October 1st, 2017, 08:31 PM  #11 
Senior Member Joined: Sep 2016 From: USA Posts: 309 Thanks: 160 Math Focus: Dynamical systems, analytic function theory, numerics 
What you are asking for is equivalent to defining area to be a linear function of the side length. The answer has nothing to do with units. It is simply due to the fact that our definition of area depends quadratically on the side length. Thus when you make a linear change of variables (such as a change in unit), you necessarily get a nonlinear change in the area. As you stated, a postit note doesn't know or care about our arbitrary choices for units. However, the fact that area is a nonlinear function of side length is based completely in reality. It can't be overcome without redefining what area means which would result in a useless definition (albeit with nice properties). 
October 1st, 2017, 11:35 PM  #12 
Member Joined: Nov 2015 From: USA Posts: 96 Thanks: 6 
Why must it be useless? Is there some reason you are certain that it would be useless?
Last edited by skipjack; October 2nd, 2017 at 04:15 AM. 
October 2nd, 2017, 12:32 AM  #13 
Senior Member Joined: Jun 2015 From: England Posts: 762 Thanks: 219 
They used to teach a whole theory of the use of these ratios and other relationships to Engineers studying engineering management under the guise of engineering economics. Engineering economics is quite different from that airy fairy nonsense subject of similar name that leads to inept politicians and worse. 
October 2nd, 2017, 04:08 AM  #14 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,228 Thanks: 2411 Math Focus: Mainly analysis and algebra 
This is not how science works. You have decided what you want the answer to be and are now searching for evidence to support you. That is how religion works and, perhaps, politics. Science observes, makes hypotheses, tests them against reality, and then discards hypotheses that don't match reality. If you try building squares from "unit squares" you will see why the ratio of area to edge cannot be constant. 
October 2nd, 2017, 05:45 AM  #15 
Senior Member Joined: May 2016 From: USA Posts: 919 Thanks: 368 
Kasner and Newman's Mathematics and the Imagination has a relevant essay by Gasking. His position is that how we elect to count and measure is arbitrary, but how we actually do count and measure is determined by what leads to the simplest physics. What makes the essay particularly relevant is that at least one of Gaskings's examples utilizes rulers and square tiles for the practical purpose of tiling a square room. So the OP can realize his ambition (perhaps just by pondering Gasking's essay) but will have to redo all of physics as a result. 
October 2nd, 2017, 07:43 AM  #16 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,290 Thanks: 93 
if Edge of postit note is 2in x 2in, Area is 4in$\displaystyle ^{2}$ and A/E =2. if Edge of postit note is (1/6)ft x (1/6)ft, Area is (1/36)ft$\displaystyle ^{2}$ and A/ E =1/6. The problem is you can't ignore units. Distances and areas are not pure numbers. A/E=4in$\displaystyle ^{2}$/2in=2in A/E=(1/36)ft^{2}/(1/6)ft=(1/6)ft. But (1/6)ft=(1/6)ft x 12in/ft=2in. A/E is the same no matter what units you use, as long as you convert to the same unit. A is ab by definition, derived by adding little squares. Number theory? Elementary algebra and geometry. 
October 8th, 2017, 02:43 AM  #17 
Member Joined: Nov 2015 From: USA Posts: 96 Thanks: 6 
You assume that the area must be measured with little units of shape. I'm not looking for evidence to support what I want to believe, rather I want to achieve a result (geometry describing math without units) and am looking to find a reliable method of achieving that result. I have since realized though, that I was asking the wrong question, a question that limited the potential solution space (much like how using formulas for flat 2d geometry will fail when used on the surface of a sphere). My original thought was to compare a single edge to total area, but any possible solutions would only work for squares. So then I thought about trying to use perimeter which applies to all shapes, but not only can that be very misleading, but some shapes, particularly fractals or shapes with disjointed edges (donuts), end up getting wonky results, from a conceptual point of view at least. So I then realized a better and more useful method is to use the circumcircle. The area is then the ratio of area of the shape to the area of the circumcircle. 

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area, conundrum, interesting, line, ratios 
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