@@Pystro we can divide the surface into 2 distinct areas. We can say either area is the “inside”, but either one we choose there will always be 270 degree interior angles.
@@NoNameAtAll2 True, this is an issue. This is actually a reason I didn't go with my original idea for the infinite square, I originally considered a square tape that twisted on 2 edges to have 1 corner under the shape, however since I couldn't think of how to make the surface continuous I ended up redesigning it.
@ the best way to describe euclidian geometry, is that the shortest path between 2 points is a straight line. If you were to take a torus and draw 2 points on opposite sides, going from one point to another would have 2 shortest paths, making it non euclidian.
@@HalfBreadOrder this video was entirely made in the blender 3D viewport, using the NLA animation editor. As such I tried to reuse the animation of the first square spinning on all of the items.
A Teacher teaching children about shapes "And this shape is called a Square-" This Guy : *Bursts into the room holding a WaveSquare* "Behold! A Square!"
To snugly fit a unit square, you would need a cylinder with a height of (root2)/2 and a radius of (root2)/pi. Interestingly, by increasing the radius by increments of (root2)/2pi, we can create square-like shapes with progressively more sides. What would you call a shape with 6 or 8 same-length sides and all right angles?
@@Hankathan I would call an 8 sided ticked polygon that circles around a cylinder a 1 1 3 3 1 1 3 3 ticked octagon. For a ticked hexagon a different surface would be needed since each 2 edges alternate the side of the cylinder the vertices are on. (so points would be bottom, bottom, top top, bottom, bottom, with the next vertex supposed to be on the top due to the sequence, but that doesn't match the position of the first vertex which is on the bottom).
I feel like you would also be able to fit it onto a sphere as well After careful consideration, it can technically fit, but corners must merge. Two perpendicular great circles.
@asheep7797 but are opposing sides parallel to each other in this case? Technically opposing lines would form a single continuous loop around the sphere. If we also drop the rule that it has to be 4 sides, and only concentrate on the rules, has to have only right angles, and all side lengths are the same, then you could make a triangular square on a sphere and a pentagonal square on an inverse square/a horn shaped space
they are non euclidian polygons. I do like math and geometry, so this is pretty cool. You can get topology to behave however you want or need to get those repeating shapes to work. You can even place that square on a mobius strip.
@@christopherfreeman2858 when I was thinking of Ideas to make the infinite square, I considered a square tape that twisted. My biggest issue with it was that the surface was not continuous.
@WaluigiGoesWa Continuous means it has no sharp angles or jumps? A cylinder does violate that, but your using the continuous side. Mobius strips are continuous along the strip side, you have to be very gradual to get the right graphical effect. A discontinuity happens in the flat geometrical representation of any curve, mostly because what makes a continuity is a smooth, infinitely precise function. A lot of shapes are continuous, so long as you meet up perfect curves and don't create any sharp angles. Say, a rounded cube is still continuous.
@@christopherfreeman2858 I was referring to the singular edge of the mobius strip. If you go towards the edge while perpendicular to it, you will hit a "crease" where you flip to the other side. Using a cylinder instead of a plane to form the strip would work though. I have actually been thinking about how twisting could be used to generate a standardized surface for almost any ticked polygon.
@@WaluigiGoesWa I drew a rampart pattern on a piece of paper, it uses consecutive 90 degree angles going left/right/right/left repeating, then joined it up like a mobius strip. If you want to, you can give it volume like a torus and keep the twist, but that seems pretty hard to recreate in a 3d program. Its also hard to show, you'd need to scale it up to get the right side lengths.
Start at the North Pole (NP). Travel in a straight line to the South Pole (SP). Turn left 90 degrees, and travel again in a straight line back to the NP. Right, then back to SP. Left, then back to NP. Turn right and you're facing back along your first line. Four straight lines, four 90 degree turns.
generally polygons aren't allowed to have multiple vertices in the same location (here, there would be two at each pole), although this is just a convention that not everyone uses (ex. Skilling's figure and some papers by Branko Grunbaum)
6:20 while I do appreciate the spinning shapes in the void, I'd appreciate more all those names, definitions, ando formulas written on the screen instead of just spelled out by voice.
@@KingShinyRotom To be honest I had made the majority of the video a couple of months ago, however I got busy with a bunch of work. Typically the video would have just become another one of my dead projects, but I decided one day to just finish up and put it out there. That’s why there is a drop in quality about 1/2-3/4 of the way through the video (reusing animations and models). I was definitely not expecting this to become my most popular video, or for so many people to care about this topic. I did actually try to write the equations for ticked polygons in grease pencil, but it didn’t look good so I removed it. I will definitely try to make my follow up video to be better.
My first instinct for the infinite square was to put the errant angle in a klein bottle type situation so it can turn kind of upside down, which is what you did
@@celestialTangle My first idea was a square tape that twisted to put the odd corner on the under side. My issue with that was that the surface was not continuous. The surface that I figured out was an extension of that.
@@thomascurley8568 I believe so, but it would be impossible to represent in 3D space. You might be able to generate the 3D equivalent to a UV to represent it though. I think I might want to go into this in a follow up video.
I tried to talk about this during the parallelogram section by showing 2 sets of lines can have a line drawn between them that is perpendicular to both lines. To be honest though due to this being non euclidean geometry, parallel lines can't exist in most spaces.
Years ago, I learned some spherical trigonometry. After trying and failling to create a python function that inputted the coordinates of three points on a sphere and outputted the area of the triangle they form, I realized that any set of three points on a sphere can be used to define exactly 16 different triangles. I have not yet created satisfying visualizations of this. If you could that would be great!
@@viquezug3936 That would be a little tricky to do. Essentially what you would need is to convert your points to the spheres UV, and then draw lines between them. Drawing the lines is the hard part due the spherical space the UV represents. I would recommend looking into flight planning for airplanes, since they have to deal with this exact problem. (They want to go in a straight line/shortest path, but their maps/mapping software is 2d).
If two of the points are antipodal there are infinitely many triangles defined, as there are infinitely many straight (shortest) paths between antipodal points
I remember seeing his video. I liked his paper models. I actually still want to 3D print out my squares, but I don't think I will have the time any time soon.
Numberphile has a video about “squares” that exist on spherical and hyperbolic surfaces, they satisfy: having equal length sides, having straight sides (geodesics), and having all 90 degree internal angles. What’s weird is that they dont have 4 sides. Pretty interesting and perhaps mentionable in a future video
As far as I understand one broad definition of mathematics is the practice of defining simple rules and explore what the consequences of those rules are in a logical and content manner. So this is definitely mathematics in my opinion
I’m glad you mentioned the “270°” part, I was so prepared to leave a comment about that. I hated that about the original meme you started the video with, too. I will agree that Non-Euclidean Geometry is awesome, and I also love thinking about it. I even made an attempt a while back to design a game with heavy use of Non-Euclidean Geometry. I… didn’t get very far. While I’m great with programming and have plenty of ideas to spare, my ability to follow through with said ideas leaves a lot to be desired. Still, fun ideas to toy around with.
Geometricians: ugh, FINE, a shape with four equal STRAIGHT lines and four right angles _that is constrained to 2-dimensional euclidean space._ I didn't think we had to spell out that last part but here we are.
@@excrubulent that is the whole point of math, to spell things out exactly. Also I would argue that euclidian squares are constrained to euclidian space, and non euclidian squares are not.
Perhaps at some point in the future, the circle inhabiting the cylinder will come to perceive the thought and recognize its futility of being by asking the following question: "Is there more to this experience?". It will then return to its favorite pastime - moving forward.
2:29 You had an opportunity to call it the Square Case and you missed it by two fucking miles. Downvoted, ratiod, L, touch grass, all that shit copypasta. Nah but really I enjoy this type of maths. Very much so
Challenge: define a square as "a closed shape made of four straight line segments with four axes of reflectional symmetry, along with four fold rotational symmetry." Now make me squares that are not, well, squares.
The entire video was made in the 3D viewport of Blender. I have another video where I move the camera around to show some of the perspective tricks I used. I usually Blender's built in video editor to edit video, and I have made some short animations in the 3D viewport, but this is my first full video in the 3D view port. It was also only my second time using grease pencil. (How I was able to draw on the objects).
It's worth noting that the meme at the beginning of the video, and your "wave square" are *the same shape*. You just presented it on a cylindrical surface, but that meme presents it on a polar projection. Both of these are completely valid ways to present them; and they're equivalent.
Of course to meaningfully define a mathematical term requires that the type of math be designated (or assumed, which is usually sufficient). The normal assumption, lacking statement otherwise, is Euclidean geometry, which requires that it be in a two-dimensional flat plane.
I love non euclidian geometry. I am still planning on making 3d non euclidian chess in the future. I have already designed the board which is a 3d chess cube wrapped around a tesseract, or as I call it the chesseract.
I Present to thee: The Skew Polygon! --> pretrial duals --> I heard about these first from Jon Misali's Regular Polygon video P.S. Squares, defined as above, are not possible on Hyperbolic or Spherical geometry.
The definition is 4 equal sides with opposite sides parallel and all interior corners are 90° Also all squares are rectangles, but not all rectangles are squares.
this is for euclidean geometry, or flat surface. For non-euclidean geometry, the definition is just that they have equal sides and equal angles. Since the sum of interior angles may not be 360 degrees in curved surfaces, the angles don't have to be 90 degrees either. However, the "squares" in the video technically have some 270 degree angles, though he mentioned this problem already. This means all angles are not equal.
@@WaluigiGoesWa the closest thing I could find would be what are called skewed polygons, which are kind of similar but they don’t curve around a non Euclidean space, they are briefly highlighted in Jan misali’s video on there being 48 regular polyhedra.
@WaluigiGoesWa Oh yeah lol, I only heard of it from a Vsauce short. It's kinda neat, I don't know how to explain it well but it has to do with a specific way you can define the sides of a triangle. so, there's the incircle, circumcircle, and 9 point circle right? the sides of a triangle have endpoints on the circumcircle, midpoints on the 9 point, tangent to incircle. a guy named b.f. Sherman realized there's a 4th line you can draw that checks all those boxes, this: the fourth side of the triangle. should be easy to look up information on it. it's pretty funny
@@TimJSwan you can can probably make a square pattern a fractal, but since a square needs to have exactly 4 sides I don’t think you could make a square a fractal.
@@WaluigiGoesWaok but consider A square resting on a fractal 3d shape where the perimeter of the square wiggles about in the 3rd dimension but looks like a square from top down Infinite perimeter square
@@WaluigiGoesWa hilbert curve is very friend shaped. Somehow always reminds me of the square shaped Slitherlink puzzle. Loved doing those on road trips.
0:09 those sides aren't curved though, we're just looking at a polar projection of a sphere. expressed in polar coordinates (magnitude and angle), all four sides are straight and even axis-aligned
ye on instinct I'd agree. I think it's expected yet generally unstated that each "angle" of a square is its interior angle, and the right angle at a changed direction is the exterior angle
Wow, I never expected my little meme square would inspire someone to make a TH-cam video.
Great work!
Are you the original creator of the meme square?!
@@WaluigiGoesWa Yes I am. I just came up with it last September and made a post to r/mathmemes
@@alltheclovers532 Cool! I’m glad I saw your meme. It was really fun to work on this.
I love when topologists make ancient philosophers roll in their graves
@@keilafleischbein59 Tis my favorite hobby.
i hope you won't go into biology to create new types of creatures that technically qualify as "human"
Behold 4 kinds of humans:
1. Standardhuman
2. Wavehuman
3. Stairhuman
4. Infinitehuman
a human is a featherless bipedal.
train your dog to walk on hind legs.
behold, a human.
@@Duckilicious or you could breed chickens without feathers.
Behold, a man
@@solaridze I will make Diogenes proud, as I have plenty of ideas for new featherless bipeds.
we started by avoiding coins, and now we're inventing shape types
@@HarryMario_ true
I knew I recognised the channel name
guess we're making stair squares now
@@DMadHacks The squarecase
@@WaluigiGoesWa closed shapes
2:28 missed opportunity to call it a "Squarecase"
Not to be confused with Squarespace, the sponsor of-
The squarecase
People have mentioned that, and I agree, Squarecase is a great name.
@@OrchidAlloy not that sponsor
@@WaluigiGoesWa what is that emoji
> “infinite square”
> look inside
> finite
People have informed me that the shape is called a lemniscate so it would be more accurate to call it a lemniscate square.
@@WaluigiGoesWa ..or a lemnisquare
@@FranticErrors or an 8 square
Well “finite” is inside “infinite”, is it not?
@@doubleking7070 yea and also ‘effective’ is in ‘ineffective’ that doesn’t mean ineffective things are effective
5:29 "If we instead ask how many interior 90° angles they have"
What's _"interior"_ anyways? Such a Euclidian question to ask...
@@Pystro we can divide the surface into 2 distinct areas. We can say either area is the “inside”, but either one we choose there will always be 270 degree interior angles.
@@WaluigiGoesWa meanwhile, square on a torus:
@@NoNameAtAll2 True, this is an issue.
This is actually a reason I didn't go with my original idea for the infinite square, I originally considered a square tape that twisted on 2 edges to have 1 corner under the shape, however since I couldn't think of how to make the surface continuous I ended up redesigning it.
Isnt a torus Euclidean? Its flat (net 0 curvature). So anything drawn on it would obey Euclidean geometry, as of my understanding
@ the best way to describe euclidian geometry, is that the shortest path between 2 points is a straight line. If you were to take a torus and draw 2 points on opposite sides, going from one point to another would have 2 shortest paths, making it non euclidian.
the way the squares were floating and spinning at the end made it seem like they're items you can collect in a videogame
@@HalfBreadOrder this video was entirely made in the blender 3D viewport, using the NLA animation editor. As such I tried to reuse the animation of the first square spinning on all of the items.
I meant to say floating *and *spinning btw, I don't know why I only said floating.
@@HalfBreadOrder btw I just uploaded a behind the scenes video as well so you can see how I keep perspective in 3D space.
New square unlocked!
Good info thx
A Teacher teaching children about shapes "And this shape is called a Square-"
This Guy : *Bursts into the room holding a WaveSquare* "Behold! A Square!"
call me Diogenes
was looking for this comment
The Koolaid Man with "straight" lines doodled all over himself.
@@NickCombs klein bottle shaped cool aid man.
I read that last line in Dr Doofenshmirtz's voice
This would be Diogenes if there's internet back in ancient greece
@@mefuri_2 behold a square!
The square of man!
7:07 The fact that you explore non-Euclidean geometry out of sheer curiosity like this _makes_ you a mathematician. Don't try to escape it 😏
@@joe_z unfortunately I am a computer scientist instead 😓
@@WaluigiGoesWa Computer scientists are mathematicians too! 😛
@@WaluigiGoesWa so find some shoehorning way to apply the theories of parameterized computational complexity to your findings
Every good computer scientist is also a mathematician, lets be honest.
@@WaluigiGoesWa You just sub-classed into the electronics skill tree. No worries!
Just remember, is random people like you screwing around until you find something cool that has led humanity to where it is now
Thank you for the encouragement.
"the only difference between science and messing around, is writing it down!" Adam Savage
Square, Squave, Stuare, and Lemnisquare
@@stellarx20 someone recommended the stair square be called the squarecase.
@WaluigiGoesWa why not call it that?
@@HalfBreadOrder 👍
lemnisquare for the win
@@WaluigiGoesWa Didn't expect to see the video creator lol, look mum
Back in my day you had to climb an infinite staircase backwards just to go on a date. Only to find out your woman is in another man’s castle.
cant you put the staircase square on the cylinder too?, youd just have to put it on there at a 45° angle
@aprcktiplaal9293 You’re right! I actually spent some time thinking about the inverse. (Putting a wave square on a surface with protrusions)
To snugly fit a unit square, you would need a cylinder with a height of (root2)/2 and a radius of (root2)/pi.
Interestingly, by increasing the radius by increments of (root2)/2pi, we can create square-like shapes with progressively more sides. What would you call a shape with 6 or 8 same-length sides and all right angles?
@@Hankathan I would call an 8 sided ticked polygon that circles around a cylinder a 1 1 3 3 1 1 3 3 ticked octagon. For a ticked hexagon a different surface would be needed since each 2 edges alternate the side of the cylinder the vertices are on. (so points would be bottom, bottom, top top, bottom, bottom, with the next vertex supposed to be on the top due to the sequence, but that doesn't match the position of the first vertex which is on the bottom).
I feel like you would also be able to fit it onto a sphere as well
After careful consideration, it can technically fit, but corners must merge.
Two perpendicular great circles.
@asheep7797 but are opposing sides parallel to each other in this case?
Technically opposing lines would form a single continuous loop around the sphere.
If we also drop the rule that it has to be 4 sides, and only concentrate on the rules, has to have only right angles, and all side lengths are the same, then you could make a triangular square on a sphere and a pentagonal square on an inverse square/a horn shaped space
they are non euclidian polygons. I do like math and geometry, so this is pretty cool. You can get topology to behave however you want or need to get those repeating shapes to work. You can even place that square on a mobius strip.
@@christopherfreeman2858 when I was thinking of Ideas to make the infinite square, I considered a square tape that twisted. My biggest issue with it was that the surface was not continuous.
@WaluigiGoesWa Continuous means it has no sharp angles or jumps? A cylinder does violate that, but your using the continuous side. Mobius strips are continuous along the strip side, you have to be very gradual to get the right graphical effect. A discontinuity happens in the flat geometrical representation of any curve, mostly because what makes a continuity is a smooth, infinitely precise function. A lot of shapes are continuous, so long as you meet up perfect curves and don't create any sharp angles. Say, a rounded cube is still continuous.
@@christopherfreeman2858 I was referring to the singular edge of the mobius strip. If you go towards the edge while perpendicular to it, you will hit a "crease" where you flip to the other side. Using a cylinder instead of a plane to form the strip would work though. I have actually been thinking about how twisting could be used to generate a standardized surface for almost any ticked polygon.
@@WaluigiGoesWa You don't have to hit the crease/edge, in the same sense you don't have to hit the edge of a piece of paper to make a square
@@WaluigiGoesWa I drew a rampart pattern on a piece of paper, it uses consecutive 90 degree angles going left/right/right/left repeating, then joined it up like a mobius strip. If you want to, you can give it volume like a torus and keep the twist, but that seems pretty hard to recreate in a 3d program. Its also hard to show, you'd need to scale it up to get the right side lengths.
Start at the North Pole (NP). Travel in a straight line to the South Pole (SP). Turn left 90 degrees, and travel again in a straight line back to the NP. Right, then back to SP. Left, then back to NP. Turn right and you're facing back along your first line. Four straight lines, four 90 degree turns.
Interesting idea, however you do not have 4 straight line and 4 90 degree corners. You have 2 straight lines perpendicular to each other
@@Ladyoftheroundtable Not if you consider each line as ending whenever it reaches a vertex (pole).
@@BooVoidCat That’s one way to make a stair square.
generally polygons aren't allowed to have multiple vertices in the same location (here, there would be two at each pole), although this is just a convention that not everyone uses (ex. Skilling's figure and some papers by Branko Grunbaum)
@ These are called degenerate polygons. I had to look
into them when I was thinking about what ticked digons would look like.
the "infinite square" shape looks kinda like an inhaler
@@Scribblersys Apparently the name of the shape is a lemniscate.
Insqualer, not to be confused with the squarecase and it’s optimal form: squarepods
6:20 while I do appreciate the spinning shapes in the void, I'd appreciate more all those names, definitions, ando formulas written on the screen instead of just spelled out by voice.
@@KingShinyRotom To be honest I had made the majority of the video a couple of months ago, however I got busy with a bunch of work. Typically the video would have just become another one of my dead projects, but I decided one day to just finish up and put it out there. That’s why there is a drop in quality about 1/2-3/4 of the way through the video (reusing animations and models). I was definitely not expecting this to become my most popular video, or for so many people to care about this topic. I did actually try to write the equations for ticked polygons in grease pencil, but it didn’t look good so I removed it.
I will definitely try to make my follow up video to be better.
My first instinct for the infinite square was to put the errant angle in a klein bottle type situation so it can turn kind of upside down, which is what you did
@@celestialTangle My first idea was a square tape that twisted to put the odd corner on the under side. My issue with that was that the surface was not continuous. The surface that I figured out was an extension of that.
Wake up baby new squares just dropped
@@usernametaken017 When they say be there or be square, you can bet your ass I’m going to be square.
me shakinf my child to wake them up and tell them abouut the greatness of horrible shapes
4:00 these people should absolutely be considered squares
I can't wait for the video of this guy making new cubes with 4D corners
I have considered it. You just have to be able manipulate 4 dimensional space.
@@WaluigiGoesWa Well can you?
@@thomascurley8568 I believe so, but it would be impossible to represent in 3D space. You might be able to generate the 3D equivalent to a UV to represent it though. I think I might want to go into this in a follow up video.
@@WaluigiGoesWa Cool :)
0:16 Not just that, the opposing sides need to be parallel. Since square are rectangles and rectangles have parallel sides
I tried to talk about this during the parallelogram section by showing 2 sets of lines can have a line drawn between them that is perpendicular to both lines. To be honest though due to this being non euclidean geometry, parallel lines can't exist in most spaces.
I dislike their reasoning as this shape is as square as all of the ones they gave. The shape it exists on is just a cone
Something similar could exist on a sphere.
2:24 you should've called it a squarecase
I agree.
Exactly what I was thinking
I will teach that to my kid so he can confuse the teacher and show his superiority
Good. >:-)
I hope the teachers don't know the actual definitions of a square then...
That infinite square’s surface looks…interesting…
I did not expect non-Euclidean geometry from a channel named WaluigiGoesWa but great video
Years ago, I learned some spherical trigonometry. After trying and failling to create a python function that inputted the coordinates of three points on a sphere and outputted the area of the triangle they form, I realized that any set of three points on a sphere can be used to define exactly 16 different triangles.
I have not yet created satisfying visualizations of this. If you could that would be great!
@@viquezug3936 That would be a little tricky to do. Essentially what you would need is to convert your points to the spheres UV, and then draw lines between them. Drawing the lines is the hard part due the spherical space the UV represents. I would recommend looking into flight planning for airplanes, since they have to deal with this exact problem. (They want to go in a straight line/shortest path, but their maps/mapping software is 2d).
If two of the points are antipodal there are infinitely many triangles defined, as there are infinitely many straight (shortest) paths between antipodal points
@@somebodyuknow2507 Well, yeah, but that had not surprized me as much
i saw the 3 as an 8 and was wondering where the 5 more squares were for the whole time lol
@imdartt lol
That's right. It goes in the square hole.
No, not the square hole.
@@WaluigiGoesWa Yes, the square hole! 😈🔥
@@xtremefps_ 😭
“Stair Square”
Missed a golden opportunity to call it a “Squarecase” smh
A lot of people have said that and I am inclined to agree.
the first square with the curved edges is the same as a projection of the cylinder one, the lines are straight but the geometry of the space is curved
I was thinking on it and I'm pretty sure it could be projected on an hourglass shape pretty well.
@@felipecesconettomartins2097 correct 👍
@@jem5636 all that really matters is that the length of the cross section curve from the top to the bottom is half the circumference.
Codeparade solved a decades old geometry question, and you, made new squares. Math is really coming along.
I remember seeing his video. I liked his paper models.
I actually still want to 3D print out my squares, but I don't think I will have the time any time soon.
All these inventions forget to take into account that the squares lines need to be parallel
That's what I was trying to get at with the parallelogram example.
Numberphile has a video about “squares” that exist on spherical and hyperbolic surfaces, they satisfy: having equal length sides, having straight sides (geodesics), and having all 90 degree internal angles. What’s weird is that they dont have 4 sides. Pretty interesting and perhaps mentionable in a future video
@@pr0hobo I remember watching that, someone recommended it when I made the original animations for Discord.
Lemniscate square instead of infinity. Lemniscate is the shape name.
@@gljames24 I didn’t know there was a shape for the curve. Thank you for telling me!
Tried to combine lemniscate and square to make lemisquare but that just sounds like lemonsquare
They all still go in the square hole.
3:30 nice airpod dude
First time someone called it an airpod. I always thought it looked like a fox if you turn it in the right direction.
airpod for the airpod case??
@@oshotz squarepod
3:05 airpod case
@@rhebucks_zh
squarepods
As far as I understand one broad definition of mathematics is the practice of defining simple rules and explore what the consequences of those rules are in a logical and content manner. So this is definitely mathematics in my opinion
@@SteinGauslaaStrindhaug
I was so confused, I thought the thumbnail was an air conditioner
It is a single frame from the initial meme that I made after seeing the original image.
I’m glad you mentioned the “270°” part, I was so prepared to leave a comment about that. I hated that about the original meme you started the video with, too. I will agree that Non-Euclidean Geometry is awesome, and I also love thinking about it.
I even made an attempt a while back to design a game with heavy use of Non-Euclidean Geometry. I… didn’t get very far. While I’m great with programming and have plenty of ideas to spare, my ability to follow through with said ideas leaves a lot to be desired. Still, fun ideas to toy around with.
Geometricians: ugh, FINE, a shape with four equal STRAIGHT lines and four right angles _that is constrained to 2-dimensional euclidean space._ I didn't think we had to spell out that last part but here we are.
@@excrubulent that is the whole point of math, to spell things out exactly. Also I would argue that euclidian squares are constrained to euclidian space, and non euclidian squares are not.
That is not a geometer*'s definition of a square but ok
I do not care if this is a joke.
@@miners_haven I don't know if you're trying to correct the word "geometrician" but either word is fine.
@@WaluigiGoesWa I think that's all fair, I was just making a joke about it.
3:27 AirPods with a stand
Perhaps at some point in the future, the circle inhabiting the cylinder will come to perceive the thought and recognize its futility of being by asking the following question:
"Is there more to this experience?".
It will then return to its favorite pastime - moving forward.
Nah the circle is an NPC living a planetoid like the characters in Super Mario Galaxy.
Behold, Plato's square
i love listening to my square waves on my square airpods
Squarepods.
5:48 "If these polygons are not squares... What are they?"
Squaren'ts.
@@chaoticsilver8442 lol
An octet of a sphere is technically a three sided rough equivalent to a square.
@@nullpoint3346 Numberphile made a great video about this. He also talks about 5 sided square equivalents in hyperbolic space.
3:01 hang on… that’s just a square tennis ball!
6:03 missed opportunity to call them squareoids, though I dont know if the name has already been used by something else
@@agentedelta2272 I meant it for poygons with n sides, not just for squares
Squaroids are just square like. Like a squircle (circle square) would be a squareoid. Also squircles are fun
Non-Euclidean? So koalas don't eat it?
That's right, it goes in the square hole!
Oh god the horrors
Polygon.
Square
2:29 You had an opportunity to call it the Square Case and you missed it by two fucking miles. Downvoted, ratiod, L, touch grass, all that shit copypasta.
Nah but really I enjoy this type of maths. Very much so
@@DGEddieDGEtm Other people have said the same thing and I personally agree.
This is the feedback I needed.
infinite square? more like squarepods case
(note : type this, go back to video, mentioned in video in under a second)
@@Be-lo_da_fluff lol
Nah man, thats a squair
@root4217 a Saquer if you will
All of these "squares" break apart when the part of the definition regarding planes gets introduced.
These are non eucldian, so they don't have to be on a plane.
Challenge: define a square as "a closed shape made of four straight line segments with four axes of reflectional symmetry, along with four fold rotational symmetry."
Now make me squares that are not, well, squares.
Sure a degenerate square with all points at the same vertex follows the rules you laid out, but does not appear as a traditional square.
"Behold, Plato's square!"
@@wompastompa3692 I think it would be Diogenes’s square.
@@WaluigiGoesWa
It was Diogenes who said, “Behold, Plato’s man!” when presenting a featherless biped (plucked chicken)
1:00 It's basically a square wave : ) ... edit: Or a "wave square" if you like!
@@SendyTheEndless that’s why I named it that.
Calling something an "Infinite Square" isn't confusing at all.
@@margaret233 People have suggested renaming it to the lemniscate square since that is the official name of the curve.
this is like something matt parker would post about
Hadn't heard of him before. Looks like stand up maths is a pretty interesting channel.
assumed to be modeled in blender
The entire video was made in the 3D viewport of Blender. I have another video where I move the camera around to show some of the perspective tricks I used.
I usually Blender's built in video editor to edit video, and I have made some short animations in the 3D viewport, but this is my first full video in the 3D view port. It was also only my second time using grease pencil. (How I was able to draw on the objects).
@@WaluigiGoesWa Yes, if you need something to rapidly develop showcasing without getting into the depth with bells and whistles, try Godot.
1:47 Drummers goin crazy with this one
love this
oh boy non-euclidean snare drills
Missed opportunity to call the stair square a squarecase and the infinity symbol is called a lemmiscate so... lemnisquare
agreed
It's worth noting that the meme at the beginning of the video, and your "wave square" are *the same shape*. You just presented it on a cylindrical surface, but that meme presents it on a polar projection. Both of these are completely valid ways to present them; and they're equivalent.
@@sodiboo that is correct. That observation is what actually inspired me to look for other squares.
This man put two stair squares side by side and thought we wouldn't notice
Huh?
i have never been so delighted and distraught to realize something is a square, or in fact any regular polygon
@@existenceispain_geekthesiren more regular polygons are getting their ticked version in the future.
@WaluigiGoesWa i am in despair but also very excited
@@existenceispain_geekthesiren
This is some excellent topologist slander.
Of course to meaningfully define a mathematical term requires that the type of math be designated (or assumed, which is usually sufficient). The normal assumption, lacking statement otherwise, is Euclidean geometry, which requires that it be in a two-dimensional flat plane.
@@crawkn thats why these are non euclidian squares
You can't just create squares and get away with it!
@@pncka ▪️▪️▪️▪️▪️▪️▪️muwahaha
bro has discovered non-euclidian geometry 💀
I love non euclidian geometry. I am still planning on making 3d non euclidian chess in the future. I have already designed the board which is a 3d chess cube wrapped around a tesseract, or as I call it the chesseract.
I saw this video and then refreshed my youtube home page, I spent way too long just trying to find it again
I am just supprised the algorithm picked it up.
The AirPods one could be cool for a videogame loading screen
maybe
Numberphile did something similar 6 years ago
@@mozzapple I think you’re referring to the three sided and five sided polygons with 90° interior angles. That was a pretty cool video tbh.
i am waiting for Vsauce to see this...
@@Titanium.22_ you’re the first person to mention Vsauce and not Matt Parker.
Topologists will look at all of these and see spheres
You could deform all of my surfaces to spheres for sure.
I like the cylinder one, but I wonder if the other two have more elegant formulations.
People have brought to my attention that the stair square can be put on a cylinder at a 45 degree angle in a triangle wave pattern.
I Present to thee: The Skew Polygon!
--> pretrial duals
--> I heard about these first from Jon Misali's Regular Polygon video
P.S. Squares, defined as above, are not possible on Hyperbolic or Spherical geometry.
The definition is 4 equal sides with opposite sides parallel and all interior corners are 90°
Also all squares are rectangles, but not all rectangles are squares.
this is for euclidean geometry, or flat surface. For non-euclidean geometry, the definition is just that they have equal sides and equal angles. Since the sum of interior angles may not be 360 degrees in curved surfaces, the angles don't have to be 90 degrees either. However, the "squares" in the video technically have some 270 degree angles, though he mentioned this problem already. This means all angles are not equal.
That is DEFINITELY an airpod case.
6:00 the wave and stair squares are apeirogons which on the right 3d surface make a closed polygon
@@morgan0 If they are in euclidian space I would say they are a pattern or something close to a tiling.
@@WaluigiGoesWa the closest thing I could find would be what are called skewed polygons, which are kind of similar but they don’t curve around a non Euclidean space, they are briefly highlighted in Jan misali’s video on there being 48 regular polyhedra.
@@Red_Ryry I can see where you are coming from, but they are pretty different since they are 3D lines, instead of 2D non euclidian lines.
Squares are typically 2 dimentional objects and not wrapped around a 3d shape
Can't wait for this to somehow get picked up by Matt Parker ^.^
Someone else mentioned him. I thought his channel looked pretty cool.
This gives "triangles have a 4th side" energy and I love that.
@@chermal7311 never heard that before
@@WaluigiGoesWalook up “Vsauce 4 sides to a triangle”
@WaluigiGoesWa Oh yeah lol, I only heard of it from a Vsauce short. It's kinda neat, I don't know how to explain it well but it has to do with a specific way you can define the sides of a triangle. so, there's the incircle, circumcircle, and 9 point circle right? the sides of a triangle have endpoints on the circumcircle, midpoints on the 9 point, tangent to incircle. a guy named b.f. Sherman realized there's a 4th line you can draw that checks all those boxes, this: the fourth side of the triangle. should be easy to look up information on it. it's pretty funny
@@IcePhoenixMusician Just watched it. It's a pretty cool fact that you can add a line that matches all of the points!
Can't you make a fractal square?
@@TimJSwan you can can probably make a square pattern a fractal, but since a square needs to have exactly 4 sides I don’t think you could make a square a fractal.
@@WaluigiGoesWaok but consider
A square resting on a fractal 3d shape where the perimeter of the square wiggles about in the 3rd dimension but looks like a square from top down
Infinite perimeter square
@@aogasd Have you ever looked into space filling curves? I am quite fond of the Hilbert curve.
@@WaluigiGoesWa hilbert curve is very friend shaped. Somehow always reminds me of the square shaped Slitherlink puzzle. Loved doing those on road trips.
The microphone quality proves that a tool is only as good as the master
I have at least gotten a little better since now I record myself using Audacity instead of Microsoft sound recorder.
0:09 those sides aren't curved though, we're just looking at a polar projection of a sphere. expressed in polar coordinates (magnitude and angle), all four sides are straight and even axis-aligned
@@sodiboo thats actually a really good idea. It explains why it maps onto the cylinder so well as the wave square.
2:35 missed opportunity to call it a "squared"
2:27 SQUARECASE WAS RIGHT THERE DUDE
People have mentioned that. I should definitely call it that in the future.
I think this actually is a form of a weird non-Euclidean trapezoidal dihedron tiling (at least the first one I think) :D
I am not sure since the 2 sides of the cylinder are not degenerate
@@WaluigiGoesWayeah so it’s a non-Euclidean dihedron I think lolll
@kro_me I don't think it would be since a dihedron is made up of 2 faces, and these surfaces have more than 2.
4:33 the lines traveling down the length of the cylinder are clearly not parallel; they intersect at the poles.
@@sodiboo I was saying that they were parallel since any straight line that is perpendicular to one line is perpendicular to the other.
3:13 woah that looks like an AirPod case
Since two of the corners turn left instead of right, wouldn't that make those 270 degrees instead of 90?
ye on instinct I'd agree. I think it's expected yet generally unstated that each "angle" of a square is its interior angle, and the right angle at a changed direction is the exterior angle
@@NorthOfEarth @metamusic64 I talk about this at the end of the video.
This video can only make me think of that “STOP DOING MATH, NUMBER WERE NOT MEANT TO BE GIVEN NAMES” image
I am the one who studies non euclidian geometry
not calling the staircase square a squarecase is a blunder of galactic proportions
@@MaximumWoahverdrive dang your right. I need to make this correction along with the infinite square being the lemniscate square.
This is some Diogenes type tomfoolery
It really is
7:26 I am interested.
you could make a stair square on a cylinder
You just have to rotate it 45 degrees.