New article today:

What Is Topology?

http://www.livescience.com/51307-topology.html

I generated the wireframe models using MATLAB. Here's the code I used along with links to all the parametric equations.
[u,v]=meshgrid(linspace(0,2*pi,25));

%cylinder
%http://mathworld.wolfram.com/Cylinder.html
% x=1.1*cos(u);
% y=1.1*sin(u);
% z=v/(2*pi);

%mobius strip
%http://mathworld.wolfram.com/MoebiusStrip.html
% r=5;
% s=(u-pi)/pi*2;
% t=v;
% x=(r+s.*cos(t/2)).*cos(t);
% y=(r+s.*cos(t/2)).*sin(t);
% z=(s.*sin(t/2));

%sphere
%http://mathworld.wolfram.com/Sphere.html
% r=5;
% theta=u;
% phi=v/2;
% x=r*cos(theta).*sin(phi);
% y=r*sin(theta).*sin(phi);
% z=r*cos(phi);

%torus
%http://mathworld.wolfram.com/Torus.html
% a=5;
% c=10;
% x=(c+a*cos(v)).*cos(u);
% y=(c+a*cos(v)).*sin(u);
% z=a*sin(v);

%klein bottle
%http://paulbourke.net/geometry/klein/
% r=4*(1-cos(u)/2);
% x=(u<pi).*(6*cos(u).*(1+sin(u))+r.*cos(u).*cos(v)) + (u>=pi).*(6*cos(u).*(1+sin(u))+r.*cos(v+pi));
% y=(u<pi).*(16*sin(u)+r.*sin(u).*cos(v)) + (u>=pi).*(16*sin(u));
% z=r.*sin(v);

%cross-cap disk
%https://en.wikipedia.org/wiki/Real_projective_plane#Cross-capped_disk
% r=5;
% x=r*(1+cos(v)).*(cos(u));
% y=r*(1+cos(v)).*(sin(u));
% z=-tanh(u-pi)*r.*sin(v);

hold on
mesh(x,y,z);
camlight left;
lighting phong;
alpha(0.4);
axis equal;
hold off

More math!

Properties of Pascal’s Triangle

http://www.livescience.com/51238-properties-of-pascals-triangle.html

My editor assured me he could find an image of a bean machine, so you can imagine my surprise when I discovered the inclusion of the following video:

I've been a fan of Numberphile videos for years, so seeing Matt Parker in was a real treat. I hope you all enjoy reading this as much as I enjoyed writing it.

Another math article!

What Is Symmetry?

http://www.livescience.com/51100-what-is-symmetry.html

It would have meant a lot to learn about a topic like this as a kid. That different kinds of symmetry had symbols associated with them was news to me as a college senior in chemical engineering.

It's a shame that the great diversity in 3-D patterns is rarely talked about outside of crystallography. What amazing sculptures are we missing out on by not exploring the topic more deeply in art?

Yet another excuse to write about the history of mathematics!

What Is Trigonometry?

http://www.livescience.com/51026-what-is-trigonometry.html

My editor left out the etymology of "sine" so I reproduce it here. From A History of Mathematics (3rd Edition) by Victor J. Katz (Section 8.7).

The English word ‘sine’ comes from a series of mistranslations of the Sanskrit ‘jyā-ardha’ meaning ‘chord-half.’ Aryabhata frequently abbreviated this term to ‘jyā’ or its synonym ‘jīvā.’ When some of the Hindu works were later translated into Arabic, the word was simply transcribed phonetically into an otherwise meaningless Arabic word ‘jiba.’ But since Arabic is written without vowels, later writers interpreted the consonants ‘jb’ as ‘jaib,’ which means bosom or breast. In the twelfth century, when an Arabic trigonometry work was translated into Latin, the translator used the equivalent Latin word ‘sinus,’ which also meant bosom, and by extension, fold (as in a toga over a breast), or a bay or gulf. This Latin word has now become our English ‘sine.’

That Malisha Dewalt over at MedievalPOC thought this was worth sharing tells me I'm doing something right in my representation of the history of math. I am happy.

Wow, I'm really behind on posting articles. Here's one from last month. I think the title scared everybody away:

What Are Logarithms?

http://www.livescience.com/50940-logarithms.html

I've been outdone on how to to explain logarithms simply to a rough approximation. From How Not to Be Wrong by Jordan Ellenberg (Chapter 11 I think).

The logarithm of a postiive number N, called log N, is the number of digits it has.
Wait, really? That's it?
No. That's not really it. We can call the number of digits the "fake logarithm," or flogarithm. It's close enough to the real thing to give the general idea of what the logarithm means in a context like this one.