Convex:
Any two points in the figure
have a line segment connecting them. If that line segment is always inside the figure, then the
figure is called "convex".
A tiling is a regular tiling if (i)it has a single tile shape
that is a regular polygon and (ii) the vertices and edges of the tiles
coincide (no overlapping edges)
A tiling is a semi-regular tiling if (ii) each tile shape is a
regular polygon, (ii) the vertices and edges of the tiles coincide (no
overlapping edges) and (iii) every vertex has the same polygon types
arranged around
it.
The numbers represent the number of sides in the poygons.
The order indcates the order in which the poygons are arranged
about a vertex.
Semiregular Tilings: global results!
Look at the results using wingeometry.
Student
lesson (Math Forum) a place for further explorations on-line.
Symmetry Ideas
Reflective symmetry: BI LATERAL
SYMMETRY
T C O
0 I A
Folding line: "axis of symmetry"
The "flip."
The "mirror."
R(P) = P' : A Transformation. Before: P
.... After : P'
If P is
on the line (axis), then R(P)=P. "P remains fixed by the reflection."
If P is not on the axis, then the line PP' is
perpendicular to the axis and if Q is the point of intersection of PP'
with
the axis then m(PQ) = m(p'Q).
Definition: We say F has a reflective symmetry wrt a line l if there is a
reflection R about the line l
where
R(P)=P' is still an element of F for every P in F....i.e.. R (F) =
F. l is called the axis of symmetry.
Examples of reflective symmetry:
Squares... People
Rotational Symmetry.
Center of rotation. "rotational pole" (usually
O) and angle/direction of rotation.
The "spin."
R(P) = P' : A transformation
If O is the center then R(O) = O.
If the angle is 360 then R(P) = P for all
P.... called the identity transformation.
If the angle is between 0 and 360 then
only the center remains fixed.
For any point P the angle POP' is the
same.
Examples of rotational symmetry.
Now... what about finding all the
reflective
and rotational symmetries of a single figure?
Symmetries of playing card.... classify the cards
by
having same symmetries.
Notice symmetry of
clubs, diamonds,
hearts, spades