Re: a tangency thought experiment

In article
<30bd5aa0-b48b-4c8f-909a-b99d81e083ed@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
calvin <crice5@xxxxxxxxxxxxxx> wrote:

Imagine
this:

You are walking in perfect three dimensional Euclidean
space,
on a perfect plane, and you are approaching a perfect
sphere,
four times your height, which is at rest on the plane,
and
tangent to it. Thus the sphere shares exactly one point
with
the floor
plane.

This sphere has some strange powers: The height of
anyone
walking underneath the plane of its equator is
continuously
made equal to half the vertical distance from the floor
plane
to the surface of the sphere; you are kept alive
and
functioning normally, no matter how small you become, as
real
air becomes replaced by breathable 'ether', and the force
of
gravity is adjusted appropriately, etc.; you are always
able
to see and distinguish between the surfaces of the sphere
and
the floor plane; and you are continuously given knowledge
of
the direction to the point of
tangency.

Your goal is to view the point of
tangency.

As you walk underneath the rim of the sphere's equator,
your
size rapidly reduces, as does your speed, as you continue
to
function 'normally' in this unusual environment. The
'speed'
of the passage of time does not change, but it seems to
slow
down because your progress becomes slower, due to
your
decreasing
size.

The rate of decrease of your size slows down as the
'ceiling'
the surface of the sphere, becomes more horizontal, and
seems
flatter as its curvature in your vicinity decreases, due
to
your decreasing
size.

As you continue approaching the point of tangency,
the
ceiling, in your vicinity, becomes increasingly flat
and
parallel to the floor, always remaining at twice your
height,
but you start to realize (if you haven't already figured
it
out) that you may never reach your goal of being in
position
to observe the point of tangency. In front of you,
the
ceiling always dips down to touch (apparently) the floor,
and
behind you the ceiling always remains completely above
eye
level, though to observe these effects, you need (and
have)
the ability to set your handy pocket telescope to
sufficient
power at all times. So even with the use of your
wonderful
telescope, and even by getting your eye down close to
floor
level, the line of sight to the point of tangency is
always
blocked by the bulge of the
sphere.

An odd thing: At any time, if you turn at a right angle
to
your direction of progress, you may then walk through a
full
circle whose center is the point of tangency. This will
take
a long time, because you're so small, but it will be a
finite
length of time, nevertheless. Yet, if you turn back in
the
direction of the point of tangency, and walk for that
same
finite length of time, you are still in the same
predicament
of being unable to approach the point of tangency. After
a
long enough time of walking, you can make the ceiling in
your
vicinity as close to flat, and as close to horizontal, as
you
like. Choose the tolerance that you desire, and you
can
achieve it by walking for a finite length of time. But
that
will never be enough to gain the vantage point which you
seek.

Finally, you give up in dispair, and just stand there and
try
to visualize your elusive destination. You know where it
is,
because you can walk in a circle around it, whenever you
like.
You can even calculate how far away it is (based on a few
very
fine measurements of the height of the ceiling, and a
little
knowledge of spherical geometry). But you must rely on
your
imagination, in order to visualize the Euclidean wonder
which
you
seek.

And what does your imagination tell you? It might tell
you
that if you were 'there', maybe by flying toward it
with
infinitely increasing speed (based on your ever shrinking
size),
you would discover that at the very place where the ceiling
becomes
most perfectly horizontal, and seemingly least capable of
sharing
only one point with the floor, that is the place where it
happens.
The exact place where horizontality of the ceiling is
achieved
is also the exact plact where it is required to deliver the
sharpest
spike conceivable, and touch the floor at only one
point.

Not only does your quest fail, but your imagination of the
nature
of your goal fails.

Imagine a Usenet post that is formatted properly.
.

Relevant Pages

  • a tangency thought experiment
    ... four times your height, which is at rest on the plane, ... Thus the sphere shares exactly one point ... tangency. ... ceiling, in your vicinity, becomes increasingly flat ...
    (sci.math)
  • Re: infinity
    ... Cartesian plane. ... The plane and the punctured sphere are. ... >> and consider lines through the topmost point of the shpere intersecting ... The topmost point does not "map" to anything. ...
    (sci.math)
  • Re: infinity
    ... > Cartesian plane. ... The plane and the punctured sphere are. ... >>> and consider lines through the topmost point of the shpere intersecting ... > The topmost point does not "map" to anything. ...
    (sci.math)
  • Re: infinity
    ... >> A one-point compactification as in both axes meeting in more than one ... > Put a sphere tangentially on top of a horizontal plane in a 3D space, ... > and consider lines through the topmost point of the shpere intersecting ...
    (sci.math)
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