The development of our modern numerals. Not as good as the comparison in Jim Al-Khalili’s book “The House Of Wisdom”, which I may have to put on here. Something that jumps out at me is how similar the character for nine has remained. Due to less frequent use maybe?
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.
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.
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.
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.
Tinted photograph from a daguerreotype of Ada Lovelace, 1844. Lovelace, daughter of Lord Byron, wrote lengthy notes to Charles Babbage who designed a theoretical mechanical digital computer, the Analytical Engine. Babbage referred to Lovelace as “the Enchantress of Numbers” and she is considered by some to be the first computer programmer.
Do other autistic people find it hard to keep up with word problems (maths)? Like you have to read it over and over and have to like write the numbers, then the operations one at a time and put them together from there?
I can’t quite remember how the teaspoon question arose except to recall that it came up in conversation with my son, a physics student.
First let me clarify the question. To state it more precisely: if you took all the molecules in a teaspoon of water and laid them end to end, how far would that thin aqueous line stretch?
The first step is to figure out how many water molecules we have.
A typical teaspoon holds about 5 millilitres (mL), which weights 5 grams. To find out how many water molecules there are in 5 grams you need to know that the molecular weight of water is 18 — the sum of the weights of one oxygen atom (16) and two hydrogen atoms (1 a piece) in H2O. What that means is 18 grams of water contains one mole of water molecules. Students of chemistry also know the mole to be a defined number of atoms or molecules (which relates the arbitrarily set scale of atomic weights — hydrogen = 1, helium = 2, and so on — to the actual weights of atoms in grams). It’s rather big: one mole is 6.022 x 1023 in scientific notation. That’s
So in 5 grams of water there would be 5/18ths of this number which is:
In other words: a lot.
But water molecules are very small; each one is only about 0.3 nanometers wide. That’s 0.0003 micrometres, or 0.0000003 millimetres or 0.0000000003 metres. These are bizarre numbers — we have no real experience of them so it’s hard to get much sense of scale. But let’s plough on anyway.
If we lay down 167,300,000,000,000,000,000,000 water molecules end to end, the total length of the line is:
167,300,000,000,000,000,000,000 molecules x 0.0000000003 metres per molecule.
Which is 50,190,000,000,000 metres.
Or 50,190,000,000 km (that’s 31,368,750,000 miles for older readers).
Which is 50 billion km. (How good was your guess?)
That’s over 10 times the width of the solar system.From a teaspoon.
Just think how far you could go with a bucket of water.
Stephen Curryis a Professor of Structural Biology at Imperial College.