1. Golden Ratio, Fractal Geometry, Tessellations, and Symmetry

Fibonacci Sequence

The sequence of numbers where each number is the sum of the two preceding ones (e.g., 0, 1, 1, 2, 3, 5, 8, 13, 21, …).

Leonardo de Pisa

The Italian mathematician who first observed the Fibonacci sequence in 1202.

Liber Abbaci

The book in which the rabbit problem, leading to the Fibonacci sequence, was posed.

1202

The year the Fibonacci sequence was first observed by Leonardo de Pisa Fibonacci.

He was investigating how fast rabbits could breed under ideal circumstances.

What was Leonardo de Pisa investigating when he first observed the Fibonacci sequence?

Begin with one male and one female rabbit.

What was the initial assumption about the rabbits in the Fibonacci sequence problem?

Rabbits mate at age one month; reproduction starts in the second month.

What is the assumption about when the rabbits can mate and when reproduction starts in the Fibonacci problem?

Rabbits never die.

What assumption was made about the lifespan of the rabbits in the Fibonacci problem?

Each female produces one male and one female every month.

What assumption was made about the production of new pairs by each female rabbit?

Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21.

What are the first nine terms of the Fibonacci sequence?

First Terms: 0 and 1.

What are the first two terms of the Fibonacci sequence?

To get the next number in the sequence, add the previous two numbers together.

What is the rule for finding the next number in the Fibonacci sequence?

Golden Ratio

The ratio of successive Fibonacci numbers, approximately equal to 1.618.

Phi

The symbol (

1.618

The approximate numerical value of the Golden Ratio.

Ratio between the sum and the larger of two numbers equals the ratio of the larger to the smaller.

What is the definition of the Golden Ratio in terms of the ratio of two numbers (a+b / a = a / b)?

Fractals

Geometric patterns characterized by self-similarity, repeated at ever smaller scales to produce irregular shapes.

Fractus

The Latin adjective, meaning to break, from which the term fractal came.

Benoit Mandelbrot

The mathematician who named and popularized fractals.

The Fractal Geometry of Nature

The book written by Benoit Mandelbrot in 1977.

1977

The year Benoit Mandelbrot wrote The Fractal Geometry of Nature.

Self-Similarity

A property of fractals where a rough or fragmented geometric shape can be split into parts, each being an approximately reduced-size copy of the whole.

Fine structure at arbitrarily small scales.

What is a key feature of a fractal regarding its structure?

Too irregular for Euclidean geometry.

How is a fractal described in relation to traditional Euclidean geometric language?

Self-similar (approximate).

What is the property of fractals where parts are reduced-size copies of the whole?

Simple recursive definition.

What is the nature of the definition for a fractal?

Sierpinski's Triangle

A popular example of a fractal with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles.

Waclaw Sierpinski

The Polish mathematician after whom the Sierpinski triangle is named.

Pascal's Triangle

A triangular array of binomial coefficients that arises in probability theory, combinatorics, and algebra; also a popular fractal.

Blaise Pascal

The French mathematician after whom Pascal's triangle is named.

Top row is row n = 0.

How are the rows of Pascal's triangle conventionally numbered, starting at the top?

Tessellation

A special type of tiling where a pattern of geometric shapes fills a two-dimensional space with no gaps and no overlaps, repeating forever in all directions.

Tessera

The Latin word meaning a small stone cube, from which the word tessellation comes.

Tessellata

The Roman mosaic pictures forming floors and tilings, made up of small stone cubes.

Tiling

A pattern of geometric shapes that fill a two-dimensional space with no gaps and no overlaps.

no gaps.

A key characteristic of a tessellation is that it has no overlaps and…?

Polygon

A two-dimensional shape with any number of straight sides.

Regular Polygon

A polygon in which all sides and all angles are equal.

Not every polygon can tesselate by themselves.

Can every polygon form a tessellation by itself?

The internal angles must add up to 360 degrees.

In a tessellation, what must be true about the internal angles of the polygons that meet at a vertex?

Equilateral Triangles

One of the three regular polygons that can form a tessellation by themselves.

Squares

One of the three regular polygons that can form a tessellation by themselves.

Regular Hexagon

One of the three regular polygons that can form a tessellation by themselves.

Circles

Curved shapes that cannot tessellate on their own because they leave gaps, but can be part of a tessellation if the gaps are viewed as shapes.

Regular Tessellation

A tessellation composed of identically sized and shaped regular polygons.

Semi-Regular Tessellation

A tessellation made from multiple regular polygons.

Irregular/Demiregular/Polymorph Tessellation

A tessellation consisting of figures that are not composed of regular polygons, which interlock without gaps or overlaps.

Vertex

A corner point in a tessellation where two or more polygons meet.

Symmetry

A concept that states a shape is identical to another after a movement (turning, flipping, or sliding).

Symmetria

The Greek word from which Symmetry comes, meaning to measure together.

Translational Symmetry

A type of symmetry where a figure is translated (moved or slid) at a set distance in the same direction, changing only its location.

Rotational/Radial Symmetry

A type of symmetry where a shape looks precisely similar to the original form after some rotation (turning).

Reflectional/Reflexive/Mirror Symmetry

A type of symmetry where one half of a figure is a mirror image of the other half, reflected over a line.

Glide Reflection Symmetry

A hybrid symmetry involving both reflection over a line and then translation along the line.

Line of Symmetry

An imaginary line or axis that passes through the center of any picture, shape or object and it is divided into two identical halves.