Maybe I should pose a simpler puzzle: using a mind’s-eye instruction-set similar to (but possibly different from) #1, #2, and #3, how would you assess whether a polygon presented to you was regular? You have no measuring equipment — just your brain and your eyes and your hands (which you can use as manipulators but not as measuring rods). If your answer is “Well, I just see at a glance whether a polygon is regular or not”, what sort of processing do you think your brain is doing at a preconscious level? Also consider U.S.-style stop-signs. Are they all regular octagons? How could you tell apart a truly regular one versus a close-but-not-quite-regular one? Jim On Friday, July 27, 2018, James Propp <jamespropp@gmail.com> wrote:
There are a few things my physical eye and mind's eye, working in tandem, can do pretty well, using an internal imaginary screen superimposed over the real scene before me. (NB: I am not using the word "real" and "imaginary" in the mathematical sense here!) I'll try to describe the operations without giving away the use of them I'll make, so forgive me if these descriptions seem a bit abstract.
#1. Given a real or imaginary point, I can "draw" an imaginary horizontal or vertical line through it. (It's important that the lines be horizontal and vertical; I can draw imaginary horizontal and vertical lines much more accurately than lines pointing in other directions. This is hardwired.)
The line doesn't really exist for long, and fades as fast as I draw it (by which I mean track it with my gaze), but it lasts long enough for me to do operation #2.
#2. Given an imaginary line and a real line, I can find the imaginary intersection point.
And:
#3. Given two real lines, I can recognize when they are symmetrical around the vertical axis (and if they are not, I can recognize which of the two lines is more vertical than the other).
I used these mind's eye primitives in devising my method of estimating the aspect ratio of a rectangle that I can hold in my hands and rotate.
Jim Propp
On Fri, Jul 27, 2018 at 10:34 AM, James Propp <jamespropp@gmail.com> wrote:
I just took a small cylindrical part to a hardware store, where the clerk, after determining that it wasn't a standard-sized (American) part and therefore was probably metric, seemed pretty unhappy about letting me borrow his metric ruler to take the part's measurements; together we were able to measure the diameter of the cylinder (1 cm) but he got pretty annoyed with me during the process so I didn't feel I could further impose on him by measuring the length.
(Side-note: he placed the small cylinder with one circular face resting on the table, positioned it between the 9cm mark and the 10cm mark, and announced that the diameter was 10 centimeters. I expressed incredulity, and said "No, it can't be 10 centimeters. ... Look, 10 minus 9 is 1." Maybe I should have been gentler about it.)
While walking home, I tried to figure out the length of the cylinder by eye, using the fact that a cylinder viewed from its side is a rectangle, and using the fact that I already knew one side of the rectangle (i.e. the measured diameter of the cylinder). I found a pretty good method; I'm wondering if any of you will think of better (or more amusing) ones.
I'll post my method after others post theirs.
Jim Propp