@@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!"
@@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.
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.
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)
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.
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.
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.
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.
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.
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
1:15 Squares are part of the closed polygons family, and I was going to say that, one important factor of closed polygons is that they enclose a finite space and yours on the non-eucliean cylinder surface doesn't, it'd divide the curved plane in two infine and symmetrical spaces. It's like drawing a circle around a torus. It isn't a circle, it's a straight line. There's no inside and outside
But unlike with a line on a torus you can separate the cylindrical surface into 2 halves, meaning you could classify one half as the inside, and one as the outside.
@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
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
i was hoping you would mention the triangle you can draw on a spherical surface that has three 90º angles, three equal sides. its like a type of equilateral triangle, that also is a non-euclidean square of sorts
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.
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.
The “behold a square” image you showed at the beginning I believe is part of a collection of other Shapes that more loosely follow the rules of being a square. It was part of one of those alignment chart images
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.
@@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
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.
Correct me if I'm wrong, but it looks like the staircase square could be embedded in a toroidal space? It wraps around normally in both directions on the flat plane, which matches up with how toroidal geometry behaves. (For those who don't know what I'm talking about, a torus is a donut shape. The reason you can get that shape from the wrapping-around thing is because if you take a flat plane, curl it around and glue one pair of opposite edges together to get a cylinder, and then curl the cylinder around and glue the other pair of edges together, you get a torus.)
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).
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.
On a cone the curved sections can not be "curves of equal height" though. (And they also aren't conical sections.) Although there certainly exist curves that are "straight" on a cone. But the top and bottom sides not being curves of equal height also means that the straight looking sides can't simply be pointed at the tip of the cone and have to be the same kind of "funny curves" as the curved looking sides. Which makes it a nightmare to calculate what kind of properties the cone would need to fit such a "square".
These look related to spherical geometry (at least as far as all these shapes are topographically spherical). What I think you are discovering is that in these spaces the interior angles of a polygon can get very screwy, for example with right side length you can draw a triangle with 3x90° angles. I think the definition of a square in such space is a 4 sided polygon with equal side-lengths and angles. A square is to quadrilaterals as an equilateral triangle is to just triangles. Your ticked polygons are I think formally called concave polygons. Have a go with solids with holes in as well.
@@petoperceptum I like your analogy comparing a square to an equilateral triangle. Since my ticked polygons have required angles I would say they are a subset of polygons that includes all regular polygons (which are convex), and some concave polygons.
to avoid curving sides, we can just cure spacetime itself also i guess you can make a time sqare, it would like a line appearing for a frame, then two dots for some time and then line again
True. You could also have the surface exist in the 4th spatial dimension for only seeing some of the lines. (I do kind of wonder if you could invent new cubes as well by manipulating 3d space in 4 dimensions)
@WaluigiGoesWa screw space time, literally funniest way to define a square i came up with so far is placing 4 points in a singularity, saying that they are 1 unit apart from each other and saying that only 90° angles are possible because it's a singularity, nothing has meaning anyway, so why cares if i made up a thing or two
In france we have an additional rule for what is a square and its that if you draw a line from corner to center, every corner intersect in the middle and their opposite. Which with yours is not possible, and so in france it wouldn't be a square
0:07 I would say the meme square counts, because you could argue that the curve is only a matter of perspective, or again, Euclidean vs non Euclidean, as you can make the shape with the edges of a piece of paper by curving one side of the paper. The paper itself is three dimensional, and therefore a rectangular prism, not a square, but the edges themselves follow all the rules of a square, but just shifted.
I typed out an entire comment about the 270 degree interior angles that disqualified all of them being squares but then you mentioned it and I felt stupid for thinking you wouldn't have already realised it
The "stair" square can actually be placed on a cylinder due to the fact that the curved surface can be unwrapped into a plane that repeats along two edges. Also, I think there might be a way to put the "infinite" square on a Möbius strip but I might be wrong.
Love this video. I love taking particular math definitions and being like... Okay, what else fits that? If you enclude the necessity of the 90degree angles being all interior, then you're finally limited to only the normal square (I think). As soon as you remove that though, you get a lot of interesting shapes to play around with.
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
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.
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
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
> “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
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
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.
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!
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
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
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
This would be Diogenes if there's internet back in ancient greece
@@mefuri_2 behold a square!
The square of man!
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.
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.
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.
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.
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
I did not expect non-Euclidean geometry from a channel named WaluigiGoesWa but great video
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 :)
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.
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
“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.
I will teach that to my kid so he can confuse the teacher and show his superiority
Good. >:-)
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.
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.
i saw the 3 as an 8 and was wondering where the 5 more squares were for the whole time lol
@imdartt lol
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.
1:47 Drummers goin crazy with this one
love this
oh boy non-euclidean snare drills
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.
That is DEFINITELY an airpod case.
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’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.
3:05 airpod case
@@rhebucks_zh
squarepods
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.
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
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
1:00 It's basically a square wave : ) ... edit: Or a "wave square" if you like!
@@SendyTheEndless that’s why I named it that.
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.
This is some excellent topologist slander.
1:15 Squares are part of the closed polygons family, and I was going to say that, one important factor of closed polygons is that they enclose a finite space and yours on the non-eucliean cylinder surface doesn't, it'd divide the curved plane in two infine and symmetrical spaces.
It's like drawing a circle around a torus. It isn't a circle, it's a straight line. There's no inside and outside
But unlike with a line on a torus you can separate the cylindrical surface into 2 halves, meaning you could classify one half as the inside, and one as the outside.
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
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.
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.
You should collab with jan Misali and create technically correct cubes
(for context, jan Misali created a video about the 48 regular polyhedra)
i was hoping you would mention the triangle you can draw on a spherical surface
that has three 90º angles, three equal sides.
its like a type of equilateral triangle, that also is a non-euclidean square of sorts
I saw Numberphile cover that. It was pretty interesting.
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.
That's right, it goes in the square hole!
Oh god the horrors
No you did not, because squares are plane shapes by deffinition.
You can't just apply some rule and ignore others to define something.
Behold, Plato's square
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.
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.
We can make a square and a circle one in the same by imposing a non-Euclidean metric on the plane.
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.
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.
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.
The “behold a square” image you showed at the beginning I believe is part of a collection of other Shapes that more loosely follow the rules of being a square. It was part of one of those alignment chart images
I just saw it posted around Discord a bunch. I don't know if there is more to the image.
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
The AirPods one could be cool for a videogame loading screen
maybe
i love listening to my square waves on my square airpods
Squarepods.
The first shape can be made by curlinh up the corners of a square
(the one in the "behold a square" that looks like a baseball field)
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
Next, make a square on a mobius strip.
@@Sturmischer you could just draw a standard square on it lol.
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.
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.
Wow, I never expected my little meme square would inspire someone to make a RUclips video.
Great work!
The square and triangle (bill and ford animatic) got some competition
Squares are typically 2 dimentional objects and not wrapped around a 3d shape
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.
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.
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 is like something matt parker would post about
Hadn't heard of him before. Looks like stand up maths is a pretty interesting channel.
Correct me if I'm wrong, but it looks like the staircase square could be embedded in a toroidal space? It wraps around normally in both directions on the flat plane, which matches up with how toroidal geometry behaves.
(For those who don't know what I'm talking about, a torus is a donut shape. The reason you can get that shape from the wrapping-around thing is because if you take a flat plane, curl it around and glue one pair of opposite edges together to get a cylinder, and then curl the cylinder around and glue the other pair of edges together, you get a torus.)
Nah man, thats a squair
@root4217 a Saquer if you will
The stair square can also be put on a cylindrical surface by making the lines at a 45° angle to the axis of the cylinder
@@sDuAvTaTjAe that is correct. A few other people have mentioned that as well. I might have to make it in the follow up video.
Topologists will look at all of these and see spheres
You could deform all of my surfaces to spheres for sure.
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.
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.
The stair square can be also represented on a sphere
You can also represent a wave square on a sphere (Both squares would be degenerate on a sphere though).
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 first square that you dismissed as not a square because some sides are curved fits neatly on a cone
@@ugielka technically it would have the same classification as the wave square that I showed right after.
On a cone the curved sections can not be "curves of equal height" though. (And they also aren't conical sections.) Although there certainly exist curves that are "straight" on a cone. But the top and bottom sides not being curves of equal height also means that the straight looking sides can't simply be pointed at the tip of the cone and have to be the same kind of "funny curves" as the curved looking sides.
Which makes it a nightmare to calculate what kind of properties the cone would need to fit such a "square".
"behold, a square"
missed opportunity to name the last one the lemnisquare
or the lemniscate square
Certified 270 degree triangle moment.
Such a waste of opportunity to name it Parker squares
The parker square already has a definition though
..and the sides need to face each other for it to be a square too
@@LunarRumour that is part of a square being a parallelogram.
These look related to spherical geometry (at least as far as all these shapes are topographically spherical). What I think you are discovering is that in these spaces the interior angles of a polygon can get very screwy, for example with right side length you can draw a triangle with 3x90° angles. I think the definition of a square in such space is a 4 sided polygon with equal side-lengths and angles. A square is to quadrilaterals as an equilateral triangle is to just triangles.
Your ticked polygons are I think formally called concave polygons.
Have a go with solids with holes in as well.
@@petoperceptum
I like your analogy comparing a square to an equilateral triangle.
Since my ticked polygons have required angles I would say they are a subset of polygons that includes all regular polygons (which are convex), and some concave polygons.
This is some Diogenes type tomfoolery
It really is
You could've made the stair square on a cylinder like the last one by wrapping a zigzag pattern around a cylinder
That is true.
to avoid curving sides, we can just cure spacetime itself
also i guess you can make a time sqare, it would like a line appearing for a frame, then two dots for some time and then line again
True. You could also have the surface exist in the 4th spatial dimension for only seeing some of the lines.
(I do kind of wonder if you could invent new cubes as well by manipulating 3d space in 4 dimensions)
@WaluigiGoesWa screw space time, literally
funniest way to define a square i came up
with so far is placing 4 points in a singularity, saying that they are 1 unit apart from each other and saying that only 90° angles are possible
because it's a singularity, nothing has meaning anyway, so why cares if i made up a thing or two
In france we have an additional rule for what is a square and its that if you draw a line from corner to center, every corner intersect in the middle and their opposite. Which with yours is not possible, and so in france it wouldn't be a square
0:07 I would say the meme square counts, because you could argue that the curve is only a matter of perspective, or again, Euclidean vs non Euclidean, as you can make the shape with the edges of a piece of paper by curving one side of the paper. The paper itself is three dimensional, and therefore a rectangular prism, not a square, but the edges themselves follow all the rules of a square, but just shifted.
“I’ll go to bed early tonight”
Me at 2 a.m.
I typed out an entire comment about the 270 degree interior angles that disqualified all of them being squares but then you mentioned it and I felt stupid for thinking you wouldn't have already realised it
The "stair" square can actually be placed on a cylinder due to the fact that the curved surface can be unwrapped into a plane that repeats along two edges.
Also, I think there might be a way to put the "infinite" square on a Möbius strip but I might be wrong.
you could make a stair square on a cylinder
You just have to rotate it 45 degrees.
Can't wait for this to somehow get picked up by Matt Parker ^.^
Someone else mentioned him. I thought his channel looked pretty cool.
I'm pretty sure the Stair Square would also work in a cylinder if the edges were at 45 degrees relative to the top/bottom of the cylinder.
It would work! A few people have brought up that design, so I think I might feature it in the next video I make on ticked polygons.
Love this video. I love taking particular math definitions and being like... Okay, what else fits that? If you enclude the necessity of the 90degree angles being all interior, then you're finally limited to only the normal square (I think). As soon as you remove that though, you get a lot of interesting shapes to play around with.
@@jem5636 It was really fun to think of these models. I am glad you enjoyed!
2:27 SQUARECASE WAS RIGHT THERE DUDE
People have mentioned that. I should definitely call it that in the future.