You talked about the King and Rook in Chess, but some of the weirdest "circles" come from the other pieces. A bishop for instance can only ever access half of the squares on the board, so it's "circles" would look similar to a Rook's, but filled with holes. And knight is even stranger. Because of its specific movement pattern, it takes four movements just to get to the square one space diagonally to it, in which time it could have also travelled to the complete other side of the board. So as you can suspect, it has some really weirdly shaped "circles."
Ok, you wanted examples, not just corrections, so: Minecraft uses a weird mix of taxicab balls, Chebyshev balls, axis-aligned cubes and actual Euclidean-distance balls. In some instances, it gets even weirder---water flow is taxicab on a flat surface but gets all kinds of weird in 3D. It has memory, so its circles don't just depend on the current state but also on what was there before. The circles for a water block that never had any block below it and one that had a single solid block under it at one point in time are very different.
Thanks for the video. This is the fun side of math that is often lost in math class. Also, the fact that you're making this video with editing and that I chose to watch it also play a big role
Thank you so much for the positive comment. I tell my students something similar, that we do a good job of taking all the fun and interesting things out of mathematics and teach the rest the students in school. A couple researchers talked about nutritious math versus tasty math. Why can't wait expose the kids to some tasty math?
Diamond does have a definition in math: 2 equilateral triangles sharing an edge. From this we get the word polyiamond, meaning a shape formed by several triangles sharing edges. This is similar to the word polyomino meaning a shape formed by several squares sharing edges, derived from the word domino.
I have to imagine the growth pattern of plants fits this circle logic in some way, based on the limits of the distance nutrients can travel through them.
Reminds me of data mining class in university. Although we used the term "Manhattan Distance" instead of taxis ;) I also think the term equidistance is a lot more suited for the non-euclidean cases than circle as the circle is just the special case of an euclidean equidistance
3:17 Ah, the cardinal sin of orienting the chessboard incorrectly. I've heard some people claim, jokingly or otherwise, that non-chess players seem to orient the board incorrectly more often than not, even though pure random guessing should yield a 1 in 2 chance. That's probably just confirmation bias speaking, but it honestly feels that way sometimes.
Aah! I'm only a casual chess player, but when I sit down and play with my kids I make sure that it is oriented the correct way. I didn't even think about checking that visual!
A while back I was thinking about how if you draw the best approximation of a circle possible with 2 color pixel art, then regardless of the resolution the circumference is always the same as the perimeter of a square of the same width/height since it's all right angles. Therefore, for pixel circles, pi = 4.
I really applied the concept of this kind of norm when i study computer vision. The pictures are just a bunch of pixels so a lot of picture manipulation sometimes involves a waffle like distance rather than normal Pythagoras.
This video reminded me of how in Dutch the word 'kring' is super weird to translate because it can refer to a ring or loop or even a quasi-concentric formation of people or things that don't necessarily look circular depending on spacial restrictions, but never an actual geometric circle, because that's a circle.
Reminds me of I had a programming assignment the other day where I had to make a diamond pattern and while I was supposed to print a certain number of characters in each row what I did instead was make the alternating grid and carve a circle from the center because it was small enough for me to get away with just using a radius and not printing anything outside it
This is intensely nerdy (also potentially incorrect) but Final Fantasy 14, the price for teleporting is determined by how many loading screens you'd encounter to get to your destination, no matter the map size or shape. Airship travel is ignored though so it's often cheaper to ride the airship first then teleport within the destination. Prices are complicated further by the fact that every player can have 3 half price locations and 1 free location (which is on a cooldown)
In Bare Knuckle Fighting Championship, the ring is referred to as the "Squared Circle." I've never understood why. I've tried looking up an explanation but found nothing. My best guess is that the circle has a particular square footage, or there is some cultural connotation.
You could create a mathematical definition for a diamond. The issue that many math teacher space is that students will see a square that is oriented with vertex up, and not consider it a square, but a different object with different properties.
Patrick, did you press the Sin button on your calculator? No, I pressed the Squine button. What do you mean Squine button? Its right here! It means the Sin of a square! “…”
Like a previous comment, I know these as a kite. The reason I emphasized that there is no diamond in mathematics is because I know mathematicians that really get bothered by people calling those kind of shapes diamonds. I don't consider myself a mathematician, since I don't have a PhD in mathematics, so I was trying to be true to the field by emphasizing that. However, consider this my apology if am incorrect. It may be the case that this is still debated in the field of mathematics and maybe I should have done more research.
@@MathTheWorld I DO have a maths degree from my university days (2nd class Honours - that was a TOUGH year). You and other commenters may be 100% right and I could be wrong. I don't specifically remember "diamond" ever being used during academic studies but growing up it was definitely used privately where I grew up - this may change over time and from place to place until things get formalized through internationally accepted education. I'm happy to also apologize if I am wrong - it was 25 years ago that I did my studies and in retrospect @jmvr's "kite" seems more correct to me now as to what I described above than what a younger me would believe and perhaps I was previously poorly educated. I do appreciate everyone's contribution. Hug?
WOW, you have actually hit a critical point right there. What is the curvature of a Manhatten space, or a Chebyshev space, or any other space that has a different metric function from the euclidean metric? Unfortunately, I have no knowledge about this, I am here to pose the right questions, not to answer them. 😅 Hopefully more knowledgeable people will enlighten us!
Your comment has left me curious about a generalised form of Pythagoras' theorem for non-euclidean geometries. But I don't think this video ever gave a definition of a square, so the relevance of Pythagoras or curvature is going to be really dependent on your definition of a square. You could say that a square has four edges, four vertices that all have an interior angle of 90°, 4 lines of symmetry, 4-fold rotational symmetry, etc., which works but isn't very generalisable. Or you could go for something more broadly useful, like the 2d case of a shape where all edges that meet at a vertex are orthogonal. The former means squares only exist on planes and doesn't do much to indicate any relation to cubes, hypercubes, etc. The latter says that squares are part of the family of hypercubes, but it also gives the freedom to work outside of Euclidean geometries: as an example, on a spherical surface you can have a triangle with all interior angles being right angles, and that's a square under this definition. Also, there are a couple of different types of squircles, but they all have circle-like properties at the 'corners' and approach straight lines at the centre of the edges, so this isn't a squircle. It looks a bit like what you'd get if you projected a square onto the surface of a sphere?
@@lawrencebates8172 I think the definition which I am going to enlist, inspired by your observations, is the one that we should use. We all know that a square in euclidean geometry is a shape with 4 edges, we absolutely don't consider a shape with other than 4 edges a square. However, of course this isn't enogh because there are many shapes with 4 sides that aren't squares like rhombus or a rectangle, what makes squares distinguishable is the act that it has same length for all its sides and same interior angles, we call a polygon with these properties regular. From this analysis, the definition of a square in general should be the following: *A regular 4-sided polygon.* The above definition works for any geometry, the only difference is that polygons edges in other geometries are the geodesics instead of straight lines.
Of course we want to see a video on your square circle drawing compass!!
please
I bet matt parker would too. If only i could get this in front of him
Ok, I will start working on that video with my team!
0:59... that's not how a compass works... the distance between 2 arms of a compass define the radius, not the diameter
I said that out loud as if the narrator could hear me.
It seems fitting that someone with "tau" in their username said "the radius, not the diameter". Is that a stretch? That might be a stretch.
@@isavenewspapers8890 well now we've got to invent the Tau compass...
@ So a standard compass?
A pi compass would be one that somehow draws a circle based on its diameter
Finally, a circle that can fit in a square hole.
😨
* crying Alison Burke noises *
splat tim..
You talked about the King and Rook in Chess, but some of the weirdest "circles" come from the other pieces. A bishop for instance can only ever access half of the squares on the board, so it's "circles" would look similar to a Rook's, but filled with holes. And knight is even stranger. Because of its specific movement pattern, it takes four movements just to get to the square one space diagonally to it, in which time it could have also travelled to the complete other side of the board. So as you can suspect, it has some really weirdly shaped "circles."
“All these squares make a circle. All these squares make a circle. All these squares make a circle. …”
if i didn't see this commented i was going to do it myself
Mr. Popo dropped a gallon of hallucinator, again.
Kami, I want you to tell me I can leave the lookout if I want to
Yes Mr. Popo-
DON'T TELL ME WHAT TO DO
TIL waffle irons can square the circle.
If only the ancient Greeks had thought of that one.
I cannot believe I missed that connection! It seems so obvious now. Nice job!
Ok, you wanted examples, not just corrections, so:
Minecraft uses a weird mix of taxicab balls, Chebyshev balls, axis-aligned cubes and actual Euclidean-distance balls. In some instances, it gets even weirder---water flow is taxicab on a flat surface but gets all kinds of weird in 3D. It has memory, so its circles don't just depend on the current state but also on what was there before. The circles for a water block that never had any block below it and one that had a single solid block under it at one point in time are very different.
Thanks for the video. This is the fun side of math that is often lost in math class. Also, the fact that you're making this video with editing and that I chose to watch it also play a big role
Thank you so much for the positive comment. I tell my students something similar, that we do a good job of taking all the fun and interesting things out of mathematics and teach the rest the students in school. A couple researchers talked about nutritious math versus tasty math. Why can't wait expose the kids to some tasty math?
DND uses chebyshev distance too so if you're paying on a grid, any spells that are spheres or circles are actually cubes or squares
Only in modern D&D. In previous versions of D&D, every even-numbered diagonal move is a double distance space.
Diamond does have a definition in math: 2 equilateral triangles sharing an edge. From this we get the word polyiamond, meaning a shape formed by several triangles sharing edges. This is similar to the word polyomino meaning a shape formed by several squares sharing edges, derived from the word domino.
I have to imagine the growth pattern of plants fits this circle logic in some way, based on the limits of the distance nutrients can travel through them.
HOW DID ALL THESE SQUARES MAKE A CIRCLE ??!!
To a topologist it's obvious lol, any closed curve is a circle
all these squares make a circle all these squares make a circle all these squares make a circle
3:22 We'd even have some alternative circles if castling were on the table.
You are right! I wish I would have thought of that!
Now that you mention the title, the answer is immediately obvious. Very cool observation.
I love when someone gives me a whole new point of view of seeing things. Subscribed Instantly
Reminds me of data mining class in university. Although we used the term "Manhattan Distance" instead of taxis ;)
I also think the term equidistance is a lot more suited for the non-euclidean cases than circle as the circle is just the special case of an euclidean equidistance
yeah ive always known it as manhattan too :)
3:17 Ah, the cardinal sin of orienting the chessboard incorrectly. I've heard some people claim, jokingly or otherwise, that non-chess players seem to orient the board incorrectly more often than not, even though pure random guessing should yield a 1 in 2 chance. That's probably just confirmation bias speaking, but it honestly feels that way sometimes.
Aah! I'm only a casual chess player, but when I sit down and play with my kids I make sure that it is oriented the correct way. I didn't even think about checking that visual!
@ Alright, understandable :)
This is why there's such a big to-do about the shape that's hit by a fireball in D&D.
squine and cosquine are my new favorite functions thank you 🙌
A while back I was thinking about how if you draw the best approximation of a circle possible with 2 color pixel art, then regardless of the resolution the circumference is always the same as the perimeter of a square of the same width/height since it's all right angles. Therefore, for pixel circles, pi = 4.
I really applied the concept of this kind of norm when i study computer vision. The pictures are just a bunch of pixels so a lot of picture manipulation sometimes involves a waffle like distance rather than normal Pythagoras.
This video reminded me of how in Dutch the word 'kring' is super weird to translate because it can refer to a ring or loop or even a quasi-concentric formation of people or things that don't necessarily look circular depending on spacial restrictions, but never an actual geometric circle, because that's a circle.
portal probably has really weird circles since you can use portals to cut through space but only on some surfaces
You had an entire waffles worth of batter! Where is your rubber spatula! Get out of the kitchen!
Oh my god...this channel is DIAMOND♦️
Ha! Thanks!
Reminds me of I had a programming assignment the other day where I had to make a diamond pattern and while I was supposed to print a certain number of characters in each row what I did instead was make the alternating grid and carve a circle from the center because it was small enough for me to get away with just using a radius and not printing anything outside it
This is intensely nerdy (also potentially incorrect) but Final Fantasy 14, the price for teleporting is determined by how many loading screens you'd encounter to get to your destination, no matter the map size or shape. Airship travel is ignored though so it's often cheaper to ride the airship first then teleport within the destination.
Prices are complicated further by the fact that every player can have 3 half price locations and 1 free location (which is on a cooldown)
Fun Fact: A district in Germany is called »Kreis«, the same word for circle.
5:22 these lifeguards must be the worst swimmers known to man
You mentioned the grid and it immediately clicked in my head because of fluids in Minecraft 😂
KAMI! I need you to tell me that I can leave the lookout if I want to
Any shape is a circle if you make the conditions precise enough
Chess board should be rotated 90 degrees. Kings start the game in e1/e8, and those squares are opposite each king's color.
The knight movement is the best example
Can you make a circular prism
Theres a map for europe where you can go by public transportation in 1h, 2h, ... colored. So they are also circles 😊
In Bare Knuckle Fighting Championship, the ring is referred to as the "Squared Circle." I've never understood why. I've tried looking up an explanation but found nothing. My best guess is that the circle has a particular square footage, or there is some cultural connotation.
Neat. I liked this video a lot
Glad you enjoyed it!
So if you put enught dough into the waffle iron, what is the shape?
2:32 It's not clear what exactly you mean by that. Nothing seems to be preventing the word diamond from being an alternate name for a rhombus.
In general a diamond has 2 pairs of equal sides and a single line of symmetry through 1 diagonal.
@ That's one possible definition, though this is conventionally called a kite.
@@isavenewspapers8890
To be fair, a diamond has many shapes based on how you manufacture it so .. a diamond shape is whatever.
You could create a mathematical definition for a diamond. The issue that many math teacher space is that students will see a square that is oriented with vertex up, and not consider it a square, but a different object with different properties.
@ Well, I'll grant you that one. I would also prefer that a square be referred to as a square regardless of orientation for teaching purposes.
Well, 9999 times out of 10000 when we talk about shapes, we talk about Euclidean geometry. So, you're not wrong, but neither are we.
Where the circle go, that's right to the square hole
*in non-euclidean topologies
manhattan/taxicab space
Calculate pi with this
All these squares make a circle
okay, now im hungry
The board is set up wrongly
Where does the circle go? That's right! It goes to the sQuArE hole!
Manhattan baby
all these squares make a circle...
THE CHESS BOARD
1:01 whyyy would you draw it like that??? 😫
SQUINE AND COSQUINE
Patrick, did you press the Sin button on your calculator?
No, I pressed the Squine button.
What do you mean Squine button?
Its right here! It means the Sin of a square!
“…”
*different to/from
7:48 Huh, I don't remember this channel being affiliated with BYU
Nice
Yep! This is from the Mathematics Education department at BYU, and the College of Computational, Mathematical, and Physical Sciences
The lifeguard thing has been advice given to me at a lifesaving competition.
Wow! I love it when we figure out real-world conclusions for our videos that is actually being used or taught!
Wouldn't the waffle technically be a disk, not a circle?
wha the
2:35 WRONG! A diamond IS a real shape - a quadrilateral with a single line of symmetry intersecting 2 opposite corners.
APOLOGIZE!!!!
I believe that is conventionally called a "kite", not a diamond.
@jmvr I also call it a diamond.
To be fair, a diamond has many shapes based on how you manufacture it so .. a diamond is whatever.
Like a previous comment, I know these as a kite. The reason I emphasized that there is no diamond in mathematics is because I know mathematicians that really get bothered by people calling those kind of shapes diamonds. I don't consider myself a mathematician, since I don't have a PhD in mathematics, so I was trying to be true to the field by emphasizing that.
However, consider this my apology if am incorrect. It may be the case that this is still debated in the field of mathematics and maybe I should have done more research.
@@MathTheWorld I DO have a maths degree from my university days (2nd class Honours - that was a TOUGH year). You and other commenters may be 100% right and I could be wrong. I don't specifically remember "diamond" ever being used during academic studies but growing up it was definitely used privately where I grew up - this may change over time and from place to place until things get formalized through internationally accepted education.
I'm happy to also apologize if I am wrong - it was 25 years ago that I did my studies and in retrospect @jmvr's "kite" seems more correct to me now as to what I described above than what a younger me would believe and perhaps I was previously poorly educated.
I do appreciate everyone's contribution.
Hug?
998th like
No square can ever be a circle due to pythogorus and curvature. Also ur compass shape isnt a square but a squircle
Are you trying to apply Euclidean logic to non-Euclidean spaces?
@isavenewspapers8890 my logic comes from the definitions of the shapes the video was referencing
WOW, you have actually hit a critical point right there.
What is the curvature of a Manhatten space, or a Chebyshev space, or any other space that has a different metric function from the euclidean metric?
Unfortunately, I have no knowledge about this, I am here to pose the right questions, not to answer them. 😅
Hopefully more knowledgeable people will enlighten us!
Your comment has left me curious about a generalised form of Pythagoras' theorem for non-euclidean geometries. But I don't think this video ever gave a definition of a square, so the relevance of Pythagoras or curvature is going to be really dependent on your definition of a square. You could say that a square has four edges, four vertices that all have an interior angle of 90°, 4 lines of symmetry, 4-fold rotational symmetry, etc., which works but isn't very generalisable. Or you could go for something more broadly useful, like the 2d case of a shape where all edges that meet at a vertex are orthogonal. The former means squares only exist on planes and doesn't do much to indicate any relation to cubes, hypercubes, etc. The latter says that squares are part of the family of hypercubes, but it also gives the freedom to work outside of Euclidean geometries: as an example, on a spherical surface you can have a triangle with all interior angles being right angles, and that's a square under this definition.
Also, there are a couple of different types of squircles, but they all have circle-like properties at the 'corners' and approach straight lines at the centre of the edges, so this isn't a squircle. It looks a bit like what you'd get if you projected a square onto the surface of a sphere?
@@lawrencebates8172
I think the definition which I am going to enlist, inspired by your observations, is the one that we should use.
We all know that a square in euclidean geometry is a shape with 4 edges, we absolutely don't consider a shape with other than 4 edges a square.
However, of course this isn't enogh because there are many shapes with 4 sides that aren't squares like rhombus or a rectangle, what makes squares distinguishable is the act that it has same length for all its sides and same interior angles, we call a polygon with these properties regular.
From this analysis, the definition of a square in general should be the following:
*A regular 4-sided polygon.*
The above definition works for any geometry, the only difference is that polygons edges in other geometries are the geodesics instead of straight lines.