Jam wrote:Two different rectangles are different from each other in the same way that a square is different than a rectangle.
No, all types of rectangles fit under the rectangle category. Squares all fit in the square category.
They all just differ by the lengths of their sides. Squares just have all equal sides. If a rectangle has side pairs different by the golden ratio is it suddenly a different shape or is it just a type of rectangle to which I've applied a significance to it's side lengths.
Yes, it is suddenly a different shape, a square instead of a rectangle. That's the one and only difference between them
Because we are discussing polygons I will limit the definition of a shape to a polygon.
Technically then, by that logic, every infinite variation of a single polygon is a "different shape." Two oblong rectangles whose sides are of different proportions are still rectangles to you, no? We must define shapes by their geometric properties - under which a square is a rectangle - and which, because of its non-similar singularity, a property unique to squares and not to other rectangles, we can name it further.
Well yes technically I would say all shapes are different, even if they're just different sizes. Just a waste of time to come up with names for all of them, so we have to stop at some point
The "some point" being geometric properties which tie together large groups of polygons. Within the context of this discussion, that means we can classify infinite quadrilaterals into a handful of groups. Grouping all quadrilaterals with four right angles is a logical classification which we call "rectangles," under which the square also falls. It follows then that a square is a rectangle.
Nope - just because rectangles and squares share a common characteristic doesn't mean they are the same thing. You and I are both members of ESOC - doesn't mean that we are thus the same person.
A rhomboid is a four sided parallelogram whose adjacent sides are not of equal length, much like a rectangle, and whose interior angles do not form 90 degrees.
A rhombus is a four sided parallelogram whose adjacent sides are of exact length, much like a square, and whose interior angles do not form 90 degrees.
HOWEVER, a rhombus is not considered a rhomboid for whatever reason.
In fact a rhomboid is defined as "either a parallelepiped or a parallelogram that is neither a rhombus nor a rectangle"
A rhomboid is a four sided parallelogram whose adjacent sides are not of equal length, much like a rectangle, and whose interior angles do not form 90 degrees.
A rhombus is a four sided parallelogram whose adjacent sides are of exact length, much like a square, and whose interior angles do not form 90 degrees.
HOWEVER, a rhombus is not considered a rhomboid for whatever reason.
In fact a rhomboid is defined as "either a parallelepiped or a parallelogram that is neither a rhombus nor a rectangle"
The plot thickens
Squares and rhombuses have more in common than squares and rectangles
Jam wrote:Two different rectangles are different from each other in the same way that a square is different than a rectangle.
No, all types of rectangles fit under the rectangle category. Squares all fit in the square category.
They all just differ by the lengths of their sides. Squares just have all equal sides. If a rectangle has side pairs different by the golden ratio is it suddenly a different shape or is it just a type of rectangle to which I've applied a significance to it's side lengths.
Yes, it is suddenly a different shape, a square instead of a rectangle. That's the one and only difference between them
Because we are discussing polygons I will limit the definition of a shape to a polygon.
Technically then, by that logic, every infinite variation of a single polygon is a "different shape." Two oblong rectangles whose sides are of different proportions are still rectangles to you, no? We must define shapes by their geometric properties - under which a square is a rectangle - and which, because of its non-similar singularity, a property unique to squares and not to other rectangles, we can name it further.
Well yes technically I would say all shapes are different, even if they're just different sizes. Just a waste of time to come up with names for all of them, so we have to stop at some point
The "some point" being geometric properties which tie together large groups of polygons. Within the context of this discussion, that means we can classify infinite quadrilaterals into a handful of groups. Grouping all quadrilaterals with four right angles is a logical classification which we call "rectangles," under which the square also falls. It follows then that a square is a rectangle.
Nope - just because rectangles and squares share a common characteristic doesn't mean they are the same thing. You and I are both members of ESOC - doesn't mean that we are thus the same person.
I have already covered this though. A rectangle is not a particular shape, it is a classification of shapes with certain characteristics that fall under it. We agree two different sized rectangles are still rectangles, or else we would have to name every single infinite rectangle as its own shape. A square falls under the definition of a rectangle therefore it is a rectangles also.
Your analogy is incorrect, as two individual human beings are unique and not the same as a rectangle (infinite non-similar variations) and a square (single non-similar variation).
A rhomboid is a four sided parallelogram whose adjacent sides are not of equal length, much like a rectangle, and whose interior angles do not form 90 degrees.
A rhombus is a four sided parallelogram whose adjacent sides are of exact length, much like a square, and whose interior angles do not form 90 degrees.
HOWEVER, a rhombus is not considered a rhomboid for whatever reason.
In fact a rhomboid is defined as "either a parallelepiped or a parallelogram that is neither a rhombus nor a rectangle"
The plot thickens
Squares and rhombuses have more in common than squares and rectangles
Quite untrue, as squares and rectangles will always share the same interior angles, whereas rhombuses could have any infinite amount of interior angles.
Jam wrote:Two different rectangles are different from each other in the same way that a square is different than a rectangle.
No, all types of rectangles fit under the rectangle category. Squares all fit in the square category.
They all just differ by the lengths of their sides. Squares just have all equal sides. If a rectangle has side pairs different by the golden ratio is it suddenly a different shape or is it just a type of rectangle to which I've applied a significance to it's side lengths.
Yes, it is suddenly a different shape, a square instead of a rectangle. That's the one and only difference between them
Because we are discussing polygons I will limit the definition of a shape to a polygon.
Technically then, by that logic, every infinite variation of a single polygon is a "different shape." Two oblong rectangles whose sides are of different proportions are still rectangles to you, no? We must define shapes by their geometric properties - under which a square is a rectangle - and which, because of its non-similar singularity, a property unique to squares and not to other rectangles, we can name it further.
Well yes technically I would say all shapes are different, even if they're just different sizes. Just a waste of time to come up with names for all of them, so we have to stop at some point
The "some point" being geometric properties which tie together large groups of polygons. Within the context of this discussion, that means we can classify infinite quadrilaterals into a handful of groups. Grouping all quadrilaterals with four right angles is a logical classification which we call "rectangles," under which the square also falls. It follows then that a square is a rectangle.
Nope - just because rectangles and squares share a common characteristic doesn't mean they are the same thing. You and I are both members of ESOC - doesn't mean that we are thus the same person.
I have already covered this though. A rectangle is not a particular shape, it is a classification of shapes with certain characteristics that fall under it. We agree two different sized rectangles are still rectangles, or else we would have to name every single infinite rectangle as its own shape. A square falls under the definition of a rectangle therefore it is a rectangles also.
Your analogy is incorrect, as two individual human beings are unique and not the same as a rectangle (infinite non-similar variations) and a square (single non-similar variation).
I think that every single size of a rectangle is different
Jam wrote:Two different rectangles are different from each other in the same way that a square is different than a rectangle.
No, all types of rectangles fit under the rectangle category. Squares all fit in the square category.
They all just differ by the lengths of their sides. Squares just have all equal sides. If a rectangle has side pairs different by the golden ratio is it suddenly a different shape or is it just a type of rectangle to which I've applied a significance to it's side lengths.
Yes, it is suddenly a different shape, a square instead of a rectangle. That's the one and only difference between them
Because we are discussing polygons I will limit the definition of a shape to a polygon.
Technically then, by that logic, every infinite variation of a single polygon is a "different shape." Two oblong rectangles whose sides are of different proportions are still rectangles to you, no? We must define shapes by their geometric properties - under which a square is a rectangle - and which, because of its non-similar singularity, a property unique to squares and not to other rectangles, we can name it further.
Well yes technically I would say all shapes are different, even if they're just different sizes. Just a waste of time to come up with names for all of them, so we have to stop at some point
The "some point" being geometric properties which tie together large groups of polygons. Within the context of this discussion, that means we can classify infinite quadrilaterals into a handful of groups. Grouping all quadrilaterals with four right angles is a logical classification which we call "rectangles," under which the square also falls. It follows then that a square is a rectangle.
Nope - just because rectangles and squares share a common characteristic doesn't mean they are the same thing. You and I are both members of ESOC - doesn't mean that we are thus the same person.
I have already covered this though. A rectangle is not a particular shape, it is a classification of shapes with certain characteristics that fall under it. We agree two different sized rectangles are still rectangles, or else we would have to name every single infinite rectangle as its own shape. A square falls under the definition of a rectangle therefore it is a rectangles also.
Your analogy is incorrect, as two individual human beings are unique and not the same as a rectangle (infinite non-similar variations) and a square (single non-similar variation).
I think that every single size of a rectangle is different
Then this discussion is pointless, as each time you mention "rectangle" you are then referring only to a single one of its infinite variations, in which case, how are we to know that that particular combination of side lengths is the true "rectangle?"
Jam wrote:Two different rectangles are different from each other in the same way that a square is different than a rectangle.
No, all types of rectangles fit under the rectangle category. Squares all fit in the square category.
They all just differ by the lengths of their sides. Squares just have all equal sides. If a rectangle has side pairs different by the golden ratio is it suddenly a different shape or is it just a type of rectangle to which I've applied a significance to it's side lengths.
Yes, it is suddenly a different shape, a square instead of a rectangle. That's the one and only difference between them
Because we are discussing polygons I will limit the definition of a shape to a polygon.
Technically then, by that logic, every infinite variation of a single polygon is a "different shape." Two oblong rectangles whose sides are of different proportions are still rectangles to you, no? We must define shapes by their geometric properties - under which a square is a rectangle - and which, because of its non-similar singularity, a property unique to squares and not to other rectangles, we can name it further.
Well yes technically I would say all shapes are different, even if they're just different sizes. Just a waste of time to come up with names for all of them, so we have to stop at some point
The "some point" being geometric properties which tie together large groups of polygons. Within the context of this discussion, that means we can classify infinite quadrilaterals into a handful of groups. Grouping all quadrilaterals with four right angles is a logical classification which we call "rectangles," under which the square also falls. It follows then that a square is a rectangle.
Nope - just because rectangles and squares share a common characteristic doesn't mean they are the same thing. You and I are both members of ESOC - doesn't mean that we are thus the same person.
I have already covered this though. A rectangle is not a particular shape, it is a classification of shapes with certain characteristics that fall under it. We agree two different sized rectangles are still rectangles, or else we would have to name every single infinite rectangle as its own shape. A square falls under the definition of a rectangle therefore it is a rectangles also.
Your analogy is incorrect, as two individual human beings are unique and not the same as a rectangle (infinite non-similar variations) and a square (single non-similar variation).
I think that every single size of a rectangle is different
Then this discussion is pointless, as each time you mention "rectangle" you are then referring only to a single one of its infinite variations, in which case, what makes that particular combination of side lengths the true "rectangle" and what exactly is it?
To bring it back on topic I would say the one true rectangle is a lasagna noodle shape
Farfalle for me because of the texture (cooked outside vs less cooked in the middle), however they don't retain heat as long as other shapes I found (could be wrong) and aren't as easy to eat. Farfalle carbonara is delicious. Ex aequo and easier to present would be tagliatelle, usually with butter/olive oil, parmesan and some lemon juice + veal escalope. Love them bolognese too but takes longer to prepare. Not sure why they aren't in the poll - my go-to pasta if I cook for someone. Second would be lasagna, because that dish is one of my favorites when properly done
"Prestige is like a powerful magnet that warps even your beliefs about what you enjoy. If you want to make ambitious people waste their time on errands, bait the hook with prestige." - Paul Graham
Back to the whole pizza slice shape thing, I am a qualified engineer so my opinion of mathematics should count. I think Ear’s argument is very abstract and we should just all agree it’s correct. The real issue hear is the mechanics of pizza consumption by shape. I will start with the rectangle vs square issue.
This is another complicated question. If your assumptions are your pizza is in the shape of a rectangle and the crust follows the long edge, my opinion is you have got some issues. Everyone agrees the crust is the worst part and this shape ensures s maximum crust to pizza ratio aside from the unfortunate scenario where you get a dreaded corner piece. In fact, when a rectangle puzza is cut, it is usually cut into 9 pieces. Usually 8 are rectangles (those around the edges) and by basics geometry the center is square. This piece my friend is the best piece as it has a crust to pizza surface area ratio of 0 which is perfection. There fore based on this logic, square pizza slices are indisputably better based on the crust to pizza surface area ratio. If one thing is better than another, it can’t be the same. Therefore a square piece of pizza can’t be the same as a rectangle piece of pizza. Take my last statement divide both parts by the quantity piece of pizza and they cancel out leaving us with a square can’t be then same as a rectangle which proves the ear’s point.
Your argument is there fore destroyed.
"Build a man a fire, he will stay warm for a night. Set a man on fire, he will be warm for the rest of his life"
Dr. D1CK wrote:Back to the whole pizza slice shape thing, I am a qualified engineer so my opinion of mathematics should count. I think Ear’s argument is very abstract and we should just all agree it’s correct. The real issue hear is the mechanics of pizza consumption by shape. I will start with the rectangle vs square issue.
This is another complicated question. If your assumptions are your pizza is in the shape of a rectangle and the crust follows the long edge, my opinion is you have got some issues. Everyone agrees the crust is the worst part and this shape ensures s maximum crust to pizza ratio aside from the unfortunate scenario where you get a dreaded corner piece. In fact, when a rectangle puzza is cut, it is usually cut into 9 pieces. Usually 8 are rectangles (those around the edges) and by basics geometry the center is square. This piece my friend is the best piece as it has a crust to pizza surface area ratio of 0 which is perfection. There fore based on this logic, square pizza slices are indisputably better based on the crust to pizza surface area ratio. If one thing is better than another, it can’t be the same. Therefore a square piece of pizza can’t be the same as a rectangle piece of pizza. Take my last statement divide both parts by the quantity piece of pizza and they cancel out leaving us with a square can’t be then same as a rectangle which proves the ear’s point.
Also I have completely lost my sense of time and with it, my mind. I legitimately thought I made this thread a couple weeks ago and apparently it's been over a year. How is that fair to me?