There I was, talking about error, and I go and make one of my own. 25 - 49 is 24, not 14.
This doesn't affect my arguments. One is that if we treat as an error a geometric specification in which c isn't long enough to connect up with another side, whether in the frustum or my example of a right angled triangle, then this suggests we could use the numerical result we do get as a means of correcting that specification, perhaps by simple addition of root 24 or 63.
Although I also suggested that such an original spec was in violation of Euclid's first two axioms, it isn't necessarily a "geometric inpossibility" (to quote the article) in the sense of being inconceivable or impossible to draw a picture of. Granted, in a 2D figure we can easily draw a line which stops short of another. In 3D this is at first consideration a bit more problematic, since a frustum or pyramid or whatever made out of say stone or clay has edges, instead of lines inked or scored into a supporting matrix like paper or sand. And how can you have such an edge which stops short and leaves a gap of empty space? Perhaps if the thing was made out of wire, or even better out of a big mass of paper mache with very thin straight tunnels in it corresponding to lines, in which ink has dried and set. If one of those tunnels didn't go all the way to another, then that could be our "impossible" frustum.
There I was, talking about error, and I go and make one of my own. 25 - 49 is 24, not 14.
This doesn't affect my arguments. One is that if we treat as an error a geometric specification in which c isn't long enough to connect up with another side, whether in the frustum or my example of a right angled triangle, then this suggests we could use the numerical result we do get as a means of correcting that specification, perhaps by simple addition of root 24 or 63.
Although I also suggested that such an original spec was in violation of Euclid's first two axioms, it isn't necessarily a "geometric inpossibility" (to quote the article) in the sense of being inconceivable or impossible to draw a picture of. Granted, in a 2D figure we can easily draw a line which stops short of another. In 3D this is at first consideration a bit more problematic, since a frustum or pyramid or whatever made out of say stone or clay has edges, instead of lines inked or scored into a supporting matrix like paper or sand. And how can you have such an edge which stops short and leaves a gap of empty space? Perhaps if the thing was made out of wire, or even better out of a big mass of paper mache with very thin straight tunnels in it corresponding to lines, in which ink has dried and set. If one of those tunnels didn't go all the way to another, then that could be our "impossible" frustum.
Chris G