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Off Topic / Re: Klein bottles: you know a glass vial is metal if it manages to defy everything.
« on: October 21, 2012, 04:38:10 PM »
How to prove a Klein bottle has volume.
A 1 dimensional shape has: length
A 2 dimensional shape has: area (length^2)
A 3 dimensional shape has: volume (length^3)
In order to get the one sided shapes such as a Möbius strip or Klein bottle, you take a shape from the dimension down and contort through the current dimension. Meaning you kind of need to subtract 1 dimension from it's potential measure of size. Take the following examples.
A 1 dimensional 1 sided shape: does not exist (kind of)
A 2 dimensional 1 sided shape (for example: a line bent in a circle, but not the area inside of the circle because that would make it two sided) has: length
A 3 dimensional 1 sided shape (Möbius strip) has: area (length^2)
A 4 dimensional 1 sided shape (Klein bottle), therefore, must have: volume (length^3)
In reality, anything that exists exists in all dimensions and therefore it's size would be measured in length^(number of dimensions). But in mathematics, shapes can exist in the amount of dimensions you want them to, and because the Klein bottle is a mathematical shape, it has volume.
EDIT: Over some thinking a realized that a Klein bottle might just be defined by the fact it has only volume. For example: A solid sphere is a three dimensional shape with only one side, so why is a Möbius strip special? I imagine it must be because the Möbius strip is a 2 dimensional shape contorted to exist in the three dimensional world, so it only has area instead of volume. The same principle applies to the Klein bottle, what defines it is that it only has volume instead length^4.
EDIT EDIT Because it came up:
I use length^dimension because we don't have a name for fourth and beyond dimensional measurements, so it keeps it consistent an easier to understand. Substituting L^2 for LW or L^3 for LWH or L^4 for LWH(IV) doesn't change my point.
A 1 dimensional shape has: length
A 2 dimensional shape has: area (length^2)
A 3 dimensional shape has: volume (length^3)
In order to get the one sided shapes such as a Möbius strip or Klein bottle, you take a shape from the dimension down and contort through the current dimension. Meaning you kind of need to subtract 1 dimension from it's potential measure of size. Take the following examples.
A 1 dimensional 1 sided shape: does not exist (kind of)
A 2 dimensional 1 sided shape (for example: a line bent in a circle, but not the area inside of the circle because that would make it two sided) has: length
A 3 dimensional 1 sided shape (Möbius strip) has: area (length^2)
A 4 dimensional 1 sided shape (Klein bottle), therefore, must have: volume (length^3)
In reality, anything that exists exists in all dimensions and therefore it's size would be measured in length^(number of dimensions). But in mathematics, shapes can exist in the amount of dimensions you want them to, and because the Klein bottle is a mathematical shape, it has volume.
EDIT: Over some thinking a realized that a Klein bottle might just be defined by the fact it has only volume. For example: A solid sphere is a three dimensional shape with only one side, so why is a Möbius strip special? I imagine it must be because the Möbius strip is a 2 dimensional shape contorted to exist in the three dimensional world, so it only has area instead of volume. The same principle applies to the Klein bottle, what defines it is that it only has volume instead length^4.
EDIT EDIT Because it came up:
I use length^dimension because we don't have a name for fourth and beyond dimensional measurements, so it keeps it consistent an easier to understand. Substituting L^2 for LW or L^3 for LWH or L^4 for LWH(IV) doesn't change my point.