[FRIAM] square land math question

Thu Jul 23 16:47:05 EDT 2020

In geometry, I find it better to think in terms of objects. A point is an object that has a location, dimension 0 (no measurable property) and no other properties; a line segment is an object with one dimension, has dimension one,  and is defined by two points and so on. For each object, we have a set of functions. A point has no functions defined for it. When you say a point is an n-tuple in R^n you are talking about the representation of a point in some space, not the geometric object. To get back to Cody’s original question. From a geometric perspective, a sequence of two dimensional objects (the squares), which can be scaled,  cannot turn into a point which is a different object  type.

Here’s a somewhat different geometric view of why you have to be wary of what the kid claimed. Suppose I start with a unit square. I divide it evenly in both directions to get four equal squares. I then throw away two diagonally opposite squares so I have half the original area. However, if I follow the edges I the distance between the opposite vertices is still 2. As you repeat this construction, the area of total of all the 2^n squares goes to zero but the distance along the edges between the original opposite vertices remains as 2. 

We can’t say this construction converges to a line connecting the two original vertices since we just showed it has a length two not sqrt(2). Or does it since if we add up the diagonals of all little cubes they do sum to sqrt 2. It gets even more interesting if we remove only one of the subcubes each time and add up the perimeters of all the subcubes thus creating an object than in the limit has no area but an infinite perimeter. Fractal geometry has nice definition of dimension that cover these issues.

Ed
__________

Ed Angel

Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab)
Professor Emeritus of Computer Science, University of New Mexico

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> On Jul 23, 2020, at 2:20 PM, Frank Wimberly <wimberly3 at gmail.com> wrote:
> 
> "While a point and a vector in R^n might be described by the same tuple,
> dividing the numeric elements of the tuple does not "partition" the
> point..."
> 
> Good point, Steve.  There are infinitely many ways of resolving a vector.  E.g. (1, 1) = (1, 0) + (0, 1/2) + (0, 1/4) + (0, 1/4) etc.
> 
>   
> 
> On Thu, Jul 23, 2020 at 2:09 PM uǝlƃ ↙↙↙ <gepropella at gmail.com <mailto:gepropella at gmail.com>> wrote:
> Nice challenge! ... Welllll, the original question was basically how Cody might respond to the kid's suggestion that a point is a square with no area. My suggestion to Cody would be to answer the kid with a discussion about the actuality or potentiality of infinity ... or intermediately, distinguishing between *definitions* of "square".
> 
> And if you define define a square geometrically, then it makes complete sense that there is no arealess square. But there are OTHER ways to define a square. And since this kid already pulled out a sophisticated mathematical argument, it's useful and interesting to see how far that kid can go.
> 
> You're free to hem and haw about the foundations of math and which foundation you like better than another. But the point of discussing the extent of a point was to answer the kid's challenge. Answering a bright kid with "because Euclid says so" is not all that useful. >8^D
> 
> On 7/23/20 1:00 PM, Steve Smith wrote:
> > Can you illuminate us as to what treating the *location* of a point as a
> > *quantity* and demonstrating that the quantity can be divided
> > arithmetically adds to the meaning of a point? 
> > 
> > While a point and a vector in R^n might be described by the same tuple,
> > dividing the numeric elements of the tuple does not "partition" the
> > point, it merely scales the vector which is quite useful, but I'm not
> > sure if in any way doing so has any meaning that could be construed as
> > having "divided" the point?
> > 
> > I think Euclid's geometry is pretty "standard math"?
> 
> -- 
> ↙↙↙ uǝlƃ
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