Patterns, Fibonacci, Golden Ratio & Mathematical Language

31 question-and-answer flashcards covering patterns, the Fibonacci sequence, the Golden Ratio, mathematical language, properties of real numbers, and basic logic symbols.

What is a pattern in the context of mathematics and nature?

A repeated visual or behavioral design.

Name the three major types of patterns discussed in the lecture.

Fractals, symmetry (bilateral & radial), and spiral patterns.

What characterizes a fractal pattern?

A self-similar geometric shape with fractional (non-integer) dimensions.

How is bilateral symmetry defined?

Left-right mirror-image symmetry across a single axis.

What is radial symmetry?

Symmetry arranged around a central point.

Where are spiral patterns commonly observed?

Shells, galaxies, flowers, and other natural formations.

State the recurrence formula for the Fibonacci sequence.

Fn = Fn-1 + Fn-2.

Who introduced the Fibonacci sequence to the Western world?

Leonardo of Pisa, known as Fibonacci.

Write the first seven Fibonacci numbers.

1, 1, 2, 3, 5, 8, 13.

How many clockwise and counter-clockwise spirals are typically found in a sunflower head?

34 and 55 spirals, both Fibonacci numbers.

Which Fibonacci numbers appear in the diagonal rows of a pineapple?

5, 8, and 13.

What is the value of F11 given F10 = 55 and F9 = 34?

F11 = 89.

Approximately what is the numerical value of the Golden Ratio (Φ)?

About 1.618.

Express the Golden Ratio using a and b in the Golden Rectangle.

a/b = (a + b)/a = Φ.

How are consecutive Fibonacci terms related to the Golden Ratio?

Their ratio approaches Φ as the terms increase.

Give one architectural example where the Golden Ratio is applied.

The Golden Rectangle in building design (e.g., classical temples, modern facades).

When a 120-m plank is cut using the Golden Ratio, what are the lengths of the two pieces?

a ≈ 74.17 m and b ≈ 45.83 m.

List three reasons why the language of mathematics is important.

It is precise, concise, and powerful for communicating ideas.

Translate into algebra: "The sum of a number and ten."

x + 10.

Translate into algebra: "Five less than a number."

x − 5.

Translate: "The square of the sum of five and a number."

(5 + x)².

Translate: "There are twice as many boys as girls."

b = 2g.

State the Commutative Property of addition.

a + b = b + a.

Write the Distributive Property.

a(b + c) = ab + ac.

What is the Additive Identity property?

a + 0 = a.

Define the Multiplicative Inverse property.

a × 1/a = 1 for a ≠ 0.

What symbol represents logical conjunction (AND)?

∧

What is a proposition in logic?

A declarative statement that is either true or false.

What is the contrapositive of "If P, then Q"?

If not Q, then not P.

What does the universal quantifier (∀) signify?

"For all" or "for every" in a given domain.

What does the existential quantifier (∃) mean?

"There exists" at least one element in the domain satisfying a condition.