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Unformatted text preview: I— l
MATHEMATICAL OBJECTS MATHEMATICAL WORLD VS. PHYSICAL
UNIVERSE Math world: numbers, functions, operations,
geometries, inﬁnities, limits. Existence of math objects. Creation of math objects:
—from sets
—by deﬁnition from existing math objects “Ideal” and perfect nature of math objects Study and exploration of math objects and their
properties by deduction (“proof”). Assumptions and
theorems (= facts) Remarkable agreement of community of
scientists and mathematicians as to correctness of
mathematical investigations and results 12 2DIMENSIONAL EUCLIDEAN GEOMETRY (E2)
and
3—DIMENSIONAL EUCLIDEAN GEOMETRY (E3) How we visualize them and make copies if them Form of the deﬁnition for E2 (one of several
approaches):
inﬁnite set of basic “atomic” objects called
oints
special sets of points to be called ﬁnes Form of the deﬁnition for E3:
inﬁnite set of basic “atomic” objects called
points
special sets of points to be called lines
special sets of points to be called planes Deﬁnitions: A given set S of points in E2 is said to be
collinear if there is some line L in E2 such that every point
in S is also in L. A given set S of points in E3 is said to be collinear if
there is some line L in E3 such that every point in S is also
in L. A given set S in E3 is said to be coplanar if there is
some plane M such that every point in S is also in M. 13 INCIDENCE RELATIONS
(Part of deﬁnition of E2 and E3) For E’: Given two distinct points, there is a unique
(one and only one) line which contains both points. For any two points P and Q, there is a measure which
gives the distance d(P,Q) between P and Q. For E3: Given two distinct points, there is a unique
(one and only one) line which contains both points. Given two distinct points in a plane M, the unique line
determined'by those points must lie entirely in (be a subset
of) M. Given three distinct noncollinear points, there is a
unique plane which contains all three points. (Equivalent
assertion: Given two intersecting lines, there is a unique
plane which contains both lines.) Given two planes which intersect (have a common
point), then their intersection (the set of all common points)
must be a line. For any two points P and Q, there is a measure which
gives the distance d(P,Q) between P and Q. Every plane in E3 is a copy of E2. 14 PARALLELISM
(Deﬁnitions) For E2: Two lines are said to be parallel if they do not
intersect. For E3: Two lines are said to be parallel if they do not
intersect but do lie in a common plane. Two lines are said to be _s_l_gg:_v_v if they do not intersect
and do not lie in a common plane. If each of two given distinct lines is parallel to the
same third line, then those two given lines must be parallel. CONGRUENCE The idea of congmence is fundamental for E2 and E3.
A mapping on E2 (or E3) is a function f which carries every
point P to some corresponding point f(P). A mapping is
said to be an isometg if for every pair of points P and Q,
d(f(P),f(Q)) = d(P,Q). A set of points S is said to be
congr_uent to a set of points 8’ if there is an isometry which
carries S exactly onto 8'. An angle consists of two rays (halflines) from a
common vertex point. An angle is said to be a right angle if
it is congruent to the “supplementary” angle formed by
reversing the direction of one of its rays. ‘ I  5
PERPENDICULARIT Y For E2: Two lines are said to be pegp endicular, if they
intersect at a right angle. For E3: Two lines are said to be peppendicular, if they
intersect at a right angle. A line and a plane are said to be pegpendicular, if they
intersect at a single point P and if the given line is
perpendicular to every line lying in the given plane and
going through P. PROJECTIONS
For E’:
Fact: For a given ﬁle L and any point P, there is a
unique line L' such that L’ contains P and is perpendicular to L. The intersection point of L’ with L is called the
projection of point P on L. For any set S of points and any line L, the set of all
projections on L of points in S is called the projection of S
on L. For E3: Fact: For a given plane M and any point P, there
is a unique line going through P and perpendicular to M.
The intersection point this line with M is called the
projection of the point P on the plane M. For any set S of
points, the set of all projections on M of points in S is
called the projection of S on M. I  6
PROJECTIONS (continued) F act: For a given line L and any point P, there is a
unique plane going through P and perpendicular to L. The
intersection point of L with this plane is called the
projection of the point P on the line L. For any set S of
points, the set of all projections on L of points in S is called
the projection of S on L. MISCELLANEOUS DEFINITIONS
(to be found in Chapter 1 of the text) parallelogram
dihedral angle
polygon
ngon
quadrilateral
planar polygon
convex polygon
regular polygon
polyhedron
nhedron
convex polyhedron
regular polyhedron
octahedron
dodecahedron
icosahedron
parallelepiped
prism I  7
DEFINITIONS continued
pyramid
sphere
cylinder
cone Also: simple mensuration formulas (area and volume) ...
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 Two '04
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