Fascinated by the intricate patterns formed by fractals, basically math processes that repeat incessantly in an ongoing feedback loop, UK physicist-turned-web developer Tom Beddard makes impossibly elaborate complexes that look like they belong in the gritty cities of a dystopian fantasy. Basically what he does is write and run programs on his computer that spit out patterns—”the best outcomes are often the least expected!” he writes—that he in turn massages (by way of shadowing and the like) into looking like faceted steel-and-concrete architecture.

So, I think the hardest thing I learned in mathematics was perseverance and patience, and this is the nature of mathematics. Math is very binary. It’s usually nothing, nothing, nothing, nothing, nothing… and then everything, you’ve got it. It’s also very humbling because, once you’ve got it, you realize it looks so obvious. So, you’ve got some humiliating experience like, ‘Oh, why didn’t I get that in the first place?’ But that’s what I learned, that’s the nature of mathematics.
The 9-81-90 Triangle

The 9-81-90 #Triangle #trigonometry #geometry #mathematics #math

In a previous post (right here), I explained the 18-72-90 triangle, derived from the regular pentagon. It looks like this:

I’m now going to attempt derivation of another “extra-special right triangle” by applying half-angle trigonometric identities to the 18º angle. After looking over the options, I’m choosing cot(θ/2) = csc(θ) + cot(θ). By this identity, cot(9°) = csc(18°) + cot(18°) = 1 +…

View On WordPress

Every mathematical induction consists of three parts or acts.

The first part is called “The Base”. The mathemagician shows you something ordinary: a claim, a theorem or a lemma. He asks you to see if it is indeed real, unaltered, true for k = 1.

The second act is called “The Hypothesis”. The mathemagician takes the ordinary claim and assumes it to hold for k = n.

Now you’re looking for the trick … but you won’t find it, because of course you’re not really looking. You don’t really want to know. You want to be fooled. But you wouldn’t clap yet. Because assuming something for k = n isn’t enough. You have to prove it for k = n + 1.
That’s why every induction proof has a third act, the hardest part, the part we call “The Inductive Step”.

—  Sanchit Agrawal

How Mandelbrot’s Fractals Changed The World

by Jack Challoner/BBC News

During the 1980s, people became familiar with fractals through those weird, colorful patterns made by computers. But few realize how the idea of fractals has revolutionized our understanding of the world, and how many fractal-based systems we depend upon.

Unfortunately, there is no definition of fractals that is both simple and accurate. Like so many things in modern science and mathematics, discussions of “fractal geometry” can quickly go over the heads of the non-mathematically-minded. This is a real shame, because there is profound beauty and power in the idea of fractals.

See more Mandelbrot Fractal images

The best way to get a feeling for what fractals are is to consider some examples. Clouds, mountains, coastlines, cauliflowers and ferns are all natural fractals. These shapes have something in common - something intuitive, accessible and aesthetic.

They are all complicated and irregular: the sort of shape that mathematicians used to shy away from in favor of regular ones, like spheres, which they could tame with equations.

Mandelbrot famously wrote: “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”

The chaos and irregularity of the world - Mandelbrot referred to it as “roughness” - is something to be celebrated. It would be a shame if clouds really were spheres, and mountains cones.

Look closely at a fractal, and you will find that the complexity is still present at a smaller scale. A small cloud is strikingly similar to the whole thing. A pine tree is composed of branches that are composed of branches - which in turn are composed of branches.

Read the entire article

Fractal images © Laguna Design / Science Source

Mandelbrodt photo © Emilio Segrè / Science Source

What makes the Julia Necklace, courtesy of Mark Newson and Boucheron, a unique, “insolent and audacious” (Boucheron’s description) is the illustrious inspiration it embodies: a blend of fine artistry, craftsmanship and mathematics. Newson’s timely and constant passion for fractals has found an expression in the design of the Julia Necklace. The necklace is a representation of Julia fractals (named after the scientist who discovered them, Julia Gaston), discovered at the dawn of the twentieth century. For the reader to make a general idea: fractals are rough geometrical shape which can be divided and subdivided endlessly in other units almost identical to them.

In spite of its impressive dimensions this piece of jewelry is not heavy or difficult to wear. Its manufacturing has probably been a blood and tears job, with the jewelers’ ambition to go by the book on representing the accurate mathematic representation of each fractal. Nevertheless, what is most important is the astonishing result which can be truly added to the world’s timeless jewelry legacy.

The Julia necklace was not intended to be a costly piece, as Newson declared, but its time, craftsmanship and minute detail requirements will probably transform it in one of Boucheron’s most expensive pieces.


Watch on ryanandmath.tumblr.com

What was the hardest thing you learned when studying math?

James Tanton sits down and talks about one of his most difficult experiences in mathematics. His difficulty wasn’t one course or topic, but rather his entire PhD experience. He spent over 2 and a half years working on a single problem with nothing to show for it, until he had a burst of inspiration one afternoon and solved it in 2 hours.

This is a great, quick video to watch for not only those of us who are doing math research, but even those who are taking high school and college math courses because it really emphasizes persistence and patience in math. I’ve talked about how I struggled for years with probability, but with enough time and dedication I was able to grasp and understand it. I believe everyone with the right focus, viewpoint, and help can get to wherever they want in math.


Formerly Unknown Mathematics Professor Receives “Genius Grant”

The MacArthur Fellows Program, commonly known as the “Genius Grant” just announced their recipients for 2014. As always, they are extremely impressive experts at the top of their respective fields - but for me, one in particular stuck out.

The Simons Foundations starts telling the story like this: 

On April 17, a paper arrived in the inbox of Annals of Mathematics, one of the discipline’s preeminent journals. Written by a mathematician virtually unknown to the experts in his field — a 50-something lecturer at the University of New Hampshire named Yitang Zhang — the paper claimed to have taken a huge step forward in understanding one of mathematics’ oldest problems, the twin primes conjecture.

Unknown ‘experts’ are always making similarly large claims to prestigious institutions, but this paper was different. The reception Zhang received was incredible: “The main results are of the first rank,” the author had proved “a landmark theorem in the distribution of prime numbers.”

Zhang was a researcher that no one seemed to know, his talents had been overlooked his entire career: “after he earned his doctorate in 1991 that he had found it difficult to get an academic job, working for several years as an accountant and even in a Subway sandwich shop.”

“Basically, no one knows him,” said Andrew Granville, a number theorist at the Université de Montréal. “Now, suddenly, he has proved one of the great results in the history of number theory.”

Read more about Zhang’s incredible discovery here and here.  

Follow Quantumaniac on Twitter


Spring Forest (5,3): embedded, unembedded, and cowl
12” x 11” x 9”
Knitted wool (Dream in Color Classy, in colors Happy Forest and Spring Tickle)
2009 and 2013
A (p,q) torus knot traverses the meridian cycle of a torus p times and the longitudinal cycle q times. Here are three instantiations of a (5,3) torus knot:
(a, middle) The knot embedded on a torus. A (p,q) torus knot may be drawn on a standard flat torus as a line of slope q/p. The challenge is to design a thickened line with constant slope on a curved surface.
(b, top) The knot projection knitted with a neighborhood of the embedding torus. The knitting proceeds meridianwise, as opposed to the embedded knot, which is knitted longitudinally. Here, one must form the knitting needle into a (5,3) torus knot prior to working rounds.
(c, bottom) The knot projection knitted into a cowl. The result looks like a skinny knotted torus.

Modern Striped Klein
2” x 14” x 7”
Knitted wool (Dream in Color Classy Firescorched in Aqua Jet with Sundown Orchid and Happy Forest)
This Klein bottle was knitted from an intrinsic-twist Mobius band with the boundary self-identified. A Klein bottle can be viewed as the connected sum of two projective planes; here, the stripes highlight the two circles that generate the fundamental groups of the individual projective planes. In some positions, this coloring of the Klein bottle resembles an ouroboros (a snake eating its own tail). The design is more than 10 years old; I recently realized that I had no high-quality example of it (only worn classroom models) and thus created one. Dream in Color veil-dyed yarn was chosen to add a color depth to the seed-stitch texture. Images of this piece graced the cover of the March-April 2013 issue of American Scientist.

Free-Range Mathematician
Sarah Lawrence College / Smith College
Hadley, MA

Over the last century-and-a-half, mathematicians found every possible multiplication table.

The largest irreducible multiplication-table, dubbed the Monster Group, contains


interlocking pieces.

That’s like the number of atoms in Jupiter.

Richard Borcherds

(modified by me)