# Visualizing Math

A blog dedicated to mathematics.

Posts We Like
Blogs We Follow

So here it is. These are the primes of ℤ[√-1], ℤ[√-2], ℤ[(1+√-3)/2], ℤ[√-5], ℤ[√-6], and ℤ[(1+√-7)/2], all in one picture (at the top) and individually (in that order, below.)

The reason these rings are interesting is that each of them consists of solutions to quadratic equations x²+bx+c=0 where b and c are integers and b²-4c is some square times -1, -2, -3, -5, -6, or -7, respectively.

(thx ebering for the idea)

Repeated barycentric subdivision results in a gorgeous fractal-ish pattern.

When people ask me how I can be a math major and still say I’m not good with numbers, I’m like ‘here, let me draw you a picture.’

All so true.

^For those that are considering majoring in math but are deterred by their lack of number skills, there’s definitely more to mathematics than simply numbers. (I’m not dissing stats or number theory or algebra though. Those are cool too!)

Land of mathematics

A roulette traced from rolling an ellipse inside a circle. [thanks] [code]

TheBookBucketChallenge is a great thing… And I thought that I should do it for you.. So here are (in no specific order) the books that influenced me, mathematical speaking…

1.’The elephant in the classroom : helping children learn and love maths’ by Jo Boaler
2.’Alex’s Adventures in Numberland’ by Alex Bellos
3.’What Counts: How Every Brain is Hardwired for Math’ by Brian Butterworth
4.’The Science of Secrecy - The secret history of codes and code-breaking’ by Simon Singh
5.’Numbers without End’ by Cornelius Lanczos
6.’What Is Mathematics? An Elementary Approach to Ideas and Methods’ by Ian Stewart, Herbert Robbins, Richard Courant
7.’It Must Be Beautiful: Great Equations of Modern Science’ edited by Graham Farmelo
8.’The Language of Mathematics - Making the Invisible Visible’
by Keith Devlin
9.’Mathematical Fallacies and Paradoxes’
and my a childhood book: 10.’Chasing Vermeer’ by Blue Balliett

I would like to see you all share with me some of your math favorite books, and I would like to nominate others for this challenge: @curiosamathematica; ifyourelisteningurtheresistance; mathlover1530; mathprofessorquotes; nobel-mathematician; visualizingmath; ljiljana147

Thanks for nominating me, ioanaiuliana21! I can say, without a doubt, that the number one book that has influenced me mathematically is The Math Book by Clifford Pickover. I read that book from cover to cover in 8th grade and I loved it, and it’s where I’d look to for inspiration back when I had time to write posts.

The book is wonderful. There are so many cool topics…Mobius Strip, Klein Bottle, Gabriel’s Horn, Boy’s Surface, Rubik’s Cube, Golden Ratio, Imaginary Numbers, Magic Squares…you name it. It’s all in chronological order according to discovery/invention. Best part: every other page is a large, high-quality picture of its corresponding topic which satisfies my need for visualizations. I remember first discovering the book in my public library and thinking, “This is what I have always been looking for…”.

Golden Ratio φ = (1+sqrt(5))/2 = 1.6180339887498948482…

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0. Two quantities a and b are said to be in the golden ratio φ if

(a+b)/a = a/b = φ

One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ:

(a+b)/a = 1+ b/a = 1+1/φ

Therefore: 1+1/φ = φ
Multiplying by φ gives: φ^2 - φ - 1 = 0

Using the quadratic formula, two solutions are obtained::

φ = (1- sqrt(5))/2 or φ = (1+sqrt(5))/2

Because φ is the ratio between positive quantities φ is necessarily positive:

φ = (1+sqrt(5))/2 = 1.6180339887498948482…

See more at Golden Ratio.

Image: Phi (golden number) by Steve Lewis.

Archimedes and the quadrature of the parabola

Archimedes of Syracuse (c. 287–212 BC) was a Greek mathematician, scientist and engineer. He is widely regarded as one of the greatest mathematicians of all time.

One of Archimedes’ works was called The Quadrature of the Parabola. This proved various results about parabolas, and explained how to find the area of a parabolic segment, which is a finite region enclosed by a parabola and a line. This is easy to do nowadays using the well-known theory of integral calculus, but this was not developed until the 17th century, about 1900 years after the time of Archimedes.

Integral calculus calculates areas by approximating the area to be measured by a union of geometric shapes whose exact areas are known, and then applying a limiting process. Archimedes’ technique was very similar to this. The key to his idea was to inscribe into the parabolic segment a triangle with the same base and height. In other words, the triangle had the original line segment as its base, and touched the curved part of the parabola at the point where the tangent line to the parabola was parallel to the line segment. Archimedes proved that if the triangle has area T, then the area A of the parabolic segment was given by 4T/3.

Archimedes described a method of filling up the rest of the parabolic segment by exhaustion, using smaller and smaller triangles. The graphic shows two lighter blue triangles, four yellow triangles, eight (barely visible) red triangles, and so on. There are twice as many triangles of each successive colour as there were of the previous colour. Archimedes proved that the area of a triangle of each successive colour is 1/8 of the area of the previous type of triangle, although this is not an obvious result. For example, each light blue triangle has an area of T/8.

These observations reduce the problem of finding the area A to evaluating the sum at the bottom of the picture, which is a geometric series. Nowadays, there is a well-known formula that applies in this situation, but Archimedes summed the series using a clever ad hoc geometric argument instead.

Archimedes made some other very significant discoveries using integration-like methods. He proved that the area of a circle of radius r is equal to πr^2, and he also discovered the formulae for the surface area and volume of a sphere, and for the volume and area of a cone. Archimedes is also known for inventing the Claw of Archimedes and the Archimedes heat ray, both of which were weapons to defend the city of Syracuse. The claw was a kind of mobile grappling hook that could lift enemy ships out of the water, and modern experiments suggest that this would have been a workable device. The heat ray was a system of mirrors to focus reflected sunlight on to enemy ships, thus setting them on fire. Modern attempts to reproduce the heat ray have concluded that it would not have worked quickly enough in typical weather conditions to be able to burn enemy ships.

Wikipedia on Archimedes: http://en.wikipedia.org/wiki/Archimedes

Wikipedia on The Quadrature of the Parabola (including the graphic here): http://en.wikipedia.org/wiki/The_Quadrature_of_the_Parabola

Picture of Archimedes from http://totallyhistory.com/archimedes/

I stole the joke in the picture from Dan McQuillan on Twitter.

Here’s another good joke about Newton and Leibniz developing calculus in the 17th century, which someone in my department has on their office door: http://xkcd.com/626/

#mathematics   #sciencesunday

(via hipster-graphs)

Macro photography of snowflakes showing complex geometric construction.

https://www.flickr.com/photos/donkom/6384704061/in/photostream/

Taylor Series Approximations

A Taylor series is a way to represent a function in terms of polynomialsSince polynomials are usually much easier to work with than complicated functions, Taylor series have numerous applications in both math and physics.

There are many equations in physics — like the one describing the motion of a pendulum — that are impossible to solve in terms of elementary functions. “Approximations using the first few terms of a Taylor series can make [these] otherwise unsolvable problems” solvable for a restricted area of interest [1].

The GIF above shows the five-term Taylor series approximation of a sine wave about x=0.

Mathematica code:

```f[x_] := Sin[x]
ts[x_, a_, nmax_] :=
Sum[(Derivative[n][f][a]/n!)*(x - a)^n, {n, 0, nmax}]
Manipulate[Plot[{f[x], ts[x, 0, nmax]}, {x, -2*Pi, 2*Pi},
PlotRange -> {-1.45, 1.45},
PlotStyle -> {{Thick, Cyan}, {Thick, Dotted, Yellow}},
AxesStyle -> LightGray, Background -> Darker[Gray, 0.8]],
{nmax, 1, 30, 1}]```

Cops and Robbers (and Zombies and Humans)

Cops and Robbers is a mathematical game in which pursuers (cops) attempt to capture evaders (robbers). The game is one of many pursuit-evasion games, each of which is governed by a different set of rules. The general goal of these problems is to determine the number of pursuers required to capture a given number of evaders.

The GIFs above show two versions of the game. The first is similar to the standard Cops and Robbers rendition, and the second is best described as “Zombies and Humans”.

In both versions, an evader moves in the direction that gets it furthest away from the pursuers (focusing more on the closer pursuers), and a pursuer moves in the direction that gets it closest to the evaders (focusing more on the closer evaders).

In the first simulation, members of both groups have a constant speed. In the second simulation, members of a group move more quickly the closer they are to members of the opposite group, and slower when further away.

`Mathematica code posted here.`

Additional sources not linked above: [1] [2]

*realizes joy division did some real quick fs photoelectron spectroscopy for their dumb album cover*

Geometry Matters:

Various nature elements that abide by geometric laws and construction patterns.