Download 11 Neutral Geometry III (Comparing geometries we`ve studied)

Comparing Geometries
Are the geometries we’ve studied compatible with the axioms we have just dictated?
Let’s compare the new geometries we’ve studied with the rules we’ve set up for “neutral”
geometry.
Let’s start with spherical geometry. All of the definitions and common notions apply—
once we identiy “straight line” with geodesics—because without these ultra basic concepts
we’d really be at a loss. So we should start with the axioms.
Spherical starts with a real bang, because it fails axiom 1! Most of the time there is
exactly one geodesic through two given points. But if you start with antipodes, you get
infinitely many lines! We must try to find a way to avoid this at some later point in time.
Spherical fails axiom 2 as well, because some lengths aren’t allowed. In particular, we
don’t allow lengths of π or longer. If we try to fix axiom 1, maybe we can fix this, too, by
making a very small sphere or something?
Axiom 3 is fine. At least, for any reasonable length and any center we can certainly
draw the requisite circle. And axiom 4 is fine as well—right angles are just fine in spherical.
Separation and Application also work fine in spherical. Lines, triangles, and circles do split
the sphere in two, and you are free to slide triangles around and they don’t change shape.
What about taxi geometry?
Axioms 1 and 2 are clearly fine this time. We did not really define lines in taxi geometry
because there isn’t a unique geodesic from one point to another unless they are aligned
horizontally or vertically. But if we take as taxi lines only the regular Euclidean lines—
which certainly are geodesics—both axioms 1 and 2 work Axiom 3 is fine as well. So is the
Separation axiom.
On the other hand, we find that the Application axiom and the all-right-angles are equal
axiom are at odds. Worse, actually, the Application axiom seems to be at odds with the
common notion that figures that coincide are equal. Consider somethig like the 2-2-2 right
triangle with vertices at A(0, 0), B(1, 1), and C(−1, 1), together with the 2-2-4 right triangle
with vertices at D(3, 0), E(5, 0), and F (3, 2). Clearly we can apply 4DEF to 4ABC with
D placed atop point A, side DE along AB and F on the same side of this common line as
C. So does E coincide with B?
If not, then either E is further away from A then B is, or it is closer. But it can’t be
further, because then AB < DE (because of the common notion that the part is less than
the whole), which says that 2 < 2. Not possible! But it can’t be closer, either, so B and E
must align.
But then because both A and D are right angles—and all right angles are the same—both
C and F must be along the same ray from A. And by the exact same argument as in the
previous paragraph, they, too, must coincide. But if points C and F coincide, and points B
and E coincide, then BC must coincide with EF , since they’re both (straight) line segments
with the same endpoints. But then by the common notion that things that coincide must
be equal, we must have the lengths BC and EF equal. Which is impossible because one of
them is 2 and the other is 4.
Maxi geometry is just as bad. And metro geometry is worse! Since they mess with the
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utterlyl basic common notions, its probably best just to not think of them as geometry at
all. They are better studied in some generalization of geometry where we loosen the rules
even more. On the other hand, we will try to rescue spherical geometry, which only doesn’t
work because we made some arbitrary choices regarding antipodes and distances, and maybe
we can figure out how to work around them.
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