Patterns and Isometries

Math is a study of patterns (can be numerical, logical, geometric)

• Geometric patterns are nice to look at because of symmetry

• Pattern has symmetry; : If you can shift, flip, or rotate the design without messing it up, that’s an isometry that keeps the pattern intact.

Transformation - process shifting points of the plane to possible new locations within the plane

• Isometry of the plane - mapping preserving distance and shape (same in shape, and size)

TYPES OF ISOMETRIES

• Translation - moves a shape in a given direction sliding it UP, DOWN, SIDEWAY or DIAGONALLY

• Reflection  - seen as a mirror image

• Rotation -  has a point about which the rotation is made and angle that says how far to rotate

• Dilation - transformation which changes the SIZE of an object

• Glide Reflection - combinations of translation and reflection

Isometries - transformation which leave dimensions of the object and its image unchanged (location wise)

• D (f(x), f(y)) = d(x,y)

• In GREEK - Isos (equal), metron (measure) = Isometric transformations/rigid transformations

Design - figure with at least one NON TRIVIAL SYMMETRY

• Pattern has a design of translation symmetry

1. Rosette patterns: finite groups of isometries containing just rotations and reflections

1. 
   

   1. Two Rosette Patterns

      1. Cn (cyclical) - has n fold rotational symmetry and NO reflectional symmetry

      2. Dn (dihedral) - has n fold rotational symmetry and reflectional symmetry (AS LONG AS WHEN U DRAW A LINE AND IS SYMMETRIC IN ALL DIMENSIONS)

1. Frieze Patterns: infinitely long strip imprinted with a design given by a repeating motif

1. the pattern repeats itself in one direction — like copying and sliding the same shape over and over

   1. F1 (HOP) - copy, paste to the right; only translation 

-> ->

2. F2 (STEP) - translation and glide reflection

-> -> 

-> ->

3. F3 (SIDLE) - Translation and vertical reflection

-> ->  <- <-

4. F4 (SPINNING HOP) - translation and rotation (half only)

-> -> ->

<- <- <-

5. F5 (SPINNING SIDLE) - translation, rotation (half), and glide reflection

<- -> <- ->

-><-

6. F6 (JUMP) - translation and horizontal reflection

-> ->

-> ->

7. F7 (SPINNING JUMPING) - translation, horizontal and vertical reflection, + rotation

<- -> <- ->

<- -> <- ->

1. Wallpaper pattern - A wallpaper pattern repeats in more than one direction — not just left to right, but also up and down.

1. To count as a wallpaper pattern, you need at least one basic shape, a copy of it beside it, and another copy above or below — so you get at least two rows and two columns of repeating shapes.

   1. Basic unit → the smallest piece of the pattern that, when copied and slid around, makes the whole design.

   2. Fundamental region → an even smaller piece inside the basic unit that can fill the whole space when you use all the pattern’s symmetries (like flips, rotations, and slides).

   3. Motif → the tiny design inside that region that doesn’t have symmetry itself, but when repeated using the pattern’s rules, creates the full design.

      1. MOTIF -> REFLECT -> ROTATE -> TRANSLATE