Symmetry, Fibonacci, Golden Ratio, and Exponential Growth

Symmetry

Rigid motion leaving an object unchanged

Reflectional Symmetry

Object folding in half with mirror images

Rotational Symmetry

Object looking the same after rotation around a fixed point

Translation Symmetry

Sliding objects without rotation, preserving orientation

Spirals

Shapes maintaining self-similarity as they grow

Tessellations

Pattern of geometric shapes without overlaps or gaps

Fractals

Pattern replication at different scales with self-similarity

Symmetria

Greek origin of 'symmetry' meaning 'the same measure'

Euler’s number

Mathematical constant used in population growth applications

Fibonacci sequence

Sequence where each number is the sum of the two preceding ones

Regularities in nature

Observable consistent patterns in the natural world

Self-similarity

Maintaining the same shape at different scales

Tessellation examples

Honeycombs, snake skin, leaf veins, cracked mud, tidal patterns

Fractal property

Never-ending pattern replication at varying scales

Order of rotation

Number of times an object can be rotated around a fixed point

Angle of rotation

Degrees through which an object is rotated and still looks the same

Translation in mathematics

Transformation sliding objects without rotation

Von Koch Curve

Fractal named after Niels Fabian Helge von Koch, created by replacing middle segment with equilateral triangles.

Minkowski Curve

Curve constructed by dividing a line segment into four parts and replacing the middle two parts with a square.

Fractals in Nature

Observed in river formations, ice crystals, and broccoli, showcasing self-similar patterns.

Symmetry in Nature

Seen in flowers, leaves, and patterns like Fibonacci spirals, exhibiting reflectional and rotational symmetries.

Fibonacci Sequence

Sequence where each number is the sum of the two preceding ones (1, 1, 2, 3, 5, 8, ...).

Golden Ratio

Ratio where longer part divided by shorter equals sum to longer, approximated by 1.618, found in Fibonacci sequence.

Fibonacci Spiral

Spiral created by drawing squares based on Fibonacci numbers, forming a pattern known as Fibonacci spiral.

Compound Interest

Interest added to original principal, earning interest.

Simple Interest

Interest paid on the principal amount only.

Continuous Compounding

Interest compounding continuously, growing exponentially.

Euler's Number

Approximately equal to 2.718, an irrational constant.

Napier's Constant

Another name for Euler's number, introduced by John Napier.

Population Growth Models

Mathematical models like Malthusian and logistic growth.

Malthusian Growth Model

Describes exponential population growth with unlimited resources.

Logistic Growth Model

Accounts for constraints in population growth.

Exponential Decay

Quantity decreases continuously over time exponentially.

Euler's Number Formula

𝐴= 𝑃𝑒𝑟𝑡, where A is the accumulated balance.

Compound Interest Formula

𝐴= 𝑃𝑒𝑟𝑡, where A is the accumulated balance.

Simple Interest Formula

Interest = Principal x Rate x Time.

Malthusian Model Formula

𝑃(𝑡) = 𝑃𝑜𝑒𝑟𝑡, where P(t) is population after time t.

Logistic Growth Model Formula

𝑃(𝑡) = 𝐾/(1+𝐴𝑒^(-𝑘𝑡)), where K is carrying capacity.

Exponential decay

A decrease in quantity over time at a rate 𝑟>0, modeled by the function 𝑃(𝑡) = 𝑃₀e^(-𝑟𝑡)

Exponential growth

An increase in quantity over time at a rate 𝑟>0, modeled by the function 𝑃(𝑡) = 𝑃₀e^(𝑟𝑡)

𝑃(𝑡)

Quantity at any time 𝑡 in exponential decay or growth models

𝑃₀

Initial quantity in exponential decay or growth models

𝑟

Rate of decay or growth in decimals in exponential models

𝑡

Time in exponential decay or growth models

𝑒

Approximately 2.718, a constant used in exponential functions

Radioactive decay

A type of exponential decay, such as in the decay of radioactive elements

Half-life

The time taken for half of the initial quantity to decay in exponential decay processes

Simple interest

Interest calculated only on the initial principal amount, not on interest already earned

Compound interest

Interest calculated on the initial principal amount and also on the accumulated interest from previous periods

Logistic growth

A type of growth that increases rapidly at first and then slows down as it reaches a maximum value

Translation

A transformation that slides objects along without rotating them, preserving orientation

Tessellation

A pattern made of geometric shapes joined without overlaps or gaps to cover a plane

Fractal

A pattern that replicates at different scales, exhibiting self-similarity

Golden ratio

A ratio where the ratio of the longer part to the shorter part is equal to the ratio of the sum to the longer part