In 5th grade I had an origami ninja star business with a classmate. It was supposed to be a 50-50 revenue split, but my classmate took more than that. In conclusion, the end.
The hiragana on top of kanji is known as furigana and is commonly used for Japanese learners who are still learning kanji. I could have used katakana, but that's usually reserved for differentiating kunyomi vs. onyomi readings As for why I wrote the Japanese in the first place, I was just nerding out a little on the origin of the word
I first misread this as Origami Cheese Patterns and I was immediately intrigued. I did have to get over my disappointment, but I am glad I did. Well done.
This is super well done! (even if I am a bit salty someone got to this topic before me)! Origami is super nostalgic for me. I remember being in grade/middle school and binging Jeremy Schaefer videos. Thanks for making the vid!
I'm glad you liked it! Unfortunate about the nerdsnipe, but I'd love to see some other proofs because my proof for |M-V|=2 esprcially is a bit rough around the edges. I got pretty deep into the weeds of origami and never saw a nice proof for these theorems. I figured this would be a perfect chance to throw my hat into the ring. Once again, glad you enjoyed!
Oh my god. Your explanations are cristal clear. The representations are 100% intuitive. And the pixel artstyle is GORGEOUS. You are going to rank very high in this Somepi, or even win!
Big fan of your aliased low-res visuals! They're beautifully done, and I can't help but imagine receiving each slide as a flipnote lol Great explanation, too! Now all I need to figure out is how to use math to make crease patterns based on existing low poly 3D models...
what a lovely video. you deserve a lot more recognition for this work, it's so good. beautifully presented and interesting at the same time. the algorithm has blessed me today, it seems.
Very cool video about some maths of origami. Simple yet direct to the point. Here's what may be a neat problem to figure out, related to origami: Given a paper with a fold at 1/n of the side, can you divide it evenly only by folding the edges together or to previously marked creases? Here's the solution for the first numbers: 1,2 are trivial; 3 can be done by folding to the mark dividing in 3 parts evenly; 4 can be done, folding edge to edge to get a half and then both edges to the center crease; 5 can be done, dividing in half twice from the starting mark then doing the same the opposite way; 6 can not be done given the rules. I hope the thought process is clear enough and enough interesting for you to give it a shot. I suspect it is related to factorization but I am too burnt out on maths to properly sit down and solve.
Cool problem! I was thinking about it yesterday and made very little progress, and I'll let you know if I find anything. Funnily enough, I thought of a very similar problem where you also can do crease-to-crease folds and also generalized to any rational p/q between 0 and 1. Solution below: You can split the paper into q pieces as long as gcd(p,q) is a power of 2. The proofs uses strong induction
Awesome video! I started origami very young, with a book that I got as a present, so it's nice seeing a video on it. And the animations are very nice, specially with your art style!
The pixel art style and subject matter take me back to when I would search up Origami patterns on my Windows 98 PC in early primary school, thanks for the nostalgia!
Definitely one of the most unique topics in this competition. The proof for the last law was a bit confusing at first, but otherwise the video was very well structured and animated. Subscribed for more content, hope you keep creating!
Very nifty, and a great introduction to some of the many facets of origami math! One small matho I wan't to point out: when you're going alternatingly around one of the vertices at 5:26, you have '180' at the bottom (the total tally of the angles) where it should be 135.
Nice catch! My brother did notice this a couple days ago, and the error is already noted in the description. Do you think I should write the errors in a pinned comment instead so more people will see it?
@@CountingTo3 Many creators do that. I think some TH-cam apps don't readily show the description when seeing the video itself or the comments, as opposed to showing it above the comments.
Interesting take! How would flat-foldability work? My guess is the paper is not exactly like a torus as a solid of revolution of a circle, but more the topologist's definition as a genus 1 object. I don't really know how to make this work yet, but I will think about it
My gut answer here (hopeful that we don't have to take it to infinity to get rid of edges) was definitely "oh, I bet it's just wraparound" (my gut is informed by a lot of pencil puzzles, mostly). I didn't even realize until I read this comment what kind of a cursed shaped paper I was suggesting folding into a crane.
@CountingTo3 Neither the sum of alternating angles nor the M-V difference rule applies at the edge, and 2-colorability is always guaranteed at any vertex on an edge. It is possible to construct any alternating angle difference 0-180, as well as any arbitrarily lagre |M-V| value, for a vertex on an edge.
@bobbob0507 Here's some interesting thoughts for you because you're getting on the right track: What if we had a paper with a 2:1 edge ratio that we folded in half to make a two sheet thick square paper and continued as normal? What does each half represent and what does the middle crease represent? How might this work for the corner? I want you to think about how we can make the behavior make sense rather than dismiss it entirely
love the pixel arts. At 4:15, I am pretty sure I can make that 7th fold in real life if it is a valley fold without making the 8th crease in the oppose side
Try it and see if it is flat-foldable. Based on the constraints, it should not work, but if it is flat-foldable, then I'd be willing to make an addendum
It does work for any point on a crease because a point in the middle of a crease has the same fold on both sides and thus you get 2-0, as said in the video. This is also the basis as to why the rest of the proof works.
i wonder how 4d origami might look like :o, then the creases could be represented in 2d and the crease pattern in 3d. In addition, i also wonder how many crease types exist in 4d origami, sounds already super interesting
Besides the engaging, well explained maths, Im so surprised at how beautiful this video is! How did you render the moving 3d models? That looks like a lot of work, great job!
All of the frames were made in aseprite by hand. There's a layer with circles or ellipses and dots to track the motion for the frames that I hide before exporting
Nice art style! But the pixelated-discrete art actually makes it harder to visualize, as it took a long time for me to actually understand what was going on at 7:28, wish it was a 3d art style would be better for this
Yeah, there was a bit of a leap there, but since I kind of get it: Imagine a fold pattern around a certain vertex and put it in the flat-folded configuration. Then cut off the corner the vertex is on. The edge of the paper basically makes the shapes shown in the video. For example the diagram at 7:44 could correspond to the pole being at a vertex with creases V-135-V-135-V-45-M-45-[return to first V]
@@nerdiconium1365 Thanks, I think I get it now. I had to go back to the cross section diagram in the previous section at 5:44 to be able to follow what is happening
Regarding the "edge" case (nice pun!), I suspect youd want to cut the paper in the middle twice and glue back together by the edges, to now be in the interior? More formally: think of the square as the fundamental domain of the torus, and thus there are no edges!
I like the idea of using a toroidal paper (folding a crane on that would be interesting), but I'm more asking a question about how to change the behavior of the paper. It also brings up arguably weirder cases where folds end randomly in the middle of the paper. I'll give you some hints: What if we taped on a second square to one of the edges and folded it on the tape first? We could continue folding it as a thicker paper and then get a new crease pattern. What does that second paper correspond to? How can we translate that behavior for the corners?
the law with the addup to 0 only aplays to planer folds, as they have an even number ov folds per vertex, but there are folding techniches, that involve patterns with 5 folds per vertex (i made a origami torus for exapmle with such a pattern) and there it dosent make sense to speak of alternating sums.
That's why flat-foldability is required and mentioned at the beginning. That being said, I would love to see your origami torus! Is there a video or blog post where you show it?
Hi, I've got a question unrealted to the video for you. I've been wanting to make math animations on youtube for a while. I know there are some software, like Manim, to facilitate making math animations, but I would really like to make the video in pixel art. I was wondering if you were doing your animations "by hand" or if you were using somthing like Manim. Thk in advance for your answer.
I make my frames using Aseprite by hand. Generally I can shortcut some frames using copy-paste, and I keep a few references open, like one for text and one for any recurring images in the video. There are probably ways to make pixel art using code, but I don't know how to make it work. Also, I am a bit of a perfectionist, so I usually want my images in the correct spots down to the pixel. The easiest way for me to do that is to do it by hand.
@@CountingTo3 Thanks, your explanations in the video are great. But my question is more: If I have a set of lines on a square paper that follow the three laws you described, can I conclude that it forms a legit crease pattern? You showed that the three laws are necessary, are they sufficient?
They are not. If you check the link in the description under "More about self-intersection", I linked a blog post that describes some other things to keep in mind. Good question!
Good video but I think some visual choices are getting in the way of understandability. In the cross-sectional diagram at 5:44, the red dot is just too tiny compared to the line's thickness, that makes it hard to see. The steps in its movement are also too large, it's hard to follow, especially when it outrigth skips the creases instead of moving through them. In that same section the line that is supposed to connect the cross-sectional view to the flat-folded view is the same thickness and color as the lines representing the paper so it's hard to parse what it means right away. In the falt-folded view, there's never any indication that the radial line from the corner is going above, beneath or in the middle of the folded piece of paper. Combining that with the related movement in the cross-sectional view being hard to read, the flat-folded view animation becomes more a distraction than a visual aid. In the next session the representation set up at 7:14 just isn't explained enough. I think I would've had an easier time had the script connected it more explicitly to what's been shown before. Explaining the legend before any of the important elements in it are present is distracting. The transformation at 7:26 happens completely isolated from any visual context and that made it completely unintelligible to me.
Thanks for the descriptive feedback! I really appreciate the specifics about clarity because there were definitely parts that were not clear, and it is my job to make them clearer. I will keep these ideas in mind for the next video
The content of this video is really great, but I think the low-res pixel art is a detriment. For example, following the paper crane crease diagram is basically impossible, and looks really unpleasant when colored in
That's understandable. The style of my videos is pixel art, and I was aiming to challenge myself with this one. Clarity is a top priority, so I try to make sure the pixel art restriction doesn't make it too hard to understand. This is something I am working on improving with every video I make
That works! We can consider the edges like a fold and then use the flipside of the paper to consider the creases from there to make it work with the three laws. However, the corners are not so easy because you will end up getting 4 more mountain folds than valley folds. You have to be a little more careful in the corners
Waterbomb base is the name of the piece that the crease pattern is for. There is an origami piece called a waterbomb and many pieces build off of the starting creases
Yes, the organizers were not planning on doing an official SoME this year. See th-cam.com/users/postUgkxNRYx6RcUmHxB_7Jr_19ZH2drRPzzaZLg?si=RA-R9lCLYXzlEM4I for more details
wow - i love this video! i never expected a pixel art style "math explainer", but it works really well, particularly paired with the overall relatively simple presentation style 🩵
At 5:26, the running total should be 135
ok
In 5th grade I had an origami ninja star business with a classmate. It was supposed to be a 50-50 revenue split, but my classmate took more than that. In conclusion, the end.
Great story
CINEMA
amazing
Sounds like steve jobs. 😂
BRO SAME, FOR ME IT WAS IN 3RD GRADE
Love how you write the kanji in hiragana as if your audience could understand that either.
It would have better readability if you did it in katakana (at least for me)
The hiragana on top of kanji is known as furigana and is commonly used for Japanese learners who are still learning kanji. I could have used katakana, but that's usually reserved for differentiating kunyomi vs. onyomi readings
As for why I wrote the Japanese in the first place, I was just nerding out a little on the origin of the word
@@CountingTo3as someone learning Japanese, thanks, I enjoy seeing kanji that I'll never be able to read.
Furigana is very helpful
I first misread this as Origami Cheese Patterns and I was immediately intrigued. I did have to get over my disappointment, but I am glad I did. Well done.
Lol
the pixel art style is so cool here !!!
Azali !
Omg ur here!!! Hiiiii ur music is amazing!
hi
This is super well done! (even if I am a bit salty someone got to this topic before me)! Origami is super nostalgic for me. I remember being in grade/middle school and binging Jeremy Schaefer videos. Thanks for making the vid!
I'm glad you liked it! Unfortunate about the nerdsnipe, but I'd love to see some other proofs because my proof for |M-V|=2 esprcially is a bit rough around the edges.
I got pretty deep into the weeds of origami and never saw a nice proof for these theorems. I figured this would be a perfect chance to throw my hat into the ring. Once again, glad you enjoyed!
For what it's worth, there's a _whole_ lot of math in origami, more than enough that you could find plenty of other good video topics. Go for it!
worth mentioning that more advanced folds: inside and reverse-inside, also result in mountain/valley creases
Yes, any origami piece can be unfolded into a crease pattern with only mountain and valley folds
Unfortunately, |M-V| does not equal 2 because when you do the subtraction what's left is | I I | which is 4, not 2.
No, 4 is IV. Unless it's a clock for some reason.
I love the art style. Looks very time consuming but definitely worth it.
Oh my god. Your explanations are cristal clear. The representations are 100% intuitive. And the pixel artstyle is GORGEOUS. You are going to rank very high in this Somepi, or even win!
This art style is soooo cooool!! Love your video :)
Big fan of your aliased low-res visuals! They're beautifully done, and I can't help but imagine receiving each slide as a flipnote lol
Great explanation, too! Now all I need to figure out is how to use math to make crease patterns based on existing low poly 3D models...
this is beautifully animated
I love your art style, its so unique. and the video itself is great!!
I love making rules and implications like this, sounds like something I'd come up with
I've been watching reels from ThePlantPsychologist and this motivated me to try some of his crease patterns for the first time!
I love the pixel aesthetic!
what a lovely video. you deserve a lot more recognition for this work, it's so good. beautifully presented and interesting at the same time. the algorithm has blessed me today, it seems.
I love your style of showing math, so unique!
Very cool video about some maths of origami. Simple yet direct to the point.
Here's what may be a neat problem to figure out, related to origami:
Given a paper with a fold at 1/n of the side, can you divide it evenly only by folding the edges together or to previously marked creases?
Here's the solution for the first numbers:
1,2 are trivial;
3 can be done by folding to the mark dividing in 3 parts evenly;
4 can be done, folding edge to edge to get a half and then both edges to the center crease;
5 can be done, dividing in half twice from the starting mark then doing the same the opposite way;
6 can not be done given the rules.
I hope the thought process is clear enough and enough interesting for you to give it a shot.
I suspect it is related to factorization but I am too burnt out on maths to properly sit down and solve.
Cool problem! I was thinking about it yesterday and made very little progress, and I'll let you know if I find anything.
Funnily enough, I thought of a very similar problem where you also can do crease-to-crease folds and also generalized to any rational p/q between 0 and 1. Solution below:
You can split the paper into q pieces as long as gcd(p,q) is a power of 2.
The proofs uses strong induction
2:22 im ready for topological origami
Awesome video! I started origami very young, with a book that I got as a present, so it's nice seeing a video on it.
And the animations are very nice, specially with your art style!
The pixel art style and subject matter take me back to when I would search up Origami patterns on my Windows 98 PC in early primary school, thanks for the nostalgia!
I love your art style! And good job on the video! :D
Definitely one of the most unique topics in this competition. The proof for the last law was a bit confusing at first, but otherwise the video was very well structured and animated. Subscribed for more content, hope you keep creating!
Love the visuals, very concise
Very nifty, and a great introduction to some of the many facets of origami math! One small matho I wan't to point out: when you're going alternatingly around one of the vertices at 5:26, you have '180' at the bottom (the total tally of the angles) where it should be 135.
Nice catch! My brother did notice this a couple days ago, and the error is already noted in the description. Do you think I should write the errors in a pinned comment instead so more people will see it?
I was going to comment that but I checked if anybody else had done it first just in case
@@CountingTo3 Many creators do that. I think some TH-cam apps don't readily show the description when seeing the video itself or the comments, as opposed to showing it above the comments.
amazing animations, really good job!
Ridiculously well made video!
i like the pixel art!
Amazing explanation and visuals!
visually like prob my fav somepi,,,, also like really good video in general ,,i love origami
to address the edge case, we can make the paper into torus shape
Interesting take! How would flat-foldability work? My guess is the paper is not exactly like a torus as a solid of revolution of a circle, but more the topologist's definition as a genus 1 object. I don't really know how to make this work yet, but I will think about it
My gut answer here (hopeful that we don't have to take it to infinity to get rid of edges) was definitely "oh, I bet it's just wraparound" (my gut is informed by a lot of pencil puzzles, mostly). I didn't even realize until I read this comment what kind of a cursed shaped paper I was suggesting folding into a crane.
@jkid1134 It poped into my head too, but I instantly realized it wouldn't work given the example on screen
@CountingTo3 Neither the sum of alternating angles nor the M-V difference rule applies at the edge, and 2-colorability is always guaranteed at any vertex on an edge. It is possible to construct any alternating angle difference 0-180, as well as any arbitrarily lagre |M-V| value, for a vertex on an edge.
@bobbob0507 Here's some interesting thoughts for you because you're getting on the right track:
What if we had a paper with a 2:1 edge ratio that we folded in half to make a two sheet thick square paper and continued as normal? What does each half represent and what does the middle crease represent? How might this work for the corner?
I want you to think about how we can make the behavior make sense rather than dismiss it entirely
love the pixel arts. At 4:15, I am pretty sure I can make that 7th fold in real life if it is a valley fold without making the 8th crease in the oppose side
Try it and see if it is flat-foldable. Based on the constraints, it should not work, but if it is flat-foldable, then I'd be willing to make an addendum
@@CountingTo3 I see
|M - V| = 2 applies not to all points on the paper / crease diagram, but only to intersections of creases.
It does work for any point on a crease because a point in the middle of a crease has the same fold on both sides and thus you get 2-0, as said in the video. This is also the basis as to why the rest of the proof works.
@@CountingTo3 To clarify: any point *on a crease* , not any point on the paper.
Well done!
august ferdinand möbius will be having a word with you
It's so lovely!
Awesome proof at 6:27
i wonder how 4d origami might look like :o, then the creases could be represented in 2d and the crease pattern in 3d. In addition, i also wonder how many crease types exist in 4d origami, sounds already super interesting
6:35 you got me here
I'm dead, somebody call r/mathMemes 💀
Very cool vid!
really cool!!!
Besides the engaging, well explained maths, Im so surprised at how beautiful this video is! How did you render the moving 3d models? That looks like a lot of work, great job!
All of the frames were made in aseprite by hand. There's a layer with circles or ellipses and dots to track the motion for the frames that I hide before exporting
great stuff 👍
Nice art style! But the pixelated-discrete art actually makes it harder to visualize, as it took a long time for me to actually understand what was going on at 7:28, wish it was a 3d art style would be better for this
Absolutely, I was working to get this done for SoMEpi up to the last day, so the animations were not as clear as I could have made them
The animation reminds me of the flipnote hetena days on the DSi 😭
7:27 what am I looking at here? I don't understand what this representation means
Yeah, there was a bit of a leap there, but since I kind of get it:
Imagine a fold pattern around a certain vertex and put it in the flat-folded configuration. Then cut off the corner the vertex is on. The edge of the paper basically makes the shapes shown in the video.
For example the diagram at 7:44 could correspond to the pole being at a vertex with creases V-135-V-135-V-45-M-45-[return to first V]
@@nerdiconium1365 Thanks, I think I get it now. I had to go back to the cross section diagram in the previous section at 5:44 to be able to follow what is happening
Same... It's hard to see the connection without looking at the comments
Regarding the "edge" case (nice pun!), I suspect youd want to cut the paper in the middle twice and glue back together by the edges, to now be in the interior?
More formally: think of the square as the fundamental domain of the torus, and thus there are no edges!
I like the idea of using a toroidal paper (folding a crane on that would be interesting), but I'm more asking a question about how to change the behavior of the paper. It also brings up arguably weirder cases where folds end randomly in the middle of the paper.
I'll give you some hints: What if we taped on a second square to one of the edges and folded it on the tape first? We could continue folding it as a thicker paper and then get a new crease pattern. What does that second paper correspond to? How can we translate that behavior for the corners?
Well done im now your 658th sub
Cool video! I did notice an error at 5:25 where the sum is labelled as 180 when it should be 135, but that's just a minor nitpick
this video is so awesome!!!!!!!
Liked and Subscribed ❤❤❤
If you make 3 M radial folds, it is not flat-foldable, unless you make additional folds, thus fails to disprove 2-colorability
Flat-foldability is a requirement for two-colorability, so if you remove the constraint, it is not always true that the laws follow
Pixels! 😃
This video is amazing! Can you tell how you did the folding animations and diagrams? Maybe share the source code as well?
Each frame is made by hand on Aseprite, so I don't have any source code
Wow!
the law with the addup to 0 only aplays to planer folds, as they have an even number ov folds per vertex, but there are folding techniches, that involve patterns with 5 folds per vertex (i made a origami torus for exapmle with such a pattern) and there it dosent make sense to speak of alternating sums.
That's why flat-foldability is required and mentioned at the beginning. That being said, I would love to see your origami torus! Is there a video or blog post where you show it?
4:30 who's gonna tell him about mobius strips?
Não sei por que assisti esse vídeo aleatório até o final. Só sei que gostei do seu estilo.
Hi, I've got a question unrealted to the video for you.
I've been wanting to make math animations on youtube for a while. I know there are some software, like Manim, to facilitate making math animations, but I would really like to make the video in pixel art. I was wondering if you were doing your animations "by hand" or if you were using somthing like Manim. Thk in advance for your answer.
I make my frames using Aseprite by hand. Generally I can shortcut some frames using copy-paste, and I keep a few references open, like one for text and one for any recurring images in the video.
There are probably ways to make pixel art using code, but I don't know how to make it work. Also, I am a bit of a perfectionist, so I usually want my images in the correct spots down to the pixel. The easiest way for me to do that is to do it by hand.
4:30 This cannot happen
Möbius strip would like a word with you
8:36 do we say that the four edges are actually the same point
I'm not sure I understand what you mean, can you elaborate more please?
8:50 make it a mathematical infinite paper. Edgecase solved xD
Is there some reciprocal theorem? Like, if you respect some rules, you automatically get a foldable crease pattern?
All 3 laws needs to be followed because they are a result of the constraints
@@CountingTo3 Thanks, your explanations in the video are great. But my question is more: If I have a set of lines on a square paper that follow the three laws you described, can I conclude that it forms a legit crease pattern? You showed that the three laws are necessary, are they sufficient?
They are not. If you check the link in the description under "More about self-intersection", I linked a blog post that describes some other things to keep in mind. Good question!
Great thank you, I will check that!
Good video but I think some visual choices are getting in the way of understandability.
In the cross-sectional diagram at 5:44, the red dot is just too tiny compared to the line's thickness, that makes it hard to see. The steps in its movement are also too large, it's hard to follow, especially when it outrigth skips the creases instead of moving through them. In that same section the line that is supposed to connect the cross-sectional view to the flat-folded view is the same thickness and color as the lines representing the paper so it's hard to parse what it means right away. In the falt-folded view, there's never any indication that the radial line from the corner is going above, beneath or in the middle of the folded piece of paper. Combining that with the related movement in the cross-sectional view being hard to read, the flat-folded view animation becomes more a distraction than a visual aid.
In the next session the representation set up at 7:14 just isn't explained enough. I think I would've had an easier time had the script connected it more explicitly to what's been shown before. Explaining the legend before any of the important elements in it are present is distracting. The transformation at 7:26 happens completely isolated from any visual context and that made it completely unintelligible to me.
Thanks for the descriptive feedback! I really appreciate the specifics about clarity because there were definitely parts that were not clear, and it is my job to make them clearer. I will keep these ideas in mind for the next video
The content of this video is really great, but I think the low-res pixel art is a detriment. For example, following the paper crane crease diagram is basically impossible, and looks really unpleasant when colored in
That's understandable. The style of my videos is pixel art, and I was aiming to challenge myself with this one. Clarity is a top priority, so I try to make sure the pixel art restriction doesn't make it too hard to understand. This is something I am working on improving with every video I make
apparently 77.5+102.5=180 and 77.5-45+102.5=180, and 180-135=0, how did this slip through the cracks XD
Errors happen, especially when the animations are very similar
@@CountingTo3 it's ok, mistakes happen
i'm terrible at recognizing voices, were you at the reu?
I haven't participated in any REU programs
but origami pieces arent always flat foldable
Not always, but this video refers to the ones that are and generally how most people create pieces because flat-foldability is a useful constraint
Are the edges of the paper mountain folds?
That works! We can consider the edges like a fold and then use the flipside of the paper to consider the creases from there to make it work with the three laws. However, the corners are not so easy because you will end up getting 4 more mountain folds than valley folds. You have to be a little more careful in the corners
@@CountingTo3 What if we consider rounded corners or round paper? Then we only have one edge all around.
That seems like it will work! Interesting solution
What does waterbomb base means?
Waterbomb base is the name of the piece that the crease pattern is for. There is an origami piece called a waterbomb and many pieces build off of the starting creases
@@CountingTo3 I see! Thanks. n_n
This year its SoME Pi?
Yes, the organizers were not planning on doing an official SoME this year. See th-cam.com/users/postUgkxNRYx6RcUmHxB_7Jr_19ZH2drRPzzaZLg?si=RA-R9lCLYXzlEM4I for more details
@@CountingTo3 Cute
(648)
wow - i love this video! i never expected a pixel art style "math explainer", but it works really well, particularly paired with the overall relatively simple presentation style 🩵