It seems completely natural that going from 2d to 3d adds a new dimension of awesomeness to fractals. Accordingly, it is quite a shame that until very recently, I was not aware of the magnificent Mandelbulb. If this is a new world for you, too, go and explore the wonders of 3d fractals using tools like Mandelbulb 3d.
In Brainiac: Science Abuse, there is a wonderful category called How Hard is Your Thing? (see ). Thaila Zucchi makes seemingly hard things undergo some of the ultimate stress tests: Can they stand the heat of thermite? The abrasion of an angle grinder? The impact of a ton of bricks?
There were several occasions in the history of mathematics when mathematicians had to answer to the question: how hard is your maths?
To me, mathematics is an ivory tower.
An ivory tower, vast, complex, and beautiful.
With its doors wide open.
At each level, helping hands are offered,
Trying to help you up the stairs.
However, the steps are steep
And there is no elevator.
No-one can carry you up.
You need to take the steps by yourself.
If you climb just a little
And open your eyes,
The view will be spectacular.
Either nature or my brain or both do a wonderful job of clustering related things for me so that I often feel coming along many related things within a short time. Recently, I read Stuart L. Pimm‘s wonderful The World According to Pimm: a Scientist Audits the Earth. It contains the following sentence:
« Robert Colwell, an ecologist from the University of Connecticut, and I were sitting in a bar drinking Antarctica and excitedly drawing lines on paper napkins, the preferred medium of serious scientific discourse worldwide. »
Shortly afterwards, I learned about the blog What’s on my blackboard? showing people’s blackboards (or whiteboards) which are often stunning.
Hence my question: what’s on your napkin? I dare you: grab your smartphones and share images of your napkins full of scientific sketches and/or formulas!
As long as they do not infect us, viruses are nothing but fascinating. One aspect of this fascination is based on the shapes of viruses. Looking at electron microscopy images of viruses or at computer models based on X-ray crystallography, my impression is that mother nature has copied from a geometry book. This impression is echoed by what you find in books and papers on virology. Why do biologists think about possible polyhedra with icosahedral symmetry? Why is it that many viruses have the shape of such polyhedra? For some years, it seemed like biologists had a very accurate theory of the construction of such regular viruses. Advances in imaging have left them less confident but with an even higher appreciation of the formation of biological shapes.
In the plane, it is relatively easy to find four convex polygons which pairwise share an edge and are otherwise disjoint. Can you find five polygons in the plane with these properties? How about polyhedra? How many polyhedra can you find such that any pair of them share a face and are otherwise disjoint?
For me, this is an example where my three-dimensional imagination fails utterly. If you have never thought about this before, I suggest to try finding as many such polyhedra as possible (why not start with seven?) before reading on.