Download How do we characterize straight?

HOW DO WE CHARACTERIZE
STRAIGHT?
WHAT IS STRAIGHT ON A ________?
The notion of (intrinsically) straightness is
characterized through symmetries!
Any path that is (intrinsically) straight on a
surface is called a geodesic.
[This would be a line on the plane or a great circle
on a sphere.]
Reflection-through-itself symmetry

This can be seen by placing a mirror on the geodesic
and “testing” that the one half of the surface is
reflected onto the other half of the surface.
Reflection-perpendicular-to-itself symmetry

A reflection through any geodesic will take any
geodesic perpendicular to the original geodesic onto
itself.
B'
P
B
Half-turn symmetry

A rotation through half of a full revolution about any
point P on a geodesic interchanges the part of the
geodesic on one side of P with the part on the other
side of P.
B'
P
B
Rigid-motion-along-itself symmetry

Translation along the geodesic – being able to move
along the geodesic without leaving the geodesic.
B'
B
Point symmetry (or Central symmetry)

Viewed intrinsically, central symmetry through a point
P on the surface sends any point A to the point A’ at
the same geodesic distance from P but on the
opposite side.
B'
P
B
Side-Angle-Side Congruency


Are two triangles congruent if two sides and the
included angle of one are congruent to two sides
and the included angle of the other?
Is SAS Congruency true on the sphere? If so, reflect
on how you might prove it for all triangles. If it is
not true for all triangles, see if you can find a set of
conditions on the triangles so that SAS Congruency
is true.
Angle-Side-Angle Congruency


Are two triangles congruent if one side and the
adjacent angles of one are congruent to one side
and the adjacent angles of another?
Is ASA Congruency true on the sphere? If so, reflect
on how you might prove it for all triangles. If it is
not true for all triangles, see if you can find a set of
conditions on the triangles so that ASA Congruency
is true.
Connection to Algebraic Thinking



Generic Problem: Find the maximum number of
regions you can create on the given surface by
using a given number of geodesic, i.e., 0 geodesics,
1 geodesic, 2 geodesics, 3 geodesics, 4 geodesics,
5 geodesics, and 6 geodesics. Record your data in
a table, and then generalize the pattern for n
geodesics.
A) You are to use lines on the plane.
B) You are to use great circles on the sphere.