Chapter 1

• Mathematics: is a branch of science, which deals with numbers and their operations.
• Patterns: are specific designed, character, behavior that is visible in nature.
• Fibonacci Sequence: is named after Leonardo of Pisa, it is a sequence of numbers obtained by adding the last two numbers of the sequence forms, 1,1,2,3,5,8,13,21,34….
• Leonardo of Pisa: whom fibonacci sequence is name after
• Sym: means together
• Metry: means measurement
• Symmetry: An object is said to be in symmetrical form if it can be divided into many parts identically. It means that every part of the whole is identical to other parts.
• Types of Geometrical Symmetry:

  1. Reflectional Symmetry
  2. Rotational Symmetry
  3. Translational Symmetry
• Reflectional Symmetry: It is a type of geometrical symmetry wherein half of the image is exactly identical to other half of the image.
• Rotational Symmetry: It is a type of geometrical symmetry wherein the object is rotated to a certain degree about at axis like 45 degrees but the image of the object does not change.
• Translational Symmetry: It is a geometrical symmetry wherein a particular pattern is followed and the object is being moved from one place to another without change in the image of the object.
• Translational Symmetry. It is a geometrical symmetry wherein a particular pattern is followed and the object is being moved from one place to another without change in the image of the object.
• Types of Rossette Pattern:

  1. Cyclic Rossettes
  2. Dihedral Rossettes
• Cyclic Rossettes: does not have any reflection symmetry like pinwheel.
• Dihedral Symmetry: does have reflection symmetry.
• A frieze pattern is a design on a two-dimensional surface that is repetitive in one direction.
• The first frieze group, F 1 , contains only translation symmetries. Mathematician John Conway created names that relate to footsteps for each of the frieze groups. According to Conway, F 1 is also called a HOP.
• The second frieze group, F 2 , contains translation and glide reflection symmetries. According to Conway, F 2 is called a STEP.
• The third frieze group, F 3 , contains translation and vertical reflection symmetries. Conway named F 3  a SIDLE.
• The fourth frieze group, F 4 , contains translation and rotation (by a half-turn) symmetries. According to Conway, F 4  is called a SPINNING HOP.
• The fifth frieze group, F 5 , contains translation, glide reflection and rotation (by a half-turn) symmetries. Conway calls F 5  a SPINNING SIDLE.
• The sixth frieze group, F 6 , contains translation and horizontal reflection symmetries. Conway named F 6  a JUMP.
• The seventh frieze group, F 7 , contains all symmetries (translation, horizontal & vertical reflection, and rotation). According to Conway, F 7  is named a SPINNING JUMP.
• A wallpaper pattern is a plane figure which has more than one direction of translation symmetry.
• The lattice of translations is the collection of all

  translated images of a point.
• Transformation: It is a process of moving an object from its original position to a new position.
• The object in the new position is called the image.
• Translation: It is the simplest type of transformation. The object moves in a fixed distance and in a fixed direction.
• Reflection. It involves “flipping” the object over a line called the line of reflection.
• Rotation: It involves “turning” the object about a point called the center of rotation.
• Dilation. It involves a resizing of the object. It could result in an increase in size (enlargement) or a decrease in size (reduction).