# Visualizing Math

A blog dedicated to mathematics.

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Each dot is only moving in a straight line, but is created by balls moving in circles through 3 dimensional space.

[Click for more interesting science facts and gifs]

Geometrical visualisation of a 4D shape from a 3D Perspective - The Tesseract

Hypercubes are shapes with n dimensions where n is greater than the 3 dimensions of a normal cube. The Tesseract, or the 4-cube, is a 4 dimensional hypercube.

where n is the dimensional number, in any hypercube:

vertices = 2^n

edges = n(2^n-1 )

faces = 2^n-3(n-1)n

The brachistochrone

This animation is about one of the most significant problems in the history of mathematics: the brachistochrone challenge.

If a ball is to roll down a ramp which connects two points, what must be the shape of the ramp’s curve be, such that the descent time is a minimum?

Intuition says that it should be a straight line. That would minimize the distance, but the minimum time happens when the ramp curve is the one shown: a cycloid.

Johann Bernoulli posed the problem to the mathematicians of Europe in 1696, and ultimately, several found the solution. However, a new branch of mathematics, calculus of variations, had to be invented to deal with such problems. Today, calculus of variations is vital in quantum mechanics and other fields.

(via curiosamathematica)

Two curves cut all circles at right angles: straight line and a tractrix.

I didn’t know what a tractrix was, so I Googled it! Tractrix definition from Wikipedia: Tractrix (from the Latin verb trahere ”pull, drag”; plural: tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a tractor (pulling) point that moves at a right angle to the initial line between the object and the puller at an infinitesimal speed

Here’s a gif from Wikipedia showing a tractrix being created from dragging a pole:

A curved variant of the Sierpiński fractal.

Submitted by Manuvstheworld:

# Fanfiction, Graphs, and PageRank

On a website called fanfiction.net, users write millions of stories about their favorite stories. They have diverse opinions about them. They love some stories, and hate others. The opinions are noisy, and it’s hard to see the big picture.

With tools from mathematics and some helpful software, however, we can visualize the underlying structure.

In the following post, we will visualize the Harry Potter, Naruto and Twilight fandoms on fanfiction.net. We will also use Google’s PageRank algorithm to rank stories, and perform collaborative filtering to make story recommendations to top fanfiction.net users.

If you’re not interested in the details, you can skip to the following:

Interactive GraphsHarry PotterNarutoTwilight

Story RankingsHarry PotterNarutoTwilight

Story RecommendationsHarry PotterNarutoTwilight

And of course, you might skim below to see the pretty pictures!

[MORE]

I recently started watching a television show called Numbers, and this interesting use of mathematics to create cool and useful software reminds me of that show.

Let $$S_n (z) = \sum_{k=0}^n z^k$$ then parametrize $$S_n$$ on circles in the complex plane centered at the origin. The animation shows the graph of the real part of $$S_{100} (r e^{2\pi i t})$$ as a function of $$t$$ and how it changes as $$r$$ ranges from $$0$$ to $$1$$.

After looking into platonic solids earlier in my project, I began to look at other forms of polyhedra in order to form 3D structures relevant to my ideas. This above image displays Wenzel Jamnitzer’s few of many explorations of solid forms known as polyhedra, which are based on the five platonic solids. Polyhedra are beautiful 3-D geometrical figures that have fascinated philosophers, mathematicians and artists for millennia. Seeing as I have been working with very simple forms throughout the project, I thought it beneficial to explore more complex 3D shapes. From left to right are variations of the icosahedron and the dodecahedron each progression more complex than the last, as iterations have been added with more geometric components. I would like to visually explore some regular polyhedrons and perhaps build some sculpture ideas based on them.

Mandelbulb variation. 3D fractals are very new. A few years ago the famous Mandelbrot set was projected in 3D with some additional calculations like folding coordinates in an imaginary 4th D plane. This resulted in Mandelbulb, an organic like structure. Mandelbox or The Amazing Box derived from this. These fractals are truly amazing. One cannot imagine the things that can be found in them. Art, architecture, landscapes, anything. The universe is most probably a natural fractal.

(via geometric-aesthetic)

"While fractal geometry is often used in high-tech science, its patterns are surprisingly common in traditional African designs," said Ron Eglash, senior lecturer in comparative studies in the humanities. Eglash is author of African Fractals: Modern Computing and Indigenous Design (Rutgers University Press, 1999).

Eglash said his work suggests that African mathematics is more complex than previously thought. He also says using African fractals in U.S. classrooms may boost interest in math among students, particularly African Americans. He has developed a Web page to help teachers use fractal geometry in the classroom. (http://www.cohums.ohio-state.edu/comp/eglash.dir/afractal.htm)

Fractals are geometric patterns that repeat on ever-shrinking scales. Many natural objects, like ferns, tree branches, and lung bronchial systems are shaped like fractals. Fractals can also be seen in many of the swirling patterns produced by computer graphics, and have become an important new tool for modeling in biology, geology, and other natural sciences.

In African Fractals, Eglash discusses fractal patterns that appear in widespread components of indigenous African culture, from braided hairstyles and kente cloth to counting systems and the design of homes and settlements.

Sources: csdt.rpi.edu aziarts.com theskylinedesigngroup.wordpress.com leahchappel.wordpress.com

This is the magic hexagon. The numbers add up to 38 along each diagonal/vertical line.

It can be called the magic hexagon rather than a magic hexagon because there are no other hexagons numbered 1,2…n with this property, no matter how many layers the arrangement has.

(well, except for the one which is just one hexagon with ‘1’ written in it, and that’s hardly magical…)

(credit: Mathematical Gems I by Ross Honsberger)

### From Pascal’s Triangle to the Bell-shaped Curve

Blaise Pascal (1623-1662) did not invent his triangle. It was at least 500 years old when he wrote it down, in 1654 or just after, in his Traité du triangle arithmétique. Its entries C(nk) appear in the expansion of (a + b)n when like powers are grouped together giving C(n, 0)an + C(n, 1)an-1b + C(n, 2)an-2b2 + … + C(nn)bn; hence binomial coefficients. The triangle now bears his name mainly because he was the first to systematically investigate its properties. For example, I believe that he discovered the formula for calculating C(nk) directly from n and k, without working recursively through the table. Pascal also pioneered the use of the binomial coefficients in the analysis of games of chance, giving the start to modern probability theory. In this column we will explore this interpretation of the coefficients, and how they are related to the normal distribution represented by the ubiquitous “bell-shaped curve.”

(via mathematica)

Submitted by The Frankens Team:

Visualize sequences without repetition - find more combinatorial examples.

The osculating circle at a point on a moving sine wave. This is the circle that best approximates the curve at the point. [more] [code]