Imagine an island (e.g: Bermuda, Hook Island, Sardinia, etc)
Draw a square or rectangle approximating all of the land not currently touching water (e.g: All pixels must not contain water)
Draw a larger red square encompassing the smaller red square or rectangle.
Subtract any brown, green, or “land” pixels, and add them to the total count of Box_1.
Remove green, blue and other “water” pixels from Box_2.
Your final result will be a red outline precisely mapping the coastline of the island in question. You can now measure distance by taking pixels and multiplying by the scale of the zoom-distance (parralax).
Over a very broad range of scales (like, from the scale of 10km down to the scale of 1mm) the number of boundary pixels of a natural shape like an island increases according to a power law as you increase the resolution.
This means that your approach doesn’t give you an objective value because it depends so strongly on the resolution.
This way of computing the length of a boundary leads to the concept of box-counting dimension. When you increase the resolution of the pixel grid, you’ll get a larger number of pixels on the boundary. Keep refining the grid many times. Graph the log of the total number of pixels against the log of the number of boundary pixels. The box counting dimension is the slope of that graph.
Why would we call this “dimension”? Because if you do this to a line, the slope is 1, and if you do it to a square, the slope is 2.
This measurement is a factor of pixel size. As resolution increases and pixel width approaches 0, the shoreline length approaches infinity.
Though I guess you’d eventually run into the problem of clearly defining the shoreline once you’re distinguishing between water molecules and grains of sand. is the water between the sand molecules part of the ocean? How concave is the boundary on the stretches between sand grains?
And of course, it’s dynamic as tides and waves change it. And how does wet sand due to rain play into it - we’re now having to differentiate based on salinity of water.
Reminds me of the conundrum of being unable to measure shoreline
Alan Davies (of qi fame) once made a documentary where he tried to measure the length of a piece of string, the shoreline issue also comes up.
https://www.bbc.co.uk/programmes/p00whwmc
Imagine an island (e.g: Bermuda, Hook Island, Sardinia, etc)
Draw a square or rectangle approximating all of the land not currently touching water (e.g: All pixels must not contain water)
Draw a larger red square encompassing the smaller red square or rectangle.
Subtract any brown, green, or “land” pixels, and add them to the total count of Box_1.
Remove green, blue and other “water” pixels from Box_2.
Your final result will be a red outline precisely mapping the coastline of the island in question. You can now measure distance by taking
pixelsand multiplying by the scale of the zoom-distance (parralax).Over a very broad range of scales (like, from the scale of 10km down to the scale of 1mm) the number of boundary pixels of a natural shape like an island increases according to a power law as you increase the resolution.
This means that your approach doesn’t give you an objective value because it depends so strongly on the resolution.
This way of computing the length of a boundary leads to the concept of box-counting dimension. When you increase the resolution of the pixel grid, you’ll get a larger number of pixels on the boundary. Keep refining the grid many times. Graph the log of the total number of pixels against the log of the number of boundary pixels. The box counting dimension is the slope of that graph.
Why would we call this “dimension”? Because if you do this to a line, the slope is 1, and if you do it to a square, the slope is 2.
More information: https://en.wikipedia.org/wiki/Fractal_dimension?wprov=sfla1
This measurement is a factor of pixel size. As resolution increases and pixel width approaches 0, the shoreline length approaches infinity.
Though I guess you’d eventually run into the problem of clearly defining the shoreline once you’re distinguishing between water molecules and grains of sand. is the water between the sand molecules part of the ocean? How concave is the boundary on the stretches between sand grains?
And of course, it’s dynamic as tides and waves change it. And how does wet sand due to rain play into it - we’re now having to differentiate based on salinity of water.
Oh that’s funny. I see what you’re doing 😆
Philosophers are no longer permitted on the beach 🚧 (/s)
https://en.wikipedia.org/wiki/Coastline_paradox
Just get a small enough ruler either you’ll measure the shoreline and disprove calculus or you’ll solve quantum physics.
Well, if you get a small enough ruler you will already disprove quantum physics. No need to use it for anything.
At best you’ll disprove shoreline.