when i was younger, every time i tried to get some bit of projective geometry, it was just a mysterious mess to me. well, maybe, i could remember something, do formulas, make some constructions, but the essence was still unaccessible. now, when some maths intuition settled in my head, i am starting to rediscover a lot of mathematical concepts, so videos like this are very useful not for getting some rigid knowledge (you may always read a textbook or a Wiki page for this purpose) but learn some motivation of a concept, get the intuition, highlight the connections with other math stuff and well, really understand what's going on. so i'd like to thank a lot the author for the video, it really uncovered some bits of information needed to order my previous knowledge. thank you and good luck in developing the channel!
Very nice! The cross-ratio is one of my favorite less well-known topics. With the Mobius transformation view, we can see that the cross-ratio of four complex numbers is real if and only if they lie on the same line or circle.
Really nice video. I've just made my final bachelors degree project about projective geometry and used de cross-ratio (which i knew as double ratio) to make an euclidean clasification of the conics.
I'm very sorry! Other people complained about the music, too. Next time, I will definitely let people watch the video beforehand. But I'm happy to hear that you lied the rest of it!
@@sumandproduct I love the music, tactical fading in and out like 3blue1brown would be more effective as you build the story of this amazing projection topic. I have been very curious about this subject with blender and fspy for a couple of years and you satisfied my curiosity. Now please do go into the 10 minute long explanation about why infinity works as 1! Btw I worked at an architecture company that used to leverage these concept with CAD to draw projective lines and create view transformations manually from other 2D horizontal cad views in the larger model space that contains multiple views by utilizing Construction lines which are infinite... Deep world of line projection is fascinating and useful in transformation of images. Linear algebra has a great way of doing these transformations but this video's content doesn't really require anything beyond what is taught in elementary school, which makes it so beautiful for the math community.
@@pa.l.2499 Thank you very much! I think you can get the most out of these projective transformations when you mix the linear algebra approach and this pen-and-paper approach. But for the video I tried to stay in the latter realm completely, as it makes it more approachable, as you mentioned.
@@sumandproduct I concur that in the future you have to set the music at much lower level, or actually not use music. Occasionally you swallow consonants, such as around 1:10 where your script says 'square length' but it ends up sounding as 'skwele'. Between that and your german accent: the viewer neets a couple of brain cycles more to get to the words you are saying. Yeah, there is a trade-off there. Pronouncing everything would sound ponderous (Related subject: video by Tom Scott : th-cam.com/video/qu4zyRqILYM/w-d-xo.html ) In the case of 3blue1brown I see the music used as part of the brand; the narration is soothing, the music is soothing. For minutephysics too I regard the music used not as part of the video as such, but as part of the brand. Personally I tend to prefer minimalism.
Great Video! The way of looking at the harmonic construction as a projective parallelogram was really nice, I wish we would have been shown that in our geometry lecture!
Thank you very much! I think, this is the third way to look at the harmonic point construction I worked with over the years. Probably my favourite so far.
I'm really happy you decided to share this video as it was very clearly presented, well animated and well paced. Love the subject and how you made it easy to follow your train of thought by using (wonderful) animations. Great job!!
When I did a perspective drawing in high school, I tried to make a street with many (a dozen or so) buildings of equal width. I used the Fibonacci sequence to measure out their respective widths and it looked good enough. I now wonder how it aligned to this reality of math.
I was following along with the progress on this video on the SoME2 discord! I'm so glad you were able to get it out before the deadline. Awesome stuff :).
I recall from a drafting & architecture class in high school that the formula for measuring along the axis in perspective was something simple involving the cosine function. I suppose that was an approximation. When I took computer graphics in college, I recall the extra dimension allowing perspective transformations, and that moving a parameter changed the perspective; but I don't recall any formula for how that relates to the vanishing point.
I can't be sure, but I don't recall any formulas using cosine or sine for this. A possibility is that it was orthographic projection; what you did in high school. And I'm sure there are plenty of explicit complete formulas for all this used by graphics programmers. But I usually stitch together everything I need using these steps.
Really, really nice! And what a great starter for any lecture about Projective Geometry. If all maths and maths ed students just knew what you showed in the video, the world would be a better one!
I was considering doing a video entry on projective geometry after listening to Prof. Richter Geberts great lecture, I but scrapped it due to time issues. What a coincidendce now to see you make this wonderful entry :)
You should make it anyway! SoME is a good excuse to get started, but not necessary, of course. Moreover, there so much more about projective geometry to talk about, and I can't do it all on my own 😉
This video came as an suggestion. It's a good video for understanding, as I find it's your first video.. I am hoping you do more of this🥳 love from Nepal 🇳🇵
Oh man! What a treat! How I wish you had been around in 1985 when I was doing 3-point perspective in technical drawing class. This is such a clear and thoughtful exposition. How long did it take you to put together??
Thank you! 🙏 I mostly worked at night; one or two hours per day. Took me about 6 weeks for the script, 2 weeks for the animations and a couple of days for recording and editing.
@@sumandproduct The script was good, in terms of outline, how much time was spent on each topic, and transitions. But I have one complaint for something that stood out: you mention that in drawing class we are told that the vanishing points lie on the horizon, but your drawing on-screen shows them on a line that is far from horizontal! The editing was excellent. The recording and general production value was better than most beginners. Good work there!
I genuinely thought this was one of those professional videos made by big math-stuff content creators, you did a great job! This was a very friendly-speaking informative video and I think you narrated everything pretty well! I'm not a native english speaker but I didn't really think you had an accent hahaha, your english is great! :)
Yea, I use all of this when drafting and/or illustrating all the time. This is core artisan measurement and partitioning. Thing is, I was forced to figure it out by myself. I ended up creating a chart of 'natural' divisions with this method (from corner to centerline) - with a really weird holdout of 3/11 which is more likely an exotic number stuck in between 1/3 and 1/5. There are some other exotic ones out there ( like 4/7 ), but not until you get to the point this method is no longer useful. You will also stumble over the link between music theory, resonance, and design theory just sitting in the middle of all of this - as these natural divisions are how the old artisans basically did everything - and it will also make things like the "imperial" measurement system make a WHOLE lot more sense and base 12 numerical systems in general. Copy all of that on top of Babylonian 60-divison clocks and how that translated into its own unique measurement system and - boom - you got the background for so much of our technological progress. I love this stuff, and its nice to see other people talking about it. I have a whole "Artisan Cheat Sheet" that I post regularly for other artists/craftsmen to use as it can be really hard to find this information collected or easily digested - as computers pretty much killed this whole way of thinking.
I'm so glad you liked it! Coming from the maths side I'm pretty much "done" once I know that all fractions can be constructed. But there's surely more insight to gain if you actually have to do these constructions by hand.
@@pacefactor That's also probably true for most things in mathematics. Theory is important, but doing something by hand a couple of times really helps to built intuition.
Bellissimo video! Grazie. Io sto scrivendo un libro che riguarda il birapporto armonico con svariate costruzioni grafiche. Mi fa piacere vedere che nel mondo ci sono altre persone che studiano e divulgano la Geometria Proiettiva! Quando finirò il libro avrò piacere di condividerlo.
Everything is clearly explained @sumandproduct, great video, congratulations! I just wonder which software did you use for the figures and animations? Thank you
Thank you! The animations are written in CindyJS and then screen-recorded via OBS. Audio recording with Audacity and editing with DaVinci Resolve. (And the thumbnail was made with Inkscape.)
great video, but we're a bit confused about the start of the "school method" section... how do you equally divide the auxiliary line in a way that doesn't also work directly on the original line?
Do you mean in the Euclidean case at the very start of the section or the start of the projective case? Let me answer both... In the Euclidean case: I used a compass. It's hard to see, but the little violet marks are circular arcs of circles with the same radius. In the projective case: I used the iterated cross-ratio construction, which was introduced at around 14:10. The idea is, that the square-splitting construction can also be used to duplicate a segment. So, from 0, 1, infinity you can get 2. Then, from 1,2, infinity you can get 3. And so on... From n, n+1, infinity you can get n+2. It's always the same construction (with different starting points, of course). But at 14:10, I re-used many of the already existing yellow lines for 3, 4 and 5. Later, in the “school” construction, I do exactly the same. But instead of starting with the segment 0 to 1, I use 0 to r because the exact numbers don't matter. Did that help?
I have a 2-Point Perspective Drawing that I'm working on for my Art Class and I've determined the position of the Cube's Bottom-Left Corner to be at 1937 units away from the Closest Edge of the Cube from the Viewer. When using a Ruler to measure 0 to 1937, I get Roughly 5.4cm. I want to find the Position of 4527 units from 0. How would I do this?
Yes, I'm implicitly assuming a photo without distortion! Depending on the type of distortion, I think there are ways to do similar measurements. But I have little to no experience with them, so I'm not sure.
3:05 This is correct if one assumes that there is no atmosphere or its refraction is zero. If you include the atmospheric refraction, the (horizontal) vanishing points should be a bit further above the horizon, right?
It's more like assuming there is no atmosphere. 😅 But yes, calculating where we see parallel lines meeting depends on much more than just the geometry itself.
Ah, that... I could share it, but it's a terrible, terrible hack. No one should do it that way and no one should even look at it. 🙈 It's just better to assume that the typing effect is not possible in CindyJS.
Reading the description, I think you've exaggerated the audio quality/accent "problems" quite a lot. I've heard a lot worse in both of these regards, and I didn't really find the audio or accent disturbing while watching. The only audio nitpick I have is the abrupt ending, I think you could've let the music ring out or fade out a bit at the end without it affecting the video negatively in any way. In either case, it was a nice video, good job!
Thank you for your kind words! Maybe I'm just too self-aware with my voice. 😅 Others mentioned the ending, too. I had no idea how to finish, so I just... quit. 🙈 Letting the music out is a nice idea!
@@sumandproduct You don't have more videos yet to show the end-screen with the two video recommendations from the same channel, yet. But you could plan for that. You could show a black screen with "Fin." typed in the center in white; then draw the grid and twist the perspective (with the writing on it) until it becomes the deeded flat line! Have an audible /beep/ when this happens flash a red "Error: division by zero. Program halted" and while that fades out also shrink the line from each end, being left with a point - clearly an homage to the original "Outer Limits" title sequence which itself is a reference to turning off old CRTs.
Ho guardato con maggiore attenzione. Al minuto 12:51 dici che, dati tre punti A, B e C, non è possibile costruire il quarto punto D del doppio rapporto (ABCD). Invece è possibile! Se vuoi ti mando la costruzione.
Nice tricks I finally understood some things that were exposed to me in art class, but not in an enough mathematical way for me. Btw, i got some difficulties understanding your speach. As a non nativue english speaker, I needed all my attention to graps the sounds you were saying. I don't have this problem with most english videos, and I think it is due to 2 things. 1. The music The music is fuzzy (not sure what that mean, but dense and chaotical) and loud. For you, used to all kind of parasite noise in english, no problem But I need all the details of the speech to understand what's going on. With a little of noise, your brain can easily recover the original sounds emitted, byt my untrained brain can used those redundancy rules. 2. The micro I can't know, but it seems that your sound is a little bith saturated. Maybe it's just that I learn english with a different accent then yours Well Thank you for the vid, I hope I could help
![SAFEPLANET - ดาวเคราะห์ [ THE PLANET ]](http://i.ytimg.com/vi/4X1XL3I_k3c/mqdefault.jpg)
when i was younger, every time i tried to get some bit of projective geometry, it was just a mysterious mess to me. well, maybe, i could remember something, do formulas, make some constructions, but the essence was still unaccessible. now, when some maths intuition settled in my head, i am starting to rediscover a lot of mathematical concepts, so videos like this are very useful not for getting some rigid knowledge (you may always read a textbook or a Wiki page for this purpose) but learn some motivation of a concept, get the intuition, highlight the connections with other math stuff and well, really understand what's going on. so i'd like to thank a lot the author for the video, it really uncovered some bits of information needed to order my previous knowledge.
thank you and good luck in developing the channel!
Very nice! The cross-ratio is one of my favorite less well-known topics. With the Mobius transformation view, we can see that the cross-ratio of four complex numbers is real if and only if they lie on the same line or circle.
Really nice video. I've just made my final bachelors degree project about projective geometry and used de cross-ratio (which i knew as double ratio) to make an euclidean clasification of the conics.
The mathematics, explanations, and animations are beautiful, but the background music is too loud and drowns out your voice for me :(
I'm very sorry! Other people complained about the music, too. Next time, I will definitely let people watch the video beforehand.
But I'm happy to hear that you lied the rest of it!
@@sumandproduct I love the music, tactical fading in and out like 3blue1brown would be more effective as you build the story of this amazing projection topic.
I have been very curious about this subject with blender and fspy for a couple of years and you satisfied my curiosity. Now please do go into the 10 minute long explanation about why infinity works as 1!
Btw I worked at an architecture company that used to leverage these concept with CAD to draw projective lines and create view transformations manually from other 2D horizontal cad views in the larger model space that contains multiple views by utilizing Construction lines which are infinite... Deep world of line projection is fascinating and useful in transformation of images. Linear algebra has a great way of doing these transformations but this video's content doesn't really require anything beyond what is taught in elementary school, which makes it so beautiful for the math community.
@@pa.l.2499 Thank you very much!
I think you can get the most out of these projective transformations when you mix the linear algebra approach and this pen-and-paper approach. But for the video I tried to stay in the latter realm completely, as it makes it more approachable, as you mentioned.
the subtitles definitely help
@@sumandproduct I concur that in the future you have to set the music at much lower level, or actually not use music.
Occasionally you swallow consonants, such as around 1:10 where your script says 'square length' but it ends up sounding as 'skwele'. Between that and your german accent: the viewer neets a couple of brain cycles more to get to the words you are saying.
Yeah, there is a trade-off there. Pronouncing everything would sound ponderous (Related subject: video by Tom Scott : th-cam.com/video/qu4zyRqILYM/w-d-xo.html )
In the case of 3blue1brown I see the music used as part of the brand; the narration is soothing, the music is soothing. For minutephysics too I regard the music used not as part of the video as such, but as part of the brand.
Personally I tend to prefer minimalism.
Great Video! The way of looking at the harmonic construction as a projective parallelogram was really nice, I wish we would have been shown that in our geometry lecture!
Thank you very much! I think, this is the third way to look at the harmonic point construction I worked with over the years. Probably my favourite so far.
I'm really happy you decided to share this video as it was very clearly presented, well animated and well paced. Love the subject and how you made it easy to follow your train of thought by using (wonderful) animations. Great job!!
Thank you very much for your kind words! 🙏
This was a pretty fun watch, thank you
Thank you!
Shocked me when this channel only had 1k subs. This is a gold mine. Keep it up!
Thank you! I'm quite busy at the moment, but I'll definitely continue!
This is a cool topic, thank you!
When I did a perspective drawing in high school, I tried to make a street with many (a dozen or so) buildings of equal width. I used the Fibonacci sequence to measure out their respective widths and it looked good enough. I now wonder how it aligned to this reality of math.
I was following along with the progress on this video on the SoME2 discord! I'm so glad you were able to get it out before the deadline. Awesome stuff :).
Thank you very much! 🙏
Both explanation and visualization are just perfect. Thank you!
I recall from a drafting & architecture class in high school that the formula for measuring along the axis in perspective was something simple involving the cosine function. I suppose that was an approximation.
When I took computer graphics in college, I recall the extra dimension allowing perspective transformations, and that moving a parameter changed the perspective; but I don't recall any formula for how that relates to the vanishing point.
I can't be sure, but I don't recall any formulas using cosine or sine for this. A possibility is that it was orthographic projection; what you did in high school.
And I'm sure there are plenty of explicit complete formulas for all this used by graphics programmers. But I usually stitch together everything I need using these steps.
BEAUTIFUL! I can finally get a proper introduction to projection geometry without the need to look up all different books on the internet! Nice!
Really, really nice! And what a great starter for any lecture about Projective Geometry. If all maths and maths ed students just knew what you showed in the video, the world would be a better one!
Thank you very much! That means a lot, coming from!
Very good exposition. Really really clear, I thoroughly enjoyed it!
This was absolutely fascinating - great job!
Great video! 🔥 Good luck in peer-review!
Thank you! 🙏
The channel looks brand new. Best of luck. It’s gonna grow! Mark my words
Thank you! Let's hope for the best!
I was considering doing a video entry on projective geometry after listening to Prof. Richter Geberts great lecture, I but scrapped it due to time issues. What a coincidendce now to see you make this wonderful entry :)
You should make it anyway! SoME is a good excuse to get started, but not necessary, of course. Moreover, there so much more about projective geometry to talk about, and I can't do it all on my own 😉
3:13 This moment is brilliant and enlightening
I so wish I had found this video 2 or 3 years ago when I kept googling for this exact problem 😅
This video came as an suggestion. It's a good video for understanding, as I find it's your first video.. I am hoping you do more of this🥳 love from Nepal 🇳🇵
Thank you! I plan to do more, but it might take a while.
Amazing video!! Love it!!
Your accent is beautiful
Nice video. It reminded me of some topics we talked about in the geometry class I was in while working on my master's degree.
Very good video! 15:38 should the right side yellow equation be (1, inf | 2, 0)?
Thank you!
Yeah, it seems I mixed up the order here...
Fantastic Video!
Please do more videos, your explanations are very clear, great job
Thank you! I will try! I have an idea for the next video, but it is a much weirder topic. Let's see how it goes...
Cool animating style
Thank you! 🙏
This video helped me figure out some constructions in Pythagorea
Where are your subscribers? This video was very well made.
Thank you! I guess, I have to make a few more video if I want more subscribers, right? 😉
Oh man! What a treat! How I wish you had been around in 1985 when I was doing 3-point perspective in technical drawing class.
This is such a clear and thoughtful exposition. How long did it take you to put together??
Thank you! 🙏
I mostly worked at night; one or two hours per day. Took me about 6 weeks for the script, 2 weeks for the animations and a couple of days for recording and editing.
@@sumandproduct The script was good, in terms of outline, how much time was spent on each topic, and transitions. But I have one complaint for something that stood out: you mention that in drawing class we are told that the vanishing points lie on the horizon, but your drawing on-screen shows them on a line that is far from horizontal!
The editing was excellent.
The recording and general production value was better than most beginners. Good work there!
I genuinely thought this was one of those professional videos made by big math-stuff content creators, you did a great job! This was a very friendly-speaking informative video and I think you narrated everything pretty well! I'm not a native english speaker but I didn't really think you had an accent hahaha, your english is great! :)
Thank you very much! 🙏
thanks ; very interesting (sehr angenehm :) )
Yea, I use all of this when drafting and/or illustrating all the time. This is core artisan measurement and partitioning.
Thing is, I was forced to figure it out by myself. I ended up creating a chart of 'natural' divisions with this method (from corner to centerline) - with a really weird holdout of 3/11 which is more likely an exotic number stuck in between 1/3 and 1/5. There are some other exotic ones out there ( like 4/7 ), but not until you get to the point this method is no longer useful.
You will also stumble over the link between music theory, resonance, and design theory just sitting in the middle of all of this - as these natural divisions are how the old artisans basically did everything - and it will also make things like the "imperial" measurement system make a WHOLE lot more sense and base 12 numerical systems in general. Copy all of that on top of Babylonian 60-divison clocks and how that translated into its own unique measurement system and - boom - you got the background for so much of our technological progress.
I love this stuff, and its nice to see other people talking about it. I have a whole "Artisan Cheat Sheet" that I post regularly for other artists/craftsmen to use as it can be really hard to find this information collected or easily digested - as computers pretty much killed this whole way of thinking.
I'm so glad you liked it!
Coming from the maths side I'm pretty much "done" once I know that all fractions can be constructed. But there's surely more insight to gain if you actually have to do these constructions by hand.
@@sumandproduct yea doing them by hand really enlightens you to a whole history and way of thinking you don't see much anymore
@@pacefactor That's also probably true for most things in mathematics. Theory is important, but doing something by hand a couple of times really helps to built intuition.
Would you be willing to share this cheat sheet?
@@herdenq yea sure, I would need to make a link for it. Mind you, it's more for refrence and understanding that a distinct lesson or explanation
Bellissimo video! Grazie. Io sto scrivendo un libro che riguarda il birapporto armonico con svariate costruzioni grafiche. Mi fa piacere vedere che nel mondo ci sono altre persone che studiano e divulgano la Geometria Proiettiva! Quando finirò il libro avrò piacere di condividerlo.
Everything is clearly explained @sumandproduct, great video, congratulations! I just wonder which software did you use for the figures and animations? Thank you
I'm using cindyjs.org
@@sumandproduct Thank you!
I like your style
Awesome video! What tools have you used to create it? It's of very high quality.
Thank you!
The animations are written in CindyJS and then screen-recorded via OBS. Audio recording with Audacity and editing with DaVinci Resolve.
(And the thumbnail was made with Inkscape.)
Amazing
great video, but we're a bit confused about the start of the "school method" section... how do you equally divide the auxiliary line in a way that doesn't also work directly on the original line?
Do you mean in the Euclidean case at the very start of the section or the start of the projective case? Let me answer both...
In the Euclidean case:
I used a compass. It's hard to see, but the little violet marks are circular arcs of circles with the same radius.
In the projective case:
I used the iterated cross-ratio construction, which was introduced at around 14:10. The idea is, that the square-splitting construction can also be used to duplicate a segment. So, from 0, 1, infinity you can get 2. Then, from 1,2, infinity you can get 3. And so on... From n, n+1, infinity you can get n+2. It's always the same construction (with different starting points, of course). But at 14:10, I re-used many of the already existing yellow lines for 3, 4 and 5.
Later, in the “school” construction, I do exactly the same. But instead of starting with the segment 0 to 1, I use 0 to r because the exact numbers don't matter.
Did that help?
Great Video.
Would love to see more like this from you.
Btw what was that geogebra while editing the ruler's pic?
can u gimme ur version please.
It's called "Cinderella" and you can find it here:
cinderella.de/tiki-index.php
@@sumandproduct You should include that in the description or a pinned comment.
@@JohnDlugosz Ah, good point! Thank you!
I have a 2-Point Perspective Drawing that I'm working on for my Art Class and I've determined the position of the Cube's Bottom-Left Corner to be at 1937 units away from the Closest Edge of the Cube from the Viewer. When using a Ruler to measure 0 to 1937, I get Roughly 5.4cm. I want to find the Position of 4527 units from 0. How would I do this?
Very nice. 09:25 doesn't how well this work with a photo depend on how much distortion is added by the camera?
wiki: Distortion_(optics)
Yes, I'm implicitly assuming a photo without distortion! Depending on the type of distortion, I think there are ways to do similar measurements. But I have little to no experience with them, so I'm not sure.
3:05 This is correct if one assumes that there is no atmosphere or its refraction is zero. If you include the atmospheric refraction, the (horizontal) vanishing points should be a bit further above the horizon, right?
It's more like assuming there is no atmosphere. 😅
But yes, calculating where we see parallel lines meeting depends on much more than just the geometry itself.
This is great. Can you tell what you use to create these nice animations and the text morphing animations?
This was all done with cindyjs.org
@@sumandproduct but what about the text, equation animations? Can you share the cindyjs source code for this?
Ah, that... I could share it, but it's a terrible, terrible hack. No one should do it that way and no one should even look at it. 🙈 It's just better to assume that the typing effect is not possible in CindyJS.
@@sumandproduct hah, now you just make me more curious about the secret 😆
Reading the description, I think you've exaggerated the audio quality/accent "problems" quite a lot. I've heard a lot worse in both of these regards, and I didn't really find the audio or accent disturbing while watching. The only audio nitpick I have is the abrupt ending, I think you could've let the music ring out or fade out a bit at the end without it affecting the video negatively in any way. In either case, it was a nice video, good job!
Thank you for your kind words! Maybe I'm just too self-aware with my voice. 😅
Others mentioned the ending, too. I had no idea how to finish, so I just... quit. 🙈 Letting the music out is a nice idea!
@@sumandproduct What I find funny is that you correctly say "differences" but for some reason say "distan-sees" :D Great video though :)
@@spitsmuis4772 🤣 Yeah, I'm speaking this weird mix of American and British English, I guess, and then everything is bulldozed by the German accent.
Yes, the video ended so abruptly that it went on to the next in the playlist before I could pause and read the comments!
@@sumandproduct You don't have more videos yet to show the end-screen with the two video recommendations from the same channel, yet. But you could plan for that.
You could show a black screen with "Fin." typed in the center in white; then draw the grid and twist the perspective (with the writing on it) until it becomes the deeded flat line! Have an audible /beep/ when this happens flash a red "Error: division by zero. Program halted" and while that fades out also shrink the line from each end, being left with a point - clearly an homage to the original "Outer Limits" title sequence which itself is a reference to turning off old CRTs.
Ho guardato con maggiore attenzione. Al minuto 12:51 dici che, dati tre punti A, B e C, non è possibile costruire il quarto punto D del doppio rapporto (ABCD). Invece è possibile! Se vuoi ti mando la costruzione.
Cool!
Nice tricks
I finally understood some things that were exposed to me in art class, but not in an enough mathematical way for me.
Btw, i got some difficulties understanding your speach. As a non nativue english speaker, I needed all my attention to graps the sounds you were saying.
I don't have this problem with most english videos, and I think it is due to 2 things.
1. The music
The music is fuzzy (not sure what that mean, but dense and chaotical) and loud.
For you, used to all kind of parasite noise in english, no problem
But I need all the details of the speech to understand what's going on. With a little of noise, your brain can easily recover the original sounds emitted, byt my untrained brain can used those redundancy rules.
2. The micro
I can't know, but it seems that your sound is a little bith saturated.
Maybe it's just that I learn english with a different accent then yours
Well
Thank you for the vid,
I hope I could help
Loved it 9.5, not a perfect 10 due to loud music
instead of using -1 and +1, it would probably be better to use a fictitious number ω and its negative, to avoid confusion with an "actual" 1.
First.
reversibility does not imply surjectivity.
You really need to work on your didactic skills, talking to an empty screen and such :-)
voice over your lecture is somewhat distracting, otherwise thanks.