4 ms·

Is antipodal land a physical property of Earth's continents, or is it mathematical fact of distribution on spheres? If we distribute chunks of land randomly on

Is antipodal land a physical property of Earth's continents, or is it mathematical fact of distribution on spheres? If we distribute chunks of land randomly on a sphere we would expect roughly the same amount of land on any randomly-chosen hemisphere. But what can we say about the average amount of land in the hemisphere with the most land? My intuition says that in an Earth-like sphere, with 29% land and 71% water, there's a high chance of there existing a hemisphere with almost all of the land. I have no idea how to prove this empirically, but it seems like a really nice maths problem.

jameshart 4y ago
It also depends on how strongly clustered the land tends to be - if the land is all going to be continuous then presumably there are more distributions of that land that are weighted towards a single hemisphere. Suspect there’s something to do with the fractal dimension of the coastlines as well. Agreed - sounds like a rich vein of math.
s1artibartfast 4y ago
My understanding is that it is a physical property, arising from the fact that crustal basically all crustal growth occurs in ocean floors . The locations of the continents spend the vast majority of the time either approaching a clustered state, or leaving it. Because the thick continents with land are a minority of the surface [2], They only briefly pass through a period of even dispersal when passing from one clustered state to another. >My intuition says that in an Earth-like sphere, with 29% land and 71% water, there's a high chance of there existing a hemisphere with almost all of the land Can you explain more? If you have 7 or so plates with land, why would you expect them to have a high chance of being in the same hemisphere? The math is the same as flipping a coin 7 times. Flip the first and see which side it lands on, then flip the remaining 6 and see how many match the same hemisphere. What am I missing? https://www.youtube.com/watch?v=gQqQhZp4uG8 https://www.youtube.com/watch?v=gQqQhZp4uG8 https://www.imperial.ac.uk/news/199334/scientists-find-link-between-tectonic-plate/ https://www.imperial.ac.uk/news/199334/scientists-find-link-...
- mFixman 4y ago
  > The math is the same as flipping a coin 7 times [...] I'm not saying that a particular hemisphere has a high change of having most of the land; I'm saying that there will be a hemisphere that has this chance. 29% of Earth is land and 80% of this land is in the land hemisphere, so this problem would be the equivalent of rolling a D48 14 times and placing the dice on a table so that any 11 of the rolled numbers are visible if I look at it from the top.
  - s1artibartfast 4y ago
    I see, so if you have the freedom to define hemispheres however you want, what are the chances you can fit a definition that captures the majority. I think that is also fairly solvable. let me think on it. I cant understand your example, but is this the same?: Roll a d100 once per continent (e.g. 7 times) and write the numbers down. What percent of the time can you pick a range 50 numbers that captures all 7 results. the range can wrap around from 100 to 0. The problem would be easy to monte carlo, but I would have to think about how to solve it in closed form. Will update if I have a solution.
    - mFixman 4y ago
      Your example is not the same for two reasons: 1. You are assuming there always are 7 continents and each of them contains 1% of all space on the Earth's surface. I assume there are 14 continents with 2% of the Earth's surface each, which is still incorrect but it's closer to reality. 2. I'm not defining hemisphere however I want: I'm using the canonical definition of hemisphere, but choosing any hemisphere for me dice. For example, two continents that appear on numbers 1 and 48 can never be in the same hemisphere. My question is the probability of being able to choose 11 of the 14 continents so that there exists a hemisphere that contains all of them, like the land hemisphere contains 80% of the Earth's land.
      - s1artibartfast 4y ago
        Thats interesting. A few questions: 1)When you say you are using the canonical definition of hemisphere, do you mean North and South? Or are you talking about East and West as well? 2) Out of curiosity, why 14 continents? If you are using North and South as pre-defined hemispheres, ignoring East and West, I do think it can be reduced down to a coin flip if you want to prove that plates are unlikely to almost all be in the same hemisphere. This is because allowing them to overlap allows more clustering, not less. 12/14 continents is >80% and 11/14 is <80%. If you drop a continent randomly, it has a 50/50 chance of being in the north or south. The probability of 12 or more being in the northern hemisphere is 0.647%. The probability of 12 or more being in the southern hemisphere is 0.647%. Together, the probability of 12 or more being in the same N/S hemisphere is 1.294% You can do the same for E and W, getting another 1.294%, for a total of 2.588% chance. This is allowing the continents to overlap, so the real chance would be lower.

jameshart 4y ago

It also depends on how strongly clustered the land tends to be - if the land is all going to be continuous then presumably there are more distributions of that land that are weighted towards a single hemisphere. Suspect there’s something to do with the fractal dimension of the coastlines as well. Agreed - sounds like a rich vein of math.

s1artibartfast 4y ago

My understanding is that it is a physical property, arising from the fact that crustal basically all crustal growth occurs in ocean floors . The locations of the continents spend the vast majority of the time either approaching a clustered state, or leaving it. Because the thick continents with land are a minority of the surface [2], They only briefly pass through a period of even dispersal when passing from one clustered state to another. >My intuition says that in an Earth-like sphere, with 29% land and 71% water, there's a high chance of there existing a hemisphere with almost all of the land Can you explain more? If you have 7 or so plates with land, why would you expect them to have a high chance of being in the same hemisphere? The math is the same as flipping a coin 7 times. Flip the first and see which side it lands on, then flip the remaining 6 and see how many match the same hemisphere. What am I missing? https://www.youtube.com/watch?v=gQqQhZp4uG8 https://www.youtube.com/watch?v=gQqQhZp4uG8 https://www.imperial.ac.uk/news/199334/scientists-find-link-between-tectonic-plate/ https://www.imperial.ac.uk/news/199334/scientists-find-link-...

mFixman 4y ago
> The math is the same as flipping a coin 7 times [...] I'm not saying that a particular hemisphere has a high change of having most of the land; I'm saying that there will be a hemisphere that has this chance. 29% of Earth is land and 80% of this land is in the land hemisphere, so this problem would be the equivalent of rolling a D48 14 times and placing the dice on a table so that any 11 of the rolled numbers are visible if I look at it from the top.
- s1artibartfast 4y ago
  I see, so if you have the freedom to define hemispheres however you want, what are the chances you can fit a definition that captures the majority. I think that is also fairly solvable. let me think on it. I cant understand your example, but is this the same?: Roll a d100 once per continent (e.g. 7 times) and write the numbers down. What percent of the time can you pick a range 50 numbers that captures all 7 results. the range can wrap around from 100 to 0. The problem would be easy to monte carlo, but I would have to think about how to solve it in closed form. Will update if I have a solution.
  - mFixman 4y ago
    Your example is not the same for two reasons: 1. You are assuming there always are 7 continents and each of them contains 1% of all space on the Earth's surface. I assume there are 14 continents with 2% of the Earth's surface each, which is still incorrect but it's closer to reality. 2. I'm not defining hemisphere however I want: I'm using the canonical definition of hemisphere, but choosing any hemisphere for me dice. For example, two continents that appear on numbers 1 and 48 can never be in the same hemisphere. My question is the probability of being able to choose 11 of the 14 continents so that there exists a hemisphere that contains all of them, like the land hemisphere contains 80% of the Earth's land.
    - s1artibartfast 4y ago
      Thats interesting. A few questions: 1)When you say you are using the canonical definition of hemisphere, do you mean North and South? Or are you talking about East and West as well? 2) Out of curiosity, why 14 continents? If you are using North and South as pre-defined hemispheres, ignoring East and West, I do think it can be reduced down to a coin flip if you want to prove that plates are unlikely to almost all be in the same hemisphere. This is because allowing them to overlap allows more clustering, not less. 12/14 continents is >80% and 11/14 is <80%. If you drop a continent randomly, it has a 50/50 chance of being in the north or south. The probability of 12 or more being in the northern hemisphere is 0.647%. The probability of 12 or more being in the southern hemisphere is 0.647%. Together, the probability of 12 or more being in the same N/S hemisphere is 1.294% You can do the same for E and W, getting another 1.294%, for a total of 2.588% chance. This is allowing the continents to overlap, so the real chance would be lower.