Mathematics and cancer

Today I want to describe a little mathematical curiosity which may have important implications for our judgement and our decisions.

Let's start with a simple numerical example and consider a group of short people and a group of tall people:

heights in the "short" group {1.45m, 1.50m, 1.55m} average 1.50m
heights in the "tall" group {1.90m, 2.00m, 2.10m} average 2.00m

Moving a single person from the "tall" group to the "short" group may cause the average height of both groups to increase. For instance, by moving the 1.90m person to the "short" group we end up with:

heights in the "short" group {1.45, 1.50, 1.55, 1.90} average 1.60m
heights in the "tall" group {2.00, 2.10} average 2.05m

This phenomenon is named after Will Rogers (see the Wikipedia article).

And now a serious application. We test some people for cancer and we divide them into two groups: "cancer" and "cancer-free". Each group has an average life expectancy, which is clearly shorter for the "cancer" group. Suppose now that an improved cancer detection technique becomes available, we re-test everyone, and some of the people in the "cancer-free" group are moved to the "cancer" group. Again, the average life expectancy of both groups increases: in the healthy group because we removed people who are actually sick, and in the "cancer" group because we have introduced people who have cancer in a very early stage.

Importantly, even though the life expectancy of the "cancer" group increases, it does *not* mean that the quality of the treatments has improved!

Snow flakes

The physics of snow crystals
by Kenneth G. Libbrecht.

And more about it on his website.

Fractals

What happens if you plot all the roots of
all the polynomials in a certain class?



Image by Sam Derbyshire.

John Baez explains the mysteries behind this image
here and here.

Structured procrastination

Today I want to celebrate the IgNobel prize
awarded to John Perry, a Stanford philosophy
professor who wrote a wonderful essay
highlighting the idea of structured procrastination.

Origami

This beatle..



..is folded from a single square of paper following the
above crease pattern. Robert Lang's origami page
is certainly worth a visit.

Sometimes art meets technology: here you can see
a satellite antenna which is origami-folded during
the launch and unfolds seamlessly once in orbit.

Gapminder World

"Our goal is to replace devastating myths
with a fact-based worldview. Our method
is to make data easy to understand."

-- Hans Rosling



Visit Gapminder World and discover
the revealing beauty of data visualization.
Don't forget you can change the horizontal
and vertical axes of the graph and visualize
dozens of different variables.

Conway's game of life

The game of life is an example of how few simple rules
can generate complex behaviors. The first minute of the
video introduces the rules. Then the show begins..



Don't miss the Wikipedia article!