  et
us begin our analysis of the puzzle with Figure
1, which shows a schematic representation of a person walking in low-heel
shoes, superimposed upon which is a person walking in high-heel shoes.
The shapes and dimensions of the foot are derived from the x-ray photograph
in the Chase Me, Catch Me illustration.
In the natural stride, the heel of the foot at its back-most position is
elevated and thus not acted upon by the high heel. In its forward-most
position, the heel of either foot (whatever its shodding) must engage the
walking surface. The high-heel dimension has been chosen to assure
that the angle at the ankle is at its limit, which keeps the shin nearly
vertical and the forward knee necessarily bent. As indicated, the
stride-angle at the hip is held constant between the two walkers.
A direct comparison in stride length in Figure 1, along
with simple geometrical relationships, will provide an immediate solution
to the puzzle: The wearer of high-heel shoes experiences
the shortening of each stride by...
That, presumably, makes the chase unfair, the catch easy. If you are curious about the three remaining choices, you will find that they are all correct -- or can be correct, depending on a few assumptions (see Vetruvian Man).
Aviation terminology has been appropriated here to describe maneuvers of the pelvis as it flies gracefully along above the surface. Thus "pitch" appeared in Figure 1 for horizontally positioning the CG. Meanwhile, "roll" and "yaw" were both postulated as zero for that case.  igure
2 addresses another aspect of the walking challenge, keeping the body moving
along at a constant elevation above the surface. The foot
wearing a high-heeled shoe passing under the body during the stride will
make lifting of the CG decidedly more pronounced. Pelvic roll
can be applied to reduce the amount of vertical motion by lowering the
hip connected to the forward-most leg. That does increase the flexing
at the knee, however, which diverts some amount of the walker's effort,
making the chase even more unfair.
Shown in Figure 3 is the high-heel walker now deploying plenty of pelvic yaw, with the apparent intention of lengthening the stride -- recovering some of the disadvantage in the chase. The note in Figure 3 is salient, however, for we are reminded that the walker postulated for the low-heel reference stride does not make use of pelvic stratagems. Indeed, all proposed improvements in the high-heel stride must be offset by the fact that for this simplified (ceteris parabus) comparison, the low-heeled walker is unfairly constrained. Ah, but there is not a dress-code on the planet that forbids the wearer of low-heeled shoes to go pitching, rolling, and yawing along the sidewalk enroute to workplace (backpacking a pair of high-heeled shoes, presumably). Background First heard in a high school locker room back in the forties, the beckoning title of the Chase Me, Catch Me puzzle is a shortened version of enticements wishfully attributed to young ladies in high-heels. The complete version appended the phrase “wed me, bed me" (in that order, of course).  The
Vetruvian Man (The
Man in Action) sometimes
called the "Canon of Proportions" is a drawing with accompanying notes
by Leonardo da Vinci made around the year 1490 depicting a human figure
in two superimposed positions with arms apart and simultaneously inscribed
in a circle and square. That drawing is at least a place to start.
Relative to height, we find thigh = shin = 26.1%;
foot
= 12.4%;
ankle-to-ball-of-foot = 9.1%.
Taking a range of heights for most likely wearers of high-heel shoes as 5' 4" to 5' 9", a median heel-height of 4", and a median low-heeled stride-length of 2' 6", we see the equivalance of.... (a) 1/4th reduction in length of stride, |
alking
with only two feet is a precarious activity, if you stop to think about
it, which few of us ever do. Walking is really what might be called
"arrested falling." Notice that Figure 1 postulated the location
of the walker's center of gravity (CG), generally at about the level of
the navel. The CG serves a vital function in upright locomotion,
for we observe that the CG must always be positioned some distance ahead
of the back-most location of the walker's foot. The incipient
fall is arrested by the action of the other foot, of course, when it strikes
the surface in its forward-most position.
Perhaps
you noticed in Figure 1 above that for the low-heel walker, the CG was
assumed to be located directly above the hips. For the walker
in high-heels, the hips will be drawn farther aft, closer to the back-most
location of the walker's foot. That shortens the "moment arm" (leverage)
for empowering forward motion, resulting in a slower walking speed.
The remedy is elementary. The wearer of high-heeled shoes merely
leans forward at the hip to move the CG to about the same horizontal location
as that enjoyed by the wearer of low-heeled shoes. And the chase
goes on.
All
aesthetic considerations have been set aside in this analysis. The
Chase Me, Catch Me puzzle is interested
only in relative stride-length, to which we now return...