Δαφνη Ευτυχια Αρθουρος

"If knowledge can create problems, it is not through ignorance that we can solve them." -- Isaac Asimov [thanks to blueeowyn]

"If knowledge can create problems, it is not through ignorance that we can solve them." -- Isaac Asimov [thanks to blueeowyn]

Those pesky meme-gophers, making it dangerous to wander around the noosphere!

Awake when I shouldn't be, and saw a joke while in a frame of mind that makes it difficult to leave ideas alone ... and as a result I'm now noticing a gap in the math I've done (though maybe it was touched on in a forgotten week of a mostly-forgotten topology course?).

I'm used to thinking of, using, and playing with, both integer and real one-dimensional spaces (i.e., number lines, "linear point sets"...). And I'm even more accustomed to pondering and manipulating real two- and three-dimensional spaces (the Cartesian plane, polar coordinates, etc.). I've even dabbled briefly in the algebra of real four-dimensional rectangular coordinates[*]. But I don't recall doing anything with integer spaces of more than one dimension.

Suddenly I'm wondering what parts of my to-do list I can ignore to go play in multidimensional integer spaces for a while to see what interesting bits pop out of them. (Does an integer space imply a metric based on Manhattan distance, or is it meaningful to have non-integer distances in a space of discrete points? My gut says there'll be valid constructions both ways, with different properties ... how about the topology[**] formed by applying a metric where the distance from (x,y) to (x+1,y+1) is the same as the distance to (x+1,y) or (x,y+1), so you can still get from (x,y) to (x+1,y+1) without passing through any other points, but all distances are still integers?)

Then again, I have the web and search engines now, which did not exist when I was in high school, and I'm confident[***] that others have found this set of problems more fascinating than I do and have written extensively about them (and some of the writings from before we had computers will have been converted and posted somewhere), so I have the option of wasting a day or three with Google and Wikipedia instead of chewing on it by myself for however long it takes me to feel satisfied or get distracted. Hmm. Would that be intellectually lazy, or just efficient?

[*] In high school, or maybe summer vacation: verifying my intuitive guess as to the formula for Pythagorean distance in 4-space; and working out formulas for angles between pairs of three-dimensional spaces in 4-space, which was part of working out the angles between the faces and solids of a pentachoron (which I didn't know that name for at the time).

[**] Or does a topology require a continuous space with an infinite number of points between any two points? I'm gonna have to hit the web to brush up on definitions if nothing else. Argh -- I'm rusty.

[***] After all, Ugol's Law isn't limited in scope to kink alone, right?

Those pesky meme-gophers, making it dangerous to wander around the noosphere!

Awake when I shouldn't be, and saw a joke while in a frame of mind that makes it difficult to leave ideas alone ... and as a result I'm now noticing a gap in the math I've done (though maybe it was touched on in a forgotten week of a mostly-forgotten topology course?).

I'm used to thinking of, using, and playing with, both integer and real one-dimensional spaces (i.e., number lines, "linear point sets"...). And I'm even more accustomed to pondering and manipulating real two- and three-dimensional spaces (the Cartesian plane, polar coordinates, etc.). I've even dabbled briefly in the algebra of real four-dimensional rectangular coordinates[*]. But I don't recall doing anything with integer spaces of more than one dimension.

Suddenly I'm wondering what parts of my to-do list I can ignore to go play in multidimensional integer spaces for a while to see what interesting bits pop out of them. (Does an integer space imply a metric based on Manhattan distance, or is it meaningful to have non-integer distances in a space of discrete points? My gut says there'll be valid constructions both ways, with different properties ... how about the topology[**] formed by applying a metric where the distance from (x,y) to (x+1,y+1) is the same as the distance to (x+1,y) or (x,y+1), so you can still get from (x,y) to (x+1,y+1) without passing through any other points, but all distances are still integers?)

Then again, I have the web and search engines now, which did not exist when I was in high school, and I'm confident[***] that others have found this set of problems more fascinating than I do and have written extensively about them (and some of the writings from before we had computers will have been converted and posted somewhere), so I have the option of wasting a day or three with Google and Wikipedia instead of chewing on it by myself for however long it takes me to feel satisfied or get distracted. Hmm. Would that be intellectually lazy, or just efficient?

[*] In high school, or maybe summer vacation: verifying my intuitive guess as to the formula for Pythagorean distance in 4-space; and working out formulas for angles between pairs of three-dimensional spaces in 4-space, which was part of working out the angles between the faces and solids of a pentachoron (which I didn't know that name for at the time).

[**] Or does a topology require a continuous space with an infinite number of points between any two points? I'm gonna have to hit the web to brush up on definitions if nothing else. Argh -- I'm rusty.

[***] After all, Ugol's Law isn't limited in scope to kink alone, right?