# How Algebra Sheds Light on Things

Egbert Brieskorn‘s mathematics books contain a lot more flesh than those structured strictly by definitions, lemmas, and theorems. What might be a hindrance to those looking for a quick grasp of a theory is a treasure trove for others who relish an extra of motivation, history, and examples. Currently, I am reading his book Plane Algebraic Curves (authored together with Horst Knörrer) which, already in the introductory chapter, contains many wonderful examples such as linkages, envelopes, and the Hopf fibration. I will come back to some of these topics in later posts. Today’s post contains a quick glimpse of caustics.

# Beautiful Binomials

While leafing through the book Geometric Trilogy I – An Axiomatic Approach to Geometry, I came across two nice geometric depictions that are probably widely known but which I would love to have seen back in school. The first one is the geometric illustration of the algebraic fact that $(a+b)^2 = a^2+2ab+b^2$, the quadratic case of the binomial theorem. This can already be found in Book II of Euclid’s Elements. From this, it is easy to come up with a three dimensional version of the construction giving $(a+b)^3=a^3+3a^2b+3ab^2+b^3$. Creating a good picture of this is somewhat tedious so I am very glad that I can use this opportunity to link to a wonderful blog where this and other wonderful mathematical illustrations and animations can be found: Hyrodium’s Graphical MathLand.

# Permanent Determinant

The determinant and the permanent of a matrix are central characters in an endeavour to bring the powerful weapons of modern geometry to a battle in the epic war of computer science: the P vs. NP problem.

JM Landsberg has recently written a wonderful introduction to geometric complexity theory which is how the corresponding research field is called. This has inspired me to borrow some of it and write about the permanent and the determinant of a matrix.

# Euler and the Kids

Being on parental leave, somewhat naturally, gave me more time to spend in the kids’ room. After a chance encounter with Mr Pythagoras there some time ago, it was now time  for a scheduled meeting with Mr Euler. The configuration in the image to this post is known to young parents as an Oball. To mathematicians, it is known as the truncated icosahedron. An interesting fact is that because it is constructed entirely from pentagons and hexagons, it has to have exactly 12 pentagons. Let’s see how Mr Euler can convince us of this necessity. Continue reading

# Pythagoras and the Kids

Yesterday, when playing with the kids, I found Mr. Pythagoras unexpectedly in their room. For a fraction of a second, I wondered why the Lego fence would fit perfectly across. Then, I saw that I had produced an instance of the 3-4-5 triangle.

# The Mandelbulb

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.

# Regular Viruses

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.

# Neighbourly Polyhedra

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.

# Beauty in Patents

In an earlier post, I wondered about the hidden beauty of scientific illustrations in books which are read only by very few specialists. A similar situation is found in the context of patents. They usually come with clarifying sketches or illustrations which range from sloppy to artistic.

Not later than from the moment Richard Buckminster Fuller filed his patent on cartography in 1944, mathematical beauty had found its way into patent applications. How many hidden gems may there be in patents? At least, having patents available online makes it a lot easier to spot some.

# Perspectiva Corporum Regularium

Most of the time, I enjoy how one book leads to another in a seemingly endless bibliophilic journey into the past (it is a pity that books usually cannot refer to future books). This time it is the book Shaping Space (see short review) referring to a book with very few words and many wonderful drawings of polyhedra: Wenzel Jamnitzer’s Perspectiva Corporum Regularium from 1568.