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C00002 00002	Geometric concepts and relationships derived
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Geometric concepts and relationships derived



Equality of Tabc and  Tdef:
	TEQ(Tabc, Tdef) if
		they are equal as list structures, i.e. Obj-equal or EQUAL,
		since Triangles are specializations of Bags.
 
Would be good to phrase this as follows:

	TEQ(Tabc, Tdef) if Tabc and Tdef are triangles, and if ALLOF:
		LEQ(Lab Lde)
	   and	LEQ(Lbc Lef)
   	   and	LEQ(Lca Lfd)

	where LEQ means line-equal (absolutely equal, not just in length).
	and Lab probably means the result of Draw-line(algs a b).

	LEQ(Lab Lde) if
		they are equal as list structures, ie EQUAL or Obj-equal.

AM knows that a generalization of LEQ is Same-llength
	which is defined as having equal values of Distance.

One way to generalize (ALL-OF (f ...) (f...)(f ...)...) is to replace f by
	a generalization of f.

So, eg, AM will replace LEQ by Same-llength in TEQ.

Have a second defn of TEQ, in terms of Angle-equal (meaning really the
same, not just in measure), and know that a genl of Angle-equal is
Same-ameasure.

Generate same-ameasure and same-llength from the concept
Measure:Angle→Number, and the knowledge that Same-size is much more
common (among numbers) than Equality is (among angles; again, identiacl, not just
same size).
Knowledge must exist which says that if we have a predicate P,
	P:XxX→{T,F}
	which is very rarely satisfied, but we have an ANALOGOUS predicate
	Q:YxY→{T,F}
	which is much nicer, much more easily satisfied, then look for an operation
	f:X→Y
	(preferably a bijection; f will probably have a domain much bigger than X)
	(try to ensure that f(X) has many different values in Y)
	Then construct the new predicate S, 
	S:XxX→{T,F}
	defined as S(a,b) ≡ Q(f(a),f(b)).
	("Q is analogous to P" means, at least, that if P(a,b) then S(a,b)).

As a first step in this analogy morass, we can give AM special powers wrt Equality.
If P has as an algorithm (EQUAL BA1 BA2), then we can be sure it is a kindof equality.
Then all we have to do vis a vis Q is to ensure that Q is reflexive: ∀x. Q(x,x).
One way to do that is of course to choose Q to be another kind of Equality.
Then all we need ensure is that f is a func -- i.e., single--valued and defined on X.

This will in fact get you Same-llength, from Line-equal, and get you
Same-ameasure, from Measure-angle, and get you Same-size from Count.
It is thus a sort of "inverse-canonizing" process.

Analog of Goldb: If you have aset of angles of prime no. of degrees (all primes
0,2,3,5,7,11,...,179, where 0 is included because it is trivial to "have" it)
then any angle (from 0 to 180 degrees) can be approximated (to within 1 degree)
by adding a pair of these angles together.
The numberof primes involved is..., and this is probably close to the minimum
numberof diff. angles you'd need to so approximate any angle bya pair from your set.
In fact, if you spend a few minutes on this problem, you'll see it is in fact hard
to find a any reasonably small set of numbers having this property!
For example, 140 degrees is approximated by adding together these angles: 37,103.
It's realtively easy to find the rigth pair; you run down the primes backwards
(starting from the given angle, 140), and see if 140 - (the angle you're at) is prime.
You go backwards because there are very few primes at higher numbers.
In fact, in this case, you only look at ... pairs before finding the rigth one.


Thus the notions of congruence of triangles and angles falls out naturally.
Also the similarity of triangles.

Another way to specialize a structure of the form (name s1 s2... sn) where
each si is a substructure of type X:  Find a predicate P
which is fairly easily satisfied predicate P:XxX→T,F,  then we can specialize
the structure by requiring that some pair of, or that all pairs of, or that
a particualr pair of s1, s2, ... satisfy P.
This  may be already contained in the general INT part of STRUCTURE.

In fact, this WILL cause AM to look for triangles whose sides are equal,
if triangles are represented as a list of 3 LINES, rather than now, as 3 points.
But it will have to modify that INT criteria, to also consder triangles
which have just one pair of equal sides.

This may come about through the op. Measure-triangle, which returns a list of
the three lengths of the 3 sides of the triangle.
Then AM may ask that we isolate triangles whose measure contains 3 identical numbers.
These would be equilateral triangles.

Note there is no real knowledge of inequality, in the sense of trichotomy, so
that BETWEEN is not that important after all!