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The Untold Story of the Line at Infinity
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Mark Hopkins
Aug 25, 1995, 3:00:00 AM8/25/95
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(The 25th Untold Story)
From go...@dcs.shef.ac.uk (O C Gopinath):
> What is the meaning of "Parallel lines meet at Infinity"?
Draw two parallel lines in perspective and they will meet at a point on
the Horizon. The Horizon is at infinity. Every parallel line you draw
pointing in the same direction will meet at that same point.
Another point on the Horizon will be the nexus for another set of parallel
lines pointing in another direction. Thus, each point on the Horizon equals
a Direction. So a Point at Infinity may be viewed as the idealization of
the concept of Direction. At the same time, every Direction represents the
tendency to point toward the Point at Infinity that represents that Direction.
Direction is meant here in the sense of undirected direction (North-South,
rather than just North, or just South). So North = South.
Actually, the Earth is curved, so the horizon is 3-4 miles away depending
on how tall you are. This phenomenon was explained in the First Testament
of Middle Earth, known as the Silmarillon. You see, in the beginning
the Earth was actually flat. In the distant West AT the Horizon (not
near it, but at it!) is where the demi-gods lived, called the Valar.
A powerful island kingdom had the arrogance to actually try and go to,
land on, and conquer the land of the Valar -- where going West beyond
a certain line had been strictly verboten.
They actually reached the land of the Valar and this got the Valar pissed.
Not just pissed, but DAMN pissed. So they destroyed the whole contigent,
sank the island, and basically trashed the entire western coast of the
neighboring continent, known as Middle Earth.
But they didn't just stop there. The Land of the Valar, which lay at the
end of the West was cleaved from Earth and from the entire spacetime
continuum and the Earth was made to curve down and around back upon itself.
And even more, it was made to do so in every direction so that thenceforth
no Direction could ever be reached without ending up bending around and
coming back where you came from. This is why the Earth is now shaped
like a squashed sphere.
So on account of this event, even though the Horizon is in reality an infinite
distance away and even though all good artists draw their parallel lines
to meet at a Point At Infinity -- in commemoration of the Golden Age --
the horizon is now only a few miles away, no longer at the End of the
Earth, but at the Bend of the Earth.
However, legend has it that every now and then it is possible for a
traveller to still find and travel along the Straight Path and reach
the Land of the Valar in the West.
But that'll just be our little secret.
The basic axioms are trivially easy because there's no longer the
hassle of worrying about distinguishing cases based on whether things are
parallel or not.
1) The Incidence Axioms
Every two points lie on a single line & every two planes intersect in a
single line.
Every two lines that intersect lie in a single plane & every two lines that
lie in a single plane intersect.
Every line has at least three points on it & every line lies in at least
three planes.
Every plane has at least three lines in it & every line contains at
least three points.
2) The Existence Axiom
There's a point and a plane not containing the point. Where? There.
3) That Harmonic whachamajiggy Axiom and the other Axioms
Blah, blah, blah
This Geometry is just Euclidean geometry in disguise. Define the plane
mentioned in Axiom 2 as the Plane at Infinity. Then define the following:
DEFINITION: Two objects are parallel if their intersection lies entirely
in the Plane At Infinity.
DEFINITION: An object is called Real if it contains at least one point not in
the Plane at Infinity, else it is called Perfect.
Notice that a Perfect point is always parallel with everything that contains
it. That's what makes it perfect. But a Real point is never parallel with
anything.
Anyway ... the result is:
THEOREM: Euclid's Fifth Postulate
Given a Real line L, every other point not on line L lies in a Real
line parallel to L.
Proof: Let L be a Real line and A one of its Real points. Let P be a
point not contained in line L. Then the following happens in order:
* A & P lie on a line, call it M.
* L and M lie in a Real plane, call it N.
* N intersects the Plane at Infinity at a line, call it H.
* H intersects line L at a point, call it W
* W and P lie on a unique line, call it X.
Thus
* X intersects L at W, which lies in the Plane at Infinity.
Thus
* X is a Real line parallel to L containing point P.
Another result, just to ensure the discussion is not vacuous:
THEOREM: Real points exist. So do Real lines and planes.
Let P be the point mentioned in Axiom 2. Then it's not contained in
the Plane at Infinity, so it's Real. Let H be a line lying in the Plane
at Infinity and W a point on H. Then P and W are contained in a line,
call it L. Let M be the plane containing both L and H.
Then L and M both contain a Real point, P. Therefore L and M are a
Real line and plane respectively.
Post-Scriptum:
I hope Vos Savant enjoys eating words with that infamous comment about
non-euclidean geometries. As far as I'm concerned, the only reason V.S.
scored tops on the IQ test is because I never took one.
From go...@dcs.shef.ac.uk (O C Gopinath):
> What is the meaning of "Parallel lines meet at Infinity"?
Draw two parallel lines in perspective and they will meet at a point on
the Horizon. The Horizon is at infinity. Every parallel line you draw
pointing in the same direction will meet at that same point.
Another point on the Horizon will be the nexus for another set of parallel
lines pointing in another direction. Thus, each point on the Horizon equals
a Direction. So a Point at Infinity may be viewed as the idealization of
the concept of Direction. At the same time, every Direction represents the
tendency to point toward the Point at Infinity that represents that Direction.
Direction is meant here in the sense of undirected direction (North-South,
rather than just North, or just South). So North = South.
Actually, the Earth is curved, so the horizon is 3-4 miles away depending
on how tall you are. This phenomenon was explained in the First Testament
of Middle Earth, known as the Silmarillon. You see, in the beginning
the Earth was actually flat. In the distant West AT the Horizon (not
near it, but at it!) is where the demi-gods lived, called the Valar.
A powerful island kingdom had the arrogance to actually try and go to,
land on, and conquer the land of the Valar -- where going West beyond
a certain line had been strictly verboten.
They actually reached the land of the Valar and this got the Valar pissed.
Not just pissed, but DAMN pissed. So they destroyed the whole contigent,
sank the island, and basically trashed the entire western coast of the
neighboring continent, known as Middle Earth.
But they didn't just stop there. The Land of the Valar, which lay at the
end of the West was cleaved from Earth and from the entire spacetime
continuum and the Earth was made to curve down and around back upon itself.
And even more, it was made to do so in every direction so that thenceforth
no Direction could ever be reached without ending up bending around and
coming back where you came from. This is why the Earth is now shaped
like a squashed sphere.
So on account of this event, even though the Horizon is in reality an infinite
distance away and even though all good artists draw their parallel lines
to meet at a Point At Infinity -- in commemoration of the Golden Age --
the horizon is now only a few miles away, no longer at the End of the
Earth, but at the Bend of the Earth.
However, legend has it that every now and then it is possible for a
traveller to still find and travel along the Straight Path and reach
the Land of the Valar in the West.
But that'll just be our little secret.
The basic axioms are trivially easy because there's no longer the
hassle of worrying about distinguishing cases based on whether things are
parallel or not.
1) The Incidence Axioms
Every two points lie on a single line & every two planes intersect in a
single line.
Every two lines that intersect lie in a single plane & every two lines that
lie in a single plane intersect.
Every line has at least three points on it & every line lies in at least
three planes.
Every plane has at least three lines in it & every line contains at
least three points.
2) The Existence Axiom
There's a point and a plane not containing the point. Where? There.
3) That Harmonic whachamajiggy Axiom and the other Axioms
Blah, blah, blah
This Geometry is just Euclidean geometry in disguise. Define the plane
mentioned in Axiom 2 as the Plane at Infinity. Then define the following:
DEFINITION: Two objects are parallel if their intersection lies entirely
in the Plane At Infinity.
DEFINITION: An object is called Real if it contains at least one point not in
the Plane at Infinity, else it is called Perfect.
Notice that a Perfect point is always parallel with everything that contains
it. That's what makes it perfect. But a Real point is never parallel with
anything.
Anyway ... the result is:
THEOREM: Euclid's Fifth Postulate
Given a Real line L, every other point not on line L lies in a Real
line parallel to L.
Proof: Let L be a Real line and A one of its Real points. Let P be a
point not contained in line L. Then the following happens in order:
* A & P lie on a line, call it M.
* L and M lie in a Real plane, call it N.
* N intersects the Plane at Infinity at a line, call it H.
* H intersects line L at a point, call it W
* W and P lie on a unique line, call it X.
Thus
* X intersects L at W, which lies in the Plane at Infinity.
Thus
* X is a Real line parallel to L containing point P.
Another result, just to ensure the discussion is not vacuous:
THEOREM: Real points exist. So do Real lines and planes.
Let P be the point mentioned in Axiom 2. Then it's not contained in
the Plane at Infinity, so it's Real. Let H be a line lying in the Plane
at Infinity and W a point on H. Then P and W are contained in a line,
call it L. Let M be the plane containing both L and H.
Then L and M both contain a Real point, P. Therefore L and M are a
Real line and plane respectively.
Post-Scriptum:
I hope Vos Savant enjoys eating words with that infamous comment about
non-euclidean geometries. As far as I'm concerned, the only reason V.S.
scored tops on the IQ test is because I never took one.
John Baez
Aug 31, 1995, 3:00:00 AM8/31/95
to
In article <41lf5d$t...@omnifest.uwm.edu>,
Mark Hopkins <ma...@omnifest.uwm.edu> wrote:
Mark Hopkins <ma...@omnifest.uwm.edu> wrote:
>They actually reached the land of the Valar and this got the Valar pissed.
>Not just pissed, but DAMN pissed. So they destroyed the whole contigent,
>sank the island, and basically trashed the entire western coast of the
>neighboring continent, known as Middle Earth.
>
>But they didn't just stop there. The Land of the Valar, which lay at the
>end of the West was cleaved from Earth and from the entire spacetime
>continuum and the Earth was made to curve down and around back upon itself.
>And even more, it was made to do so in every direction so that thenceforth
>no Direction could ever be reached without ending up bending around and
>coming back where you came from. This is why the Earth is now shaped
>like a squashed sphere.
But wait: you said that originally the Valar lived *at* the horizon.
So the horizon was originally part of the earth: perhaps prime
real estate not zoned for mortals, but still in some sense part
of the earth. So you must be saying that originally the earth was a
projective plane, and the Valar punished us by lifting us up to a double
cover?
> I hope Vos Savant enjoys eating words with that infamous comment about
>non-euclidean geometries. As far as I'm concerned, the only reason V.S.
>scored tops on the IQ test is because I never took one.
Nobody with any brains would take an IQ test unless they were
forced to.
Mark Hopkins
Sep 3, 1995, 3:00:00 AM9/3/95
to
JB = ba...@math.mit.edu (John Baez):
JB> But wait: you said that originally the Valar lived *at* the horizon.
So the horizon was originally part of the earth: perhaps prime
real estate not zoned for mortals, but still in some sense part
of the earth. So you must be saying that originally the earth was a
projective plane, and the Valar punished us by lifting us up to a double
cover?
A misquote. The quote you're referring to is this:
"Actually, the Earth is curved, so the horizon is 3-4 miles away depending
on how tall you are. This phenomenon was explained in the First Testament
of Middle Earth, known as the Silmarillion. You see, in the beginning the
Earth was actually flat. In the distant West AT the Horizon (not near it,
but at it!) is where the demi-gods lived, called the Valar."
_The Unwritten Book_, Untold Story 25, Paragraph 4
from which it should be clear that the Horizon was, and still is, at Infinity,
whereas the horizon is merely a few miles away. The term projective plane is
appropriate [1],[2] but not your characterization of the alteration: the
Horizon is nowhere to be found on the Earth, "... the loremasters of Men said
that a Straight Road must still be, for those that were permitted to find it.
And they taught that, while the new world fell away, the old road and the path
of the memory of the West still went on, as it were a mighty bridge invisible
that passed through the air of breath and of flight (which were bent now as
the world was bent), and traversed Ilmen [3] which flesh unaided cannot
endure, until it came to Tol Eressea, the Lonely Isle, and maybe even beyond,
to Valinor, where the Valar still dwell..." (_Akallabeth_ p. 349).
(Also, the Valar did not really effect the change themselves, but called upon
Iluvatar to change the foundation of the Earth.)
Technically, the account is part of the _Akallabeth_, not the _Silmarillion_.
The authorship of both should be well-known by everyone since more than a
million copies of the _Silmarillion_ were sold. The _Akallabeth_ is the
account of the rise and fall (and sinking) of Numenor and the catastrophic
change in Middle Earth, and is bound with the _Silmarillion_.
Notes:
[1] "But soon the Sun was drawn down by the servants of Ulmo, and went then
in haste under the Earth, and so came unseen to the east and there
mounted the heaven again..." (p. 116 "Of the Sun and Moon",
_Silmarillion_) is commonly interpreted as stating that East and West
were at a common nexus which when crossed by the Sun began a new day.
By isotropy, all other pairs of opposing directions were inferred to have
the same property of having their respective points on the Horizon be one
and the same.
[2] The non-orientability of the universe implies that the neutrino has a
positive rest mass since otherwise parity would be violated. It also
implies that the superiority of left-handed people is merely a biological
contingency having to do with our inability to traverse the Straight Path
and cross the Horizon, rather than a feature inherent to the physical
structure of the universe.
[3] Defined as the region above the air where the stars are (p. 419
"Index of Names", _Silmarillion_).
JB> But wait: you said that originally the Valar lived *at* the horizon.
So the horizon was originally part of the earth: perhaps prime
real estate not zoned for mortals, but still in some sense part
of the earth. So you must be saying that originally the earth was a
projective plane, and the Valar punished us by lifting us up to a double
cover?
"Actually, the Earth is curved, so the horizon is 3-4 miles away depending
on how tall you are. This phenomenon was explained in the First Testament
Earth was actually flat. In the distant West AT the Horizon (not near it,
but at it!) is where the demi-gods lived, called the Valar."
from which it should be clear that the Horizon was, and still is, at Infinity,
whereas the horizon is merely a few miles away. The term projective plane is
appropriate [1],[2] but not your characterization of the alteration: the
Horizon is nowhere to be found on the Earth, "... the loremasters of Men said
that a Straight Road must still be, for those that were permitted to find it.
And they taught that, while the new world fell away, the old road and the path
of the memory of the West still went on, as it were a mighty bridge invisible
that passed through the air of breath and of flight (which were bent now as
the world was bent), and traversed Ilmen [3] which flesh unaided cannot
endure, until it came to Tol Eressea, the Lonely Isle, and maybe even beyond,
to Valinor, where the Valar still dwell..." (_Akallabeth_ p. 349).
(Also, the Valar did not really effect the change themselves, but called upon
Iluvatar to change the foundation of the Earth.)
Technically, the account is part of the _Akallabeth_, not the _Silmarillion_.
The authorship of both should be well-known by everyone since more than a
million copies of the _Silmarillion_ were sold. The _Akallabeth_ is the
account of the rise and fall (and sinking) of Numenor and the catastrophic
change in Middle Earth, and is bound with the _Silmarillion_.
Notes:
[1] "But soon the Sun was drawn down by the servants of Ulmo, and went then
in haste under the Earth, and so came unseen to the east and there
mounted the heaven again..." (p. 116 "Of the Sun and Moon",
_Silmarillion_) is commonly interpreted as stating that East and West
were at a common nexus which when crossed by the Sun began a new day.
By isotropy, all other pairs of opposing directions were inferred to have
the same property of having their respective points on the Horizon be one
and the same.
[2] The non-orientability of the universe implies that the neutrino has a
positive rest mass since otherwise parity would be violated. It also
implies that the superiority of left-handed people is merely a biological
contingency having to do with our inability to traverse the Straight Path
and cross the Horizon, rather than a feature inherent to the physical
structure of the universe.
[3] Defined as the region above the air where the stars are (p. 419
"Index of Names", _Silmarillion_).
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