Geometry, Section 3, Transformations & Tessellations

How to do a rotation around (0, 0) on a coordinate plane.

90 degrees and 270 degrees means to switch the x and y coordinates, then use the signs of the quadrant. 180 degrees means to keep the same x and y coordinates, then use the sign of the quadrant.

3-Uniform Tilings

Are tilings that have three kinds of vertices made with regular polygons. There are thirty-nine known.

How to do a dilation on a coordinate plane.

Multiply every x and y coordinate by the scale factor.

Line of Symmetry

The line that determines when a shape is an exact reflection to itself.

scale factor of a dilation

The ratio of a side length of the image to the corresponding side length of the original figure.

Nonrigid Transformations

Transformations that alter the size or shape of a figure: Dilations or stretching a shape are examples.

Rigid Transformations (Isometries)

Transformations that do not alter the size or shape of a figure: Reflections, translations, and rotations are examples.

Reflection

all points of an image "flip" over a line.

Translation

all points of an image "slide" horizontally and/or vertically.

Rotation

all points of an image "turn" about a fixed point.

Dilations

are shapes that are similar to each other and are a ratio proportion to each other. Their only difference is their size. The corresponding angles in polygons remain congurent.

Tessellating Glide Reflections

are tiles that appear to be copies of one another that are both shifted from one place to another and flipped without tilting or resizing them. They fit together like a puzzle without any gaps or overlapping.

Tessellating Rotations

are tiles that are copies of one another that are simply rotated or turned around a point without flipping or resizing them. They fit together like a puzzle without any gaps or overlapping.

Tessellating Translations

are tiles that are copies of one another that are simply shifted from one place to another without flipping, tilting, or resizing them. They fit together like a puzzle without any gaps or overlapping.

2-Uniform Tilings

are tilings that have two kinds of vertices made with regular polygons. There are twenty known.

Tessellating Pentagons

can not be tiled with regular shapes, but can tiled with convex or concave forms.

Archimedean Tilings (1-uniform tilings)

include the three regular and eight semiregular tessellations. There are eleven known.

Image

is a figure after a transformation is done and is named with the same corresponding letters with a prime symbol.

Tesselation (Tiling)

is a pattern of identical shapes that must fit together without any gaps and should not overlap.

Vertex Arrangement (Numerical Name) Tiling

is a set of points in space described by their relative positions. For example, go around a vertex and write down how many sides each polygon has in order and write them separated with a point like 3.3.4.3.4 is a vertex with a triangle, triangle, quadrilateral, triangle, quadrilateral pattern around it.

Monohedral Tiling

is a tiling in which all tiles are congruent.

Regular Tessellation

is a tiling made by repeating a regular polygon. There are only three known shapes that can be used; equilateral triangle, square, and regular hexagon.

Semiregular Tessellation

is a tiling made by two or more regular polygons. There are eight known patterns.

Point Symmetry (Center of Rotation)

is the point a shape seems to be rotating around in a clockwise or counterclockwise direction.

Prime Symbol

looks like an apostrophe after a letter used in geometry to denote the point, line, or shape is not the original, but instead, its image after a transformation.

Reflection Symmetry

occurs when a shape can be folded across a line and is exactly the same shape on either side of the line. This can occur within a shape or across a line.

Symmetry

occurs when a shape can be folded or the entire shape rotated to create an image that appears to look like a reflection of the original.

Rotational Symmetry

occurs when a shape can be turned clockwise or counterclockwise a number of degrees and the resulting image looks identical to the original. On a coordinate plane, it appears the entire shape was moved in a clockwise or counterclockwise direction around a center of rotation and the image resides in a new location that is measured in degrees from the original.

Translation Vector

occurs when a shape translates on a coordinate plane.

Composition of Transformations

occurs when two or more transformations are combined to form a new transformation.

How to do a translation on a coordinate plane.

up means "add to y" down means "subtract from y" right means "add to x" left means "subtract from x"

How to do a reflection over an axis on a coordinate plane.

Flip over x-axis, x coordinate stays the same, y coordinate becomes its opposite. Flip over y-axis, y coordinate stays the same, x coordinate becomes its opposite.