2nd Addendum: Partial Mathematization of the Walking Decision Making Article

.

By Herb Wiggins, M.D.; Clinical Neurosciences; Discoverer/Creator of the Comparison Process/CP Theory/Model; 14 Mar. 2014.

Copyright © 2019

.

.

The method of Hypotenuses for walking can be partly mathematized. It’s still Cx Sys, so it cannot be completely expressed due to the complexity of options & interacting choices. It’s based upon the short cut method of taking the hypotenuse instead of a 90 deg. angle. The shortest distance between 2 points, IOW. And that is least energy, least distance and often least time, relatively to walking the two legs of the 90 deg. angle method. As we cannot buy time, the shortest distance/time often gives savings when walking. It depends on the routes, however. It’s complex system because it’s distance saved, energy thus saved, and often time saved. Walking on firm surfaces is faster, & takes less energy than walking on sand, or a grassy area, most cases. So while the distance is shorter, the time is longer. But it depends upon outcomes, largely, which cut through the Cx Sys and gives solutions to those highly complex problems.

.

Now, the method of longest hypotenuses gives us a basic way round all of this. For instance if walking from place to place, often we see (& easily show, mathematically) that a near 45 deg. hypotenuse is shorter/Shortest possible choice, than walking to the corner, then left or right. We short cut, that is, Least Energy (LE) routes the distance. IFF other factors are equal, then taking shorter distances also saves time, AND energy. But being the complex of the n=/>3, it’s complex system. However, it can be partly mathematized using the method of hypotenuses, as referred to, but not developed in the walking article or the First Addendum.

.

.

Clearly, the longest hypotenuse of 45 deg. is the best approach, or in many cases, the least time, energy, distance altogether. Let’s think about what happens when we walk a 45 deg. isosceles right triangle. This hypotenuse is clearly the shortest possible distance, and the greatest savings to walk a hypotenuse for obvious reasons. If we make one leg shorter, then the other is longer and vice versa. and it’s clear walking the longer hypotenuse loses distance because if we take it out to nearly the length of the other side, the short leg gets shorter and shorter and thus the hypotenuse length approximates that of the most distance, time, LE. So the 45 degree right triangle with the two legs being equal then the maximum distance between two point along the hypotenuse is saved, about 30% of the total of doing the two 90 deg. legs. If we test a 3, 4, 5 right triangle, we find a savings of only 28% so it’s clear the more the angles differ from 45 deg. the less distance is saved.

.

Thus when we go to cross a distance we do so at a 45 deg. angle from the exit of the 1 leg on the hypotenuse to the entry of the 2nd leg’s end.

.

A simple mathematical analysis of this geometry shows why this is the case, AND why the longest hypotenuse, 45 degree short cut is the best, AKA shortest, least time, least energy. Short cuts are simply another of the big Kategoria of least distance, being the best, because it also give least energy and least time, too, other factors being equal.

.

Now, we note that if we take sides of 1 next to the right angle, we have the right triangle rule, that the hypotenuse is the square root of the sums of the other two sides, which is 2 & a hypotenuse of 1.414. Thus if we walk the two legs we have walked 2 units. If we walk the hypotenuse we have saved 2 minus 1.414, or 0.586 units. If this is a mile it’s a lot of time, distance saved, is not thus the minimum % distance which can be saved being 0.586/2 or very close to 30%. or about 30% of a mile!!

.

However, what is the maximum % distance that can be saved when using the hypotenuse? And this is the key here. When the hypotenuse is nearly the length of the longest side, what’s saved is very close to, only the width of that short side. & that is just a NOT shorter to the lengths of the distance of the longest side, if a 45 deg. triangle. So if we want to save the most distance, time, LE, we take the Longest hypotenuse, all other walking conditions/factors being equal. That can be figured out exactly, idealistically, but not empirically, creating close approximations, but not exactly, using trigonometry. and won’t go into that, here.

.

Further to extend and summarize:

.

It turns out this method is complex system. The most savings is in the 45 deg. isosceles right triangle. However, that choice/option is not always there. IN those cases we take as close to that as we can, but also the Long rectangles we must use as the only options. Thus we do those knowing we save only the length of the shorter side. So we prefer the wide isosceles triangle hypotenuse, and walk at a 45 deg angle to the entry and exits when we can. Thus there is NO ideal solution because we cannot walk straight lines ideally and the math does NOT apply at all, but as approximations.

.

“Beyond the absolute”, AKA, understanding the limits of the idealisms in our words & descriptions, And our maths & measurements of all sorts!!!

.