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I think about squares often, well rectangles really. Practically everything I see is a rectangle. I go to a rectangular building and poke little rectangles to

I think about squares often, well rectangles really. Practically everything I see is a rectangle. I go to a rectangular building and poke little rectangles to fill my rectangular screen with glyphs bounded by rectangles and tesselating on the screen like rectangles would--and I see it all through my rectangular glasses. Why? Well, recycles tesselate--so that's one property that rules out pentagons, say, as our shape of choice, but why not triangles? Or hexagons? This question, I think, is similar to another one: why was euclidean geometry developed more than a thousand years earlier than the non-euclidean ones? Is there something more human about right angles than some other one? (look up the playfair's axiom, you'll see the angles I'm taking about) It's everywhere. Prime numbers are those special ones that can't be expressed as a rectangle of dots on the page. Why do we define multiplication this way? Is there something perhaps arbitrary and anthropocentric in our preference for right angles? If so it would explain why the distribution of the primes is so unpredictable. I think it is because we live at the bottom of a gravity well and have an adaptation (building things) which makes that matter. I also think that if we could rid our mathematics of that bias, it would sort of trickle up into other domains and allow our constructs to take on more richness, like you see in the fractal side of the fractal vs rectangle images presented in the article.

rtkwe 7y ago
> why was euclidean geometry developed more than a thousand years earlier than the non-euclidean ones? Because euclidean geometry is the geometry we experience at a human scale. > Why do we define multiplication this way? It's not actually multiplication though, it's just a convenient representation of the repetitive addition that is actually multiplication. It's a clear way to show to children that we've taking 9 dots 3 times to get 27. It's a tool to teach and interface to a mathematical concept. Much the same as why we use rectangular screens instead of hexagons or anything else. It's the most convenient to manufacture and address because of how repetitive and similar each row is when you think about the arrays of pixels that make up the screen. Desks are mostly rectangles because it's easy to cut straight lines and they match well. Then when you have a lot of rectangular items it makes sense for the containers of those items (ie buildings) to be rectangular too so they fit and pack well without any weird gaps and wasted space. Also I'm not sure where you're coming from with the "bias in our mathematics" most higher level math (as early as Calc3 when you start dealing with multivariate integrals one of the most important skills is converting from cartesian coordinates to spherical or cylindrical because it often drastically simplifies the problem) includes a lot of different coordinate switching to simplify representations to simplify the math of solving a particular equation or reasoning about a system.
- __MatrixMan__ 7y ago
  I'm not convinced that it's solely our scale that makes euclidean geometry the preferred one. I think it also has to do with the fact the the ancient Greeks did their math with pointy sticks in beds of sand--beds that took on a certain shape due to environmental factors beyond just scale. This caused us to associate the concept of orthogonality with a 90 degree angle, which is (in most cases) a property of the representations we use--not a property of the underlying mathematical objects. It's not a bias of the coordinate systems, it's a bias of how we conceive of the space, uh, coordinated by those systems. And it's a convenient one--it evolved in our environment for a reason--it works well here. But so did the idea that the Earth was in the center of the universe. I realize that it's a kooky theory, but it takes time to turn a kooky theory into a serious one, and so far I think it has some merits so I'm going to keep working on it. Particularly, it would explain the seemingly arbitrary values of certain mathematical constants. If our concept of "straight line" or "flat surface" were to be not special at all, but just some weird quirk of how humans do things, then constants like pi would tell us less about circles and more about our own habits of perception, and I think that would be a much more satisfying way of framing things. Or think about how the derivative of e^x is e^x. That's a cool property, but if you look at how the difference quotient is constructed you'll see that it implies axes on a flat surface set at 90 degrees to each other. So maybe what's special about e says more about the spaces implied by the coordinate system. I realize that e is not _defined_ this way. That's where the theory needs work. Somehow our concept of the natural numbers is bound up with our perspective on flatness, and of how independent things ought to be represented and compared. I've yet to be able to construct something number-like that I can map to alternative values of e and pi for different geometries, but I think it can be done.
ajuc 7y ago
> Well, recycles tesselate--so that's one property that rules out pentagons, say, as our shape of choice, but why not triangles? Or hexagons? Try to sleep on a triangular bed :) Try to read a triangular book or write a letter on a triangular page :) Try to watch TV on a triangular screen :) Try to hang clothes in a triangular cabinet :) Notice how much space is wasted in the corners of these (preasumably 60 degree) triangles. Our bodies are roughly rectangles and interact well with rectangular shapes. Our field of vision is a rectangle too.

rtkwe 7y ago

> why was euclidean geometry developed more than a thousand years earlier than the non-euclidean ones? Because euclidean geometry is the geometry we experience at a human scale. > Why do we define multiplication this way? It's not actually multiplication though, it's just a convenient representation of the repetitive addition that is actually multiplication. It's a clear way to show to children that we've taking 9 dots 3 times to get 27. It's a tool to teach and interface to a mathematical concept. Much the same as why we use rectangular screens instead of hexagons or anything else. It's the most convenient to manufacture and address because of how repetitive and similar each row is when you think about the arrays of pixels that make up the screen. Desks are mostly rectangles because it's easy to cut straight lines and they match well. Then when you have a lot of rectangular items it makes sense for the containers of those items (ie buildings) to be rectangular too so they fit and pack well without any weird gaps and wasted space. Also I'm not sure where you're coming from with the "bias in our mathematics" most higher level math (as early as Calc3 when you start dealing with multivariate integrals one of the most important skills is converting from cartesian coordinates to spherical or cylindrical because it often drastically simplifies the problem) includes a lot of different coordinate switching to simplify representations to simplify the math of solving a particular equation or reasoning about a system.

__MatrixMan__ 7y ago
I'm not convinced that it's solely our scale that makes euclidean geometry the preferred one. I think it also has to do with the fact the the ancient Greeks did their math with pointy sticks in beds of sand--beds that took on a certain shape due to environmental factors beyond just scale. This caused us to associate the concept of orthogonality with a 90 degree angle, which is (in most cases) a property of the representations we use--not a property of the underlying mathematical objects. It's not a bias of the coordinate systems, it's a bias of how we conceive of the space, uh, coordinated by those systems. And it's a convenient one--it evolved in our environment for a reason--it works well here. But so did the idea that the Earth was in the center of the universe. I realize that it's a kooky theory, but it takes time to turn a kooky theory into a serious one, and so far I think it has some merits so I'm going to keep working on it. Particularly, it would explain the seemingly arbitrary values of certain mathematical constants. If our concept of "straight line" or "flat surface" were to be not special at all, but just some weird quirk of how humans do things, then constants like pi would tell us less about circles and more about our own habits of perception, and I think that would be a much more satisfying way of framing things. Or think about how the derivative of e^x is e^x. That's a cool property, but if you look at how the difference quotient is constructed you'll see that it implies axes on a flat surface set at 90 degrees to each other. So maybe what's special about e says more about the spaces implied by the coordinate system. I realize that e is not _defined_ this way. That's where the theory needs work. Somehow our concept of the natural numbers is bound up with our perspective on flatness, and of how independent things ought to be represented and compared. I've yet to be able to construct something number-like that I can map to alternative values of e and pi for different geometries, but I think it can be done.

ajuc 7y ago

> Well, recycles tesselate--so that's one property that rules out pentagons, say, as our shape of choice, but why not triangles? Or hexagons? Try to sleep on a triangular bed :) Try to read a triangular book or write a letter on a triangular page :) Try to watch TV on a triangular screen :) Try to hang clothes in a triangular cabinet :) Notice how much space is wasted in the corners of these (preasumably 60 degree) triangles. Our bodies are roughly rectangles and interact well with rectangular shapes. Our field of vision is a rectangle too.