
#ohgodwhy
I'm a physics student and I think you're doing an amazing job. This channel encourages me to learn animation and use them to explain physics just like you. Thank you!
Thanks.
Learn blender. Free 3d modeling, animation and etc software. It's amazing
What programs would be good for learning how to animate?
@@dingdong475 blender it's free. I repeat myself
@@imdone8243 yes but i thought maybe something new may have come up since you last posted.
You can help translate this video by adding subtitles in other languages. To add a translation, click on the following link:
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You will then be able to add translations for all the subtitles. You will also be able to provide a translation for the title of the video. Please remember to hit the submit buttons for both the title and for the subtitles, as they are submitted separately.
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Thanks.
Nice
Tell us more about phasor diagram.
I can make arbic subtitles , just need time to do it
Thank you for the great vídeos!
Amazing presentation. The accuracy of the graphics and narration kinda split at the end, which is understandable 😇 it’s also easier to learn when our minds start to fill in the blanks.
We will never forget you, brown geodesic spiraling to -infinity 😭
Change your perspective and you will see it tending to positive infinity ;)
@@darkseid856 U are mr gay
Was that suppose to be a metaphor for...
@@pointlesslylukesplainingpo1200 super mario galaxy?
9:29
Newton was the genius that explained why things fall
And Einstein the genius that explained why they don't!
You sir, have won the internet
@@feynstein1004 I'll take overly used statements for 200$
NUKE things still do fall, the REASON why is different
@@-_Nuke_- I hope you didn't type all that out...
@@canyadigit6274 Motion is relative...
From the perspective of the fallen object, the entire Earth is rushing up and hitting it and from the perspective of the Earth, the falling object is moving towards the ground and hitting the Earth... No reference frame is more true; than the other.
Let me ask you this:
Say you are inside a rotating room (an empty room with no windows or doors that just rotates around its central axes)
Now inside that room you find a tennis ball. You pick it up and you throw it to the opposite wall, what would happen?
Will the tennis ball go straight and hit the wall in front of you (since you threw it straight) or will it take a 90 degrees curved path to the right and hit the wall to your right? (or left depending on how the room is turning...)
The answer is the latter. The tennis ball will take a curved trajectory and it will not go straight.
And you might ask: "why does the tennis ball go to the right when I'm throwing it straight?"
That's of course a bad question, because the tennis ball IS NOT actually taking a curved path, its YOU inside the room that is, that see the tennis ball take that curved path!
The tennis ball was always going straight, it's only your own motion that made it look like it was curving.
Like here:
ua-cam.com/video/bJ_seXo-Enc/v-deo.html
Now imagine the same idea but with "gravity" and in 4 intrinsically curved dimensions of spacetime and that motion that you have (the one that is equivalent with the motion of the room spinning) is in TIME and not in space!
Imagine all of that and you have Einstein's picture of "gravity" and you also have the answer to why Einstein considered this his greatest thought:
"A man in freefall will not feel his own weight"
No, objects in free fall are not falling, they are not moving in space because there are no forces acting on them at all.
The reason why we *see* them fall, is because we exist inside a curved spacetime but we "don't know it", our senses "don't know it" and they in fact interpret our curved minkowski spacetime as a non curved euclidean spacetime.
The only way for a human to see reality as it really is, is to go somewhere, where spacetime isn't curved. And we have such a place, it's called "outer space" away from any Star or other Planet sized massive object.
There spacetime has a very low curvature, and there, when you let something fall you can clearly see that is not moving in space.
Here on Earth is still not moving in space, but our senses by interpreting this world as an euclidean one, (and not as a mikowsky curved one as they should) see the object fall when in fact it's not falling.
ua-cam.com/video/QyeF-_QPSbk/v-deo.html
:)
The best geodesic explanation I've seen is with paper strips. Orient the strip so that neither side folds or leaves the surface - then you have a geodesic.
Putting a straight line around a cone won't work, the side of the strip towards the narrower cone end will not make contact with its surface.
It's effectively the same idea as the car wheels, but easier to simulate hands-on!
The car analogy attached on the surface gave it away for me! The wheels rotating with the same speed, cover different distances due to the curvature of the surface and thus the car turns accordingly but always moving fwd thus covering the least distance!
thanosAIAS I read a book that introduced this topic with an analogy of ants on an apple, but the car with the wheels worked better for me too
@@wilcovdberg In geodesics the speed of a car doesn't matter, it's just drawing lines over a surface.
@@wilcovdberg You are right however that if you model gravity as a 'pit' in a surface, less velocity in a direction that escapes it will cause an object to fall earlier
Me after watching this video: what is straight????
Great now youre gay because of mathemathics
Now all straight lines will rotate infinitely.
"Straight" is an _ad hoc_ qualification that you have to specify when doing differential geometry; e.g., if you're doing the ordinary Euclidian geometry, two "straight" lines are "parallel" if they conform to Euclid's fifth postulate, and vice-versa...
@@PhillipAmthor Nah you may look gay but you are actually straight. This is what geodesics tell u, BRIGHTSIDE.
Yui
I still like this even though I don't understand a single thing.
Cua Manh Dong....Yep. Same here. Probably have to watch it a few times.
@Mondo Duke it's hard when u don't know english and there's not subs in your lenguage ;P
Must be a smart 6 year old. Let’s try and not shame people for learning at different speeds, or having a less comprehensive background
@@ASOUE you are the man
@Maselek wha?
I love this channel ❤️ !!!
Finally we get something after a long time !!
The presentation and animation is beautiful.
Thanks for the compliment.
Ikr
Wow. Just wow. This was something I always wanted to visualize. Thank you so much. I love all of your videos.
Thanks. I am glad you enjoy my videos.
"Special considerations must be taken when applying this concept to the higher fourth dimensional space-time of general relativity and this will discussed later in the video"
*is only a 10 minute video*
It's gonna be a good one.
thank you kira and eugene, i love this material and I love the music and animations. I hope you still make videos for years to come
Thanks.
You are doing an amazing job @Eugene! Thanks for making these wonderful videos!
6:50 "The geodesics that are swallowed continue to spiral downwards" that's only if there is no bottom to that hole. If there's a bottom at any point, the geodesics will return. So, if we ever see light emerge from a black hole, could we use that to calculate the spacetime structure?
Or the lines can be in other plane he dont show, parallel line... universes
Something similar to this could be a pulsar which is an object that collapses but doesn't have enough mass to create a black hole, spitting it's matter out in polarized jets and spinning hundreds of times a second, but are 20km across and almost the density of a black hole. Check out the NASA mission IXPE cause we'll get some insane data we've never gotten to explore before when it launches in about a year.
@@hobbyhobbyhobbyhobby haha neutron star go brrrr
But scientist says the information that gets into blackhole will be destroyed, so if you fire a beam light in to a blackhole maybe when it exits the blackhole it will be in another form or something...Idk
FILE NAME Scientists say that information potentially gets observationally copied/xeroxed, rather than actually destroyed.
Interesting learning style, repeating things twice, not too quickly, and in the exact same way. Is there a name for that?
Also known as the deja-vu method. I agree it draws you in.
@@clearz3600 😂
Interesting learning style, repeating things twice, not too quickly, and in the exact same way. Is there a name for that?
@@numspacsym ha ha
@@sciencemathematics I found that very funny 😂
Your videos are fantastic. The visuals, the narration, the music. True works of educational art by masters of their craft. Well worth waiting for.
Thank you.
Thanks for the compliments about my videos.
@@EugeneKhutoryansky You've earned it.
So nice . Please keep making these videos.
More videos are on their way. Thanks.
I 💖 this channel.
The presentation is so effortless 🌸
Thanks.
*It's 3 p.m.*
UA-cam Recommended: *IT'S A GEODESIC!*
3pm or am?
It's 3am for me
Mind-blowing, I also love your background music as well.
This channel is really great! The explanation is really dynamic with the use of animation and calm explanation....Thank you for the good work and effort!
Thanks. Glad you liked my video.
me *having an exam on kinematics*
also me *writing about geodesics in that exam*
Finding a geodesic is equivalent to finding a path where the potential is zero
@@cameronspalding9792 but if it's a *path* then shouldn't it be POTENTIAL DIFFERENCE ? Or am I wrong somewhere ?
The Dark Knight when the potential is zero: the only force present is the force normal to the surface: this doesn’t affect the Kinetic Energy hence the Kinetic energy remains constant: hence the speed remains constant
@ ol
How do you create these animations? They are simply stunning. Imagine if we had these showing in every school, and if all physics was taught in this way.
You are creating something powerful, hats off to your efforts!
I make my 3D animations with "Poser." Though, I had to write my own algorithm for calculating the paths of the Geodesics. Thanks.
Amazing, very beautiful! Thanks for another great video, Eugene and Kira!
Thanks for the compliment.
Absolutely amazing. Thanks for your hard work on this channel!!
Thanks.
Last time I was this early people thought time was absolute! I love the title of this video, it shows you don't only appreciate math as a tool but also as something that can be aesthetic and fascinating
Welcome back!!!
Eugene thank you for another wonderful and informative vid. Boy how I wish you were around with these vids back when I was in high school!
Thanks.
You are doing a great job in explaining the physical meaning of concepts in physics
Thank you so much for it !
Thanks.
As a geometer I'm totally biased towards differential geometry, but I still have to say this was your best video ^^
8:21 The car's Vehicle registration plate! :DD
Your channel is SO AWESOME.
Thanks.
9:00: The tea level of my teapot after I spill the tea in astonishment after hearing the beautiful music in one of your videos...
I've tried for months to compute the geodesics on a general ellipsoide (x/a)^2 + (y/b)^2 + (z/c)^2 =1 with a b c all different between 2 arbitrary generic points. Your animation shows the geodesic to wrap multiple times around. I had not considered that.
Nerd
6:20
That's how a black hole "sucks" in light.
Great work Eugene! Thank youso much! Keep it up!
Glad you liked my video. Thanks.
Wow. Just wow. I had a hard time understanding this way of thinking. Now I do. Thank you.
Glad my video was helpful. Thanks.
6:17 can we get an F for those fallen geodesics
Pouring one out for the boys, the fallen geodesics
ha, you thought i was gonna reply with "F"? well i like to make people wrong.
heh.
F
Certainly you have the rare gift to help visualize some of the most complex theories of modern physics in an easy way. Thanks Eugene, thanks Kira.
Thanks for the compliment.
You always had the best explanatory videos ❤️
Thanks for the compliment.
Helped make sense of my differential geometry tutorial, thanks!
"Vegan for Animals"
beauty of ethics and morality
Go vegan
😂😂😂😂😂
8:18
"Meat-eater for Plants"
"Gravitational Time Dilation casues Gravitational Attraction."
Is it one causes the other, or is there some sort of duality?
Former is a fundamental fact. Latter is its consequence, and basically an "illusion".
Gravitational time dilation tells you that distant frames in a gravitational field will begin to appear to move relative to you (they will accelerate relative to you) because rather than moving entirely in the time direction through spacetime, they will lose some of their 4-velocity in the time direction, relative to you, and transfer that 4-velocity to a spatial direction, this all appearing to you as a gravitational force pulling these object through space, and slowing them down in time.
Since both happen simultaneously, it is not entirely accurate to say that one causes the other. It might be more accurate to say that gravitational attraction CAN be explained using time dilation. It may be possible to do the reverse, but we do not know of any such way. So in the models that we have built, it appears that time dilation is more fundamental. But, there might be other viable theories (which we don't know of) that could explain time dilation as a consequence of gravity. Or maybe, both are emergent from something else entirely.
@@Mr.Not_Sure so it means time dilation is an effect ? Sorry but I didn't really understand your comment ....
@@darkseid856 Watch this short video as a first step: ua-cam.com/video/gcvq1DAM-DE/v-deo.html
Great explanation as always. Keep up the good work!
Thanks.
Thank you for these videos Eugene.☺👍
Thanks. I am glad you like my videos.
0:37
Oh yes it is only deflected by a small amount
I totally don’t see two angles opposite of each other
Ps 5:54 The reason why they end up converging and diverging because one is positively curve and one is negatively curved.
i've convinced myself that i must be gay because i don't know what's straight and what's not anymore
lmaoooo
Your videos are so calming. The music fits perfectly with the amazing animations. And your voice is also calming. You'd think that would make the videos less interesting, but it really doesn't. It makes them a good way to get in a good mood while learning about math and physics.
This video really made the concept click. Thank you!
Thanks.
I actually misread the title as "the beauty of genocides"
*turkey and Armenia would like to know your location*
Wait is this UA-cam or Bitchute?
Me too 🤣
😂🤣
o.O
This town is a part of us all. A part of us all. A part of us all!
Stranger things?
I find this absolutely fascinating.
Wow! Beautiful work
Thanks for the compliment.
*why am I into this stuff-*
Noone:
My UA-cam reccomendations: wanna learn some physics?
Well, of course I do.
Awesome video! Thank you!
I am glad you liked my video. Thanks.
Einstein would feel proud watching this production. Great music. Great content.
vegan for animals is on the license plate of the car lol 8:19
6:16
I think that sheet can imagined as a distortion in Space created by black hole and that lines as a light
And I think this explain why light cannot escape from black hole but some of them can escape after spinning around it ??
*IDK I am right or wrong ?*
I though the same thing
Eh sure. Also as Earth and etc..
We have hope of survival just be green
@@abdullahx8118 eww no. I wanna be blue
Imagine if you had to observe physics from a two dimensional plane. I feel like that's what's going on with black holes from our dimensional perspective. There's a direction we just don't understand 🤷♂️
This is amazing thank you eugene
Thanks.
This video helped me adjust my understanding of geodesics. Without saying what I had wrong, I had them wrong. Thank you.
I am glad my video was helpful. Thanks.
I'd love to know how this applies to lenses and photons
Watch code parades videos on it, its astounding
The beauty of “vegan for animals”. lol
Car’s license plate on the back.
Another FANTASTIC video! What blows my mind is the animation starting at 3:45 - by the end, all the geodesics looks remarkably like the valence electron shell of an atom! Co-incidence?......
great man you give me the vibes !
6:17 this actually reminds me of Junji Ito’s Uzumaki manga. The whole plot of the manga revolves around this concept, especially around the ending.
I dont get it
Eugene Khutoryansky, we want to know more about you. Please give some information about where you seek your knowledge!
Are you a professor? Do you own a degree? What do you do for a living?
Very curious fan here just wondering
I think he got his masters/PhD in the 80s
he is professor from Texas. The narrator is Kira. He is inspiration.
Shirshak, actually I am not a professor. But I do live in Houston, Texas. And yes, Kira is the Narrator.
Wang Kevin, I am not that old. I was born in 1975.
Gustovo, I would prefer not to go into personal details about myself on my UA-cam channel, at least not yet. Though, it is not much of a secret, if you Google my name...
I really love these videos.
Glad to hear that. Thanks.
really great stuff as always
Thanks.
imagine the geodesics on a klein bottle
Black holes described around 7 minutes, without realizing it
also we can escape black holes lol
Another great vid.
Thanks.
Amazing animation
I have no words to appreciate uh
Thanks for the video
Thanks for the compliments.
phew, thought it read “Beauty of Genocides”. i’m gonna go now.
You clicked on it though
Saya meow...
.
22/10/20..
Hi meow
thank you so much for the video! really really appreciate it
Thanks. Glad you liked my video.
Thanks for your work!
Thanks.
*"Vegan for animals"*
Incredible! Wonderful!
Thanks.
Nice video one of the best UA-cam channel for me
Thanks. I am glad you like my videos.
@@EugeneKhutoryansky thanks ,yes I got a reply, it made my day 😊
Thank you, this explained it rather well!
Thanks. I am glad you liked my explanation.
Beautifully displayed.....
Thanks for the compliment.
Very informative video. I think I learned so much, I don't even remember what this video taught.
Absolutely stunning and enjoyable :-)
Thanks. Glad you liked it.
Very happy I subscribed!
I am happy you subscribed too.
New great video!
Thanks. Glad you liked it.
Great eugene, thanks .
Thanks.
Wonderful
I ve been watching since I was in higher secondary school.
Best intro for geodesics ever! Eliminating our misconceptions before they arise!
Thanks for the compliment about my video.
Which software used for 3d animation ? Please some one help me
I make my 3D animations with "Poser", but I had to write my own program to calculate the paths of the Geodesics.
thank you for nice work!
Thanks.
I loved how the car number plate contained a powerful message!!
9:10 I REALLY wish I would've known this before. Thank you!
Thanks for the video!
Thanks.
Never heard an english that clear and understandable, great job
Wonderful beautiful graphics.
Thanks for the compliment.
@@EugeneKhutoryansky love what you do really makes it easier for me get a perspective, as I am more of a visual thinker. Cheers
Very good explanation, thanks you
Thanks.