# Mathematics in Nature

**Slope of a Mountain**

**Using the derivative of a line to find the slope of a mountain at a certain point**

While I was wandering through the interwebs looking for various ways math can be represented in nature, I came across a photograph that I found very intriguing.

Turns out this girl is a photography as well as math student, who is finding creative ways to combine her passions. The first thing I thought when I saw this photo was how cool it would be to do this with some of my own photos, knowing I have a lot of photos of the Adirondacks I figured it would be easy to find a picture of the local mountains to do this with, only lets take it up a notch, and find the slope of these mountains.

Using desmos we could play around to find the an equation that matched up pretty well with the mountains in this image.

From here, it is not difficult to figure out the slope of any one of these mountains by using the derivatives of the equations we made. Pick find the derivative, pick a point along the x-axis and you’re good to go. Plug your point in for the x value in your equation and there you have it, the approximated slope of any mountain you wish.

Colden

f(x)=sin(.52x-.8)+1

Colden’s Sister Peaks

f(x)=cos(x/1.8)+1.8 0 ≤ x < 5.999

f(x)=.4sin(.89x)+1.1422 5.999 ≤ x ≤ 10

Colden’s Reflection

f(x)=-sin(.52x-.8)-1

Sister Peaks’ Reflection

f(x)=-cos(x/1.8)-1.8 0 ≤ x < 5.999

f(x)=-.4sin(.89x)-1.1422 5.999 ≤ x ≤ 10

Slope of Colden at (6, 1.732)

f(x)=sin(.52x-.8)+1

f’(x)=.52cos(.8-.52x)

x=6

f’(6)=.52cos(.8-.52(6))

f’(6)=-.290339

Slope: -.290339

**Fibonacci in a Pine Cone**

**Ways the Golden Ratio is seen in Nature**

We see math represented in nature more often than we think, and a biggy is the Golden Ratio. Most everyone has heard of the Fibonacci Sequence, but what really is the Golden Ratio?

Simply put it is when the ratio of two parts is equal to the ratio of both parts combined to the larger of the two, a visual often helps.

The Golden Ratio is seen everywhere from flowers, to hurricanes, to the proportions of the human body. One of the places it’s seen in fibonacci spiral form is on pinecones. You can easily see the spirals going in one direction

but they go in another as well!

This is where Fibonacci numbers, which satisfy the Golden Ratio come into play. The amount of spirals in each of the two directions are a set of Fibonacci Numbers.

In this case we have 13 and 8

so…

13 / 8 = 1.625

and

13 + 8 = 21

21 / 13 = 1.615385

1.625 =~ 1.615385 !

**Rates Ripples Grow**

**What really happens when you (or a ****fairly uniform flying projectile****) make a splash**

So, everybody loves making a splash. Literally. It’s really fun to play with water, and it’s everywhere in nature. One of the coolest things to watch is a stone skipping over water or a ripple spreading out from where a big stone just got spelunked. Anyways, why can’t we use math to at least try to understand that process better? Trick question: We can!

So, we’re going to use calculus to study ripples, but we decided to collect our own real life data first. There were four necessary ingredients to begin making mischief:

- A uniform container with a fixed diameter
- Water!! (duh)
- A stopwatch
- A fairly uniform flying projectile
- (plus we needed a meterstick to get a measure of the container, but that’s esoteria)

↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓

Next we assumed that the rate at which the diameter grows is directly proportional to the length of the diameter itself at any given moment…

**SO!**

- We dropped a ball from a constant height into the specific container at its center (or as close to the center as possible)
- We measured the time it took for the initial ripple to reach the walls of the container
- We took multiple trials

Diameter of projectile = 1.68 inches

Diameter of container – 10.875 inches

Trial 1 | .43 seconds | |

Trial 2 | .48 seconds | |

Trial 3 | .55 seconds | |

Trial 4 | .58 seconds | |

Average: | .51 seconds |

↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓

Then, we tried to derive the equation for the rate and rate constant of ripples caused by golf balls at any specific time. Disclaimer!: We have assumed the rate at which the diameter of a ripple grows is directly proportional to the size of the diameter at any given moment theoretically. We haven’t taken into account the rate at which the amplitude of each ripple decreases, making the ripple seem to diminish to nothing eventually. Theoretically, the diameter will continue to grow ad infinitum.

dydx=ky

1/y*dy/dt*dt=kdt

ln(y)=kt+c

ektec=y

ec=1.68

ekt(1.68)=y

e.51k(1.68)=10.875

k(.51)lne=6.473

k=12.6926

SO!

(For the diameter in inches)

P.S. — Remember that this is in inches. Don’t try to use this equation for metric units!!

Moral of the story: Theoretically, things get enormous very quickly.

**Fractals: THEY’RE EVERYWHERE!!!**

**Branching Fractals seen in Leaves**

Soooooo…. You know how lightning leaves branching highlight impressions on your retinas when you look directly at it? Those patterns are distinctive, and though you can’t necessarily distinguish the minutia between different lightning formations, they all seem to form a pretty generic branching shape. Sort of like how all river deltas seem to form triangular tributary shapes that appear the same from respective distances. Or how blood vessels seem as multifaceted and chaotic in a closeup as from far away. Get where I’m going with this? I’m talking about never ending patterns… Repetitions that are infinitely complex and self similar on different scales….A.K.A.

**…..****FRACTALS!!!!**

Now, fractals aren’t easy to write, and they lend themselves more to programming. Enter simple Rachel program…

Pretty neat, right?

So fractals can be contractive and iterant (approaching a certain point) or they can grow ad infinitum. For the sake of simplicity, we’ll assume the fractals we are working with are iterant.

Sort of like galaxy fractals!

But I digress…

Don’t worry though! Programming isn’t the only way to represent fractal growth. There are several notations that tell an awesome fractal story including:

- the Hutchinson system with a Hutchinson operator and attractor

This works only with contractile fractals

- the Lindenmayer system

**Example:**

**G** = (*V*, ω, *P*) → V=elements that can be replaced, symbols from v describing initial state, P=production rules describing how variables can be replaced

**Variables:**F,G

**Constants:**+,-,[,]

**Start:**(F → F-[G+F]+[G-F])(G → GG)

**Rules:**F means “draw forward”, − means “turn left 30°”, and + means “turn right 30°”. G does not correspond to any drawing action and is used to control the evolution of the curve.**[**corresponds to saving the current values for position and angle, which are restored when the corresponding**]**is executed.

**Angle:**30°

BONUS!: It was originally designed to illustrate the growth and patterning in plants!!! PERFECT!!

**MATH IS EVERYWHERE! We just have to stop and take a look around to see it.**