Mathematics in the Modern World - Patterns and Nature

Flashcards reviewing key concepts, definitions, formulas, and examples from the Nature of Mathematics lecture.

What is the origin and literal meaning of the word 'Mathematics'?

It comes from the Greek word mathema, which means 'knowledge, study, learning'.

What is symmetry in the context of natural patterns?

Symmetry indicates that an imaginary line can be drawn across an object such that the resulting parts are mirror images of each other.

How is the angle of rotation calculated for an object with $n$-fold rotational symmetry?

It is calculated using the formula \text{Angle of rotation} = \frac{360^\bottom}{n}, where $n$ is the number of folds in the rotational symmetry.

What is a fractal?

A fractal is a curve or geometric figure in which each part has the same statistical characteristic as the whole, exhibiting self-similarity across multiple scales.

Who first described the logarithmic spiral, and who later investigated it?

It was first described by René Descartes and later investigated by Jacob Bernoulli.

What is dendrochronology?

Dendrochronology is the scientific method of dating based on the amount and arrangement of growth rings found in the core of a tree trunk.

What are 'scutes' on a turtle's shell, and how do they relate to its age?

Scutes are hexagonal growth rings on a turtle's shell. The smallest scute in the center is the oldest, while the larger outer scutes are the newer ones used to estimate age.

How does a river create a meander pattern?

A meander is formed as a body of water erodes sediments along an outer concave bank and deposits them downstream on an inner convex bank.

What percentage of a plane's area is covered by circles using square packing versus hexagonal packing?

Square packing covers $78.5\times$ of the area, whereas hexagonal packing covers $90.7\times$ of the area.

In the exponential growth model $A = P e^{r t}$, what does each variable represent?

$A$ is the population size after growth, $P$ is the initial population size, $r$ is the growth rate, $t$ is time, and $e$ is Euler's constant ($\text{approximately } 2.718$).

How do you rearrange $A = P e^{r t}$ to solve for the initial population $P$?

$P = \frac{A}{e^{r t}}$ or $P = A e^{-r t}$

How do you rearrange $A = P e^{r t}$ to solve for the growth rate $r$?

$$r = \frac{\text{ln}\big(\frac{A}{P}\big)}{t}$$

How do you rearrange $A = P e^{r t}$ to solve for time $t$?

$$t = \frac{\text{ln}\big(\frac{A}{P}\big)}{r}$$

What book written in 1202 popularized the Hindu–Arabic numeral system in Europe and introduced the Fibonacci sequence?

Liber Abaci (Book of Calculation), written by Leonardo Fibonacci.

In Fibonacci's rabbit breeding problem, how many total pairs of rabbits exist at Month 12?

$377$ total pairs.

What are the petal counts for a white calla lily, Euphorbia, spiderwort, periwinkle, bloodroot, black-eyed susan, and Shasta daisy?

White calla lily has $1$, Euphorbia has $2$, spiderwort has $3$, periwinkle has $5$, bloodroot has $8$, black-eyed susan has $13$, and Shasta daisy has $21$ petals.

What is phyllotaxis?

Phyllotaxis is the arrangement of leaves on a plant stem, where the distances between consecutive leaves correlate with the Fibonacci sequence.

Who authored the 1509 work De Divina Proportione, and who provided the illustrations for the 5 Platonic solids within it?

It was written by Luca Pacioli and illustrated by Leonardo da Vinci.

What are the names and face counts of the 5 Platonic solids shown here?

Tetrahedron ($4$ triangular faces), Hexahedron ($6$ faces), Octahedron ($8$ triangular faces), Dodecahedron ($12$ pentagonal faces), and Icosahedron ($20$ triangular faces).

What is the exact formula and approximate value of the Golden Ratio ($\text{φ}$)?

$$\text{φ} = \frac{1 + \text{√}{5}}{2} \text{≈} 1.618033987\text{…}$$

What is the reciprocal of the Golden Ratio ($\text{φ} = 1.618034\text{…}$)?

$$\frac{1}{\text{φ}} = 0.618034\text{…}$$

What is the Golden Angle?

It is an angle of approximately 137.5^\bottom, corresponding to the smaller portion of a full rotation divided by the Golden Ratio.

How does a pentagram display the Golden Ratio between its line segments?

The ratios of consecutive segment lengths $a$, $b$, $c$, and $d$ satisfy $\frac{a}{b} = \frac{b}{c} = \frac{c}{d} \text{≈} 1.618\text{…}$

What three main purposes does mathematics serve in our world?

Mathematics for Organization, Mathematics for Prediction, and Mathematics for Control.

Given the study score equation $Y = 10 X + 45$, where $Y$ is the exam score and $X$ is study hours, what is the estimated score for $4$ hours of study?

$$Y = 10 (4) + 45 = 85$$

An airplane starts from rest ($v_0 = 0\text{ m/s}$) and accelerates at $a = 3.5\text{ m/s}^2$ for $t = 34.5\text{ s}$. What distance $d$ does it travel before takeoff using $d = v_0 t + \frac{1}{2} a t^2$?

$$d = 0(34.5) + \frac{1}{2} (3.5) (34.5)^2 = 2082.9375\text{ m}$$

In Vlad's sweet pack measurements (212, 206, 203, 206, 199, 196, 197, 197, 209, 206, 198, 191, 196, 206, 207), which weight is the most frequent?

$206\text{ g}$ (it appears $4$ times).