Finite Math: Symmetry, Transformations, Tessellations, and Fractals – Vocabulary Flashcards

A comprehensive set of vocabulary flashcards covering symmetry, transformations, tessellations, and fractals based on the provided lecture notes.

Finite Math Symmetry

Balance and proportion in parts of an object; symmetry arises when one half mirrors the other.

Reflection

A transformation that flips a figure over a line (the line of reflection) to produce a mirror image.

Line of Symmetry (Axis of Symmetry)

A line that divides a figure into two identical parts.

Line of Symmetry Orientation: Vertical

A line of symmetry that runs up-and-down (vertical axis).

Line of Symmetry Orientation: Horizontal

A line of symmetry that runs left-to-right (horizontal axis).

Line of Symmetry Orientation: Diagonal

A line of symmetry that runs along a diagonal direction. Florida.

One line of symmetry

A shape that is symmetrical about exactly one line.

Two lines of symmetry

A shape that is symmetrical about two lines.

Infinite lines of symmetry

A shape with infinitely many lines of symmetry (e.g., a circle).

Rotational symmetry

A shape can be rotated about a center by a nonzero angle less than 360° and still look the same.

Center of rotation

The fixed point about which a figure is rotated.

Angle of rotation

The smallest positive angle through which the figure can be rotated to coincide with itself.

Scalene triangle

A triangle with all sides of different lengths and no axis of symmetry.

Geometric Transformations

Operations that change position, size, or orientation of a shape: translations, rotations, reflections.

Translation

Moving a shape without rotating or resizing.

Rotation

Turning a shape around a fixed point without changing its size or shape.

Rigid Transformation

A transformation that preserves size and shape in the new position.

Distance preserved

Under a rigid transformation, distances between points stay the same.

Angle measure preserved

Angles keep their measures after a rigid transformation.

Parallelism preserved

Parallel lines remain parallel under a rigid transformation.

Collinearity preserved

Points on a line stay on a line after transformation.

Midpoint preserved

Midpoints remain the same under a rigid transformation.

Glide Reflection

A combination of translation and reflection across a line; the translation is parallel to the line of reflection.

Dilation

A transformation that resizes an object without changing its shape.

Enlargement

A dilation that makes the image larger.

Reduction

A dilation that makes the image smaller.

Scale factor

Ratio of the size of the image to the size of the original object.

Center of dilation

The fixed point in the plane that stays fixed during dilation.

Scale factor > 1

Stretching; enlarges the figure.

0 < scale factor < 1

Shrinking; reduces the figure.

Scale factor = 1

No change in size; same size.

Composition of transformations

Two or more transformations applied in sequence to form a new transformation.

Tessellations

Tiling the plane with shapes that cover it without gaps or overlaps.

Tiling

Another term for tessellation; repeating shapes to cover a plane.

Vertex rule (Angle rule)

Interior angles at a vertex must sum to 360° for tessellations.

Regular polygons

Polygons with all sides and angles equal (e.g., equilateral triangle, square, hexagon).

Semi-regular tessellation

Tessellations using two or more types of regular polygons meeting at a vertex.

Irregular tessellation

A tessellation using irregular or mixed shapes.

Edge-to-edge rule

Tessellations must join along full edges without gaps.

Escher

Maurits Cornelis Escher, artist known for tessellations and interlocking patterns.

Frieze patterns

Patterns that extend infinitely left and right, mapped onto themselves by a horizontal translation.

Golden Ratio (φ)

φ ≈ 1.618; the ratio a/b where a is the longer part and b the shorter; φ = a/b; 1/φ = b/a.

Golden Rectangle

A rectangle with aspect ratio φ (length/width = φ).

Golden Angle

The angle ~137.5° used in seed patterns; the complementary angle is ~222.5°.

Fibonacci sequence

A sequence where each term is the sum of the two previous ones (1,1,2,3,5,8,13,…).

Fractal

A geometric pattern with self-similarity across scales and often non-integer dimension.

Self-similarity

A property where a pattern looks the same at different scales.

Sierpinski Triangle

A triangle recursively subdivided; central triangle removed; self-similar.

Koch Snowflake

Start with an equilateral triangle; add smaller triangles on each side; infinite perimeter, finite area.

Cantor Set

Remove the middle third repeatedly from a line segment; leaves a dust-like set.

Sierpinski Carpet

2D Cantor-like set: start with a square, subdivide into 9 and remove the center; repeat.

Pythagoras Tree

Recursive structure of squares; each square splits into two smaller ones at an angle.

Menger Sponge

3D fractal: start with a cube, remove middle cube and center of each face, repeat.

Hilbert Curve

A space-filling curve that visits every point in a square grid.

Hexaflake

Self-similar fractal formed by subdividing a hexagon into seven hexagons.

Algebraic fractals

Fractals generated by algebraic equations, often involving complex numbers.

Mandelbrot Set

Classic algebraic fractal from the quadratic complex map; famous for its cardioid and bulbs.

Julia Set

Family of fractals parameterized by c in the equation z{n+1}=zn^2+c; varies with c.

Multibrot Set

Generalization of Mandelbrot using higher-degree polynomials.

Newton Fractal

Fractal patterns from applying Newton's method to root-finding in the complex plane.

Burning Ship Fractal

Escape-time fractal with ship-like, jagged edges; derived from a modified iteration.

Phoenix Fractal

Recurrence-based fractal influenced by previous iterations.