A new proof marks the first progress in decades on important cases of the so-called kissing problem. Getting there meant doing away with traditional approaches.
I can only answer partially but I’ll give it my best shot. Dimensions don’t have to represent physical space, or anything related to it. Sometimes a fourth dimension is used to represent time, but really it could represent anything.
An example I saw frequently in school used different recipes for, say bread. They all have similar ingredients: flour, water, yeast. Each recipe can be represented as a vector in three dimensions. X might be flour, Y might be water, Z might be yeast. Easy to visualize. You can imagine a center at some arbitrary point in space, and we can say that any point in that space relative to it’s center is another recipe. If it helps to see what I mean, any point that lies in an axis would only have one ingredient. The very center represents the bread recipe with no amount of any ingredient. Those would be terrible recipes, since they wouldn’t actually make any bread, but that’s what those points would represent.
Since we’re imagining things as points in space, we can even use things like the distance formulas you learned in geometry to determine how far apart the recipes are in the imagined space. Well, obviously recipes can have more than three ingredients. Suppose we add salt and sugar to the recipes. Now the vectors representing those recipes have five dimensions. The cool thing is, those formulas we used still work in any number of dimensions. We just use a more general form of them.
I hope that helps sort of get you acquainted with the idea of higher dimensions in mathematics. I learned by taking two terms of linear algebra. The second term was pretty difficult.We were all very confused 😂
I can only answer partially but I’ll give it my best shot. Dimensions don’t have to represent physical space, or anything related to it. Sometimes a fourth dimension is used to represent time, but really it could represent anything.
An example I saw frequently in school used different recipes for, say bread. They all have similar ingredients: flour, water, yeast. Each recipe can be represented as a vector in three dimensions. X might be flour, Y might be water, Z might be yeast. Easy to visualize. You can imagine a center at some arbitrary point in space, and we can say that any point in that space relative to it’s center is another recipe. If it helps to see what I mean, any point that lies in an axis would only have one ingredient. The very center represents the bread recipe with no amount of any ingredient. Those would be terrible recipes, since they wouldn’t actually make any bread, but that’s what those points would represent.
Since we’re imagining things as points in space, we can even use things like the distance formulas you learned in geometry to determine how far apart the recipes are in the imagined space. Well, obviously recipes can have more than three ingredients. Suppose we add salt and sugar to the recipes. Now the vectors representing those recipes have five dimensions. The cool thing is, those formulas we used still work in any number of dimensions. We just use a more general form of them.
I hope that helps sort of get you acquainted with the idea of higher dimensions in mathematics. I learned by taking two terms of linear algebra. The second term was pretty difficult.We were all very confused 😂